Ocean-terminating glaciers in Arctic regions have undergone rapid dynamic changes in recent years, which have been related to a dramatic increase in calving rates. Iceberg calving is a dynamical process strongly influenced by the geometry at the terminus of tidewater glaciers. We investigate the effect of varying water level, calving front slope and basal sliding on the state of stress and flow regime for an idealized grounded ocean-terminating glacier and scale these results with ice thickness and velocity. Results show that water depth and calving front slope strongly affect the stress state while the effect from spatially uniform variations in basal sliding is much smaller. An increased relative water level or a reclining calving front slope strongly decrease the stresses and velocities in the vicinity of the terminus and hence have a stabilizing effect on the calving front. We find that surface stress magnitude and distribution for simple geometries are determined solely by the water depth relative to ice thickness. Based on this scaled relationship for the stress peak at the surface, and assuming a critical stress for damage initiation, we propose a simple and new parametrization for calving rates for grounded tidewater glaciers that is calibrated with observations.

Many ocean-terminating glaciers in the Arctic are currently undergoing
rapid retreat, thinning and strong acceleration in flow. These dynamic
mass losses contribute to about half of the Greenland ice sheet's
contribution to sea level rise

Tidewater glacier evolution is the result of an interplay between mass
flux from upstream and the rate and size of calving events

Iceberg calving is a dynamical process of material failure which occurs when the local stress field in the vicinity of the calving front exceeds the fracture strength of ice, driving the formation and propagation of cracks and eventually leading to the detachment of a block of ice from the glacier front. The local geometry and water level at the terminus determine the stress field and thereby the fracture processes and the geometry evolution. Further, buoyancy forces of submerged ice and erosion from subaqueous melt are expected to enhance near-terminus stress intensity and hence calving rates, while a reclining terminus should reduce extensional stresses.

Model parameters, notations, units and values for constant parameters.

Several empirical and semi-empirical parametrizations of the calving
rate for different terminus geometries have been proposed. A simple
empirical relationship of linearly increasing calving rate with water
depth, based on observations of tidewater glaciers in Alaska, has been
established, used and extended for different regions

For near-vertical calving fronts, the main driver for calving is the
horizontal deviatoric stress

Using the above longitudinal stress at the front, the maximum height
for which a grounded glacier with a dry calving front can sustain
a stable vertical front is approximately 110

Calving termini can also be over-steepened by melt undercutting, which
leads to higher stress intensities

The state of stress near the calving front is determined by ice
geometry and water depth and controls the intensity and location of
material degradation processes. Material creep and fracture processes
in turn change the geometry of the glacier front. Observations and
theoretical considerations indicate a tendency of increasing relative
water level with increasing thickness

The relationship between water depth, stress state, front geometry and
related calving type is well illustrated at the example of Eqip
Sermia, a medium-sized ocean-terminating outlet glacier on the West
Greenland coast. Figure

Calving front of Eqip Sermia glacier in July 2016. The boxes in the picture describe the geometrical properties of the two distinct parts of the calving front.

Motivated partly by the case of contrasting calving front geometries at Eqip Sermia, the aim of this study is to better understand the detailed flow and stress regimes in the vicinity of the calving front of tidewater glaciers, including those that are far from flotation. Using a numerical model that solves the full equations for ice flow, we investigate the sensitivity to variations in front thickness and slope, the water depth and the strength of the coupling to the bed which results from sliding processes. We perform these model experiments on idealized geometries of grounded glacier termini and succeed to explicitly express the results as function of relative water depth.

Based on these model results, we derive a novel parametrization of calving rate that is calibrated with observations from Arctic tidewater glaciers. This parametrization only requires the relative water level and is based on a fit to the modeled stress field at the surface and an isotropic damage evolution relation.

We used the finite-element library libMesh

The model domain was discretized with second-order nine-node quadrangle
elements with Galerkin weighting. Model variables are approximated
with a second-order approximation for the velocities

We used a two-dimensional version of the model to conduct the
geometrical tests, as illustrated in Fig.

All numerical results are scalable with reference values for ice
thickness

Geometry of the idealized grounded glacier.

The upper surface of the glacier was described as a traction-free
surface boundary. Basal motion was parametrized with a slipperiness
coefficient

At the calving front a normal stress boundary condition was imposed
below the water level, while the surface above water was kept
stress-free. The stress boundary condition thus reads

At the upstream boundary of the glacier domain velocities were fixed to zero. Additional modeling experiments showed that different values for this upstream boundary condition do not affect the results of the analysis.

The stress state and flow field near the calving front is analyzed in
three suites of numerical experiments that investigate the effect of
variations in relative water level

The water level sensitivity experiments were performed for relative
water levels

The calving front slope sensitivity experiments were performed on
a geometry with the upper part of the calving front reclining at
various angles. The lower 25

The bed slipperiness sensitivity experiments were performed on a block
geometry with a vertical calving front and a relative water level

Any criterion for fracture propagation or damage evolution should be
independent of the choice of coordinate system and can therefore be
expressed as a function of the invariants and eigenvalues of the
stress tensor.

Sensitivity experiment results for varying water depth.

Scaled velocities along the vertical face of the calving front (solid lines) for different relative water levels. Horizontal line markers show the relative water level for each curve.

Sensitivity experiment results for varying calving front
slope. Panels

To investigate the full spectrum of possible stress states that lead
to the initiation of damage, we investigated linear combinations of
five stress invariants:

Sensitivity experiment results for varying bed slipperiness

All sensitivity experiment results shown in Figs.

The depth of the water at the calving front significantly impacts the
stress regime and consequently the ice flow pattern and magnitude near
the terminus. The effect of different water depths on the stress
field is displayed as Hayhurst stress in Fig.

For a reduction in the relative water level from

Figures

Extrusion flow, a velocity pattern for which maximum horizontal
velocity occurs below the surface

In summary, increasing relative water depth leads to decreased flow velocities and lower stresses and moves the peak of the Hayhurst stress at the surface closer to the front.

Maximum scaled Hayhurst stress and velocity for water depth experiments. The s and f letters indicate whether the scaled Hayhurst stress maxima were found at the surface or at the bottom of the calving front, respectively.

Maximum Hayhurst stress and velocity for calving front slope experiments.

Maximum Hayhurst stress and velocity for bed slipperiness coefficient experiments.

Results from the sensitivity experiment on calving front slope
displayed in Fig.

Sensitivity experiments result for an inclined surface

The flow and stress regimes of the idealized glacier are less
sensitive to an increase of bed slipperiness coefficient.
Figure

In the modeling presented so far we used a glacier geometry with
horizontal surface and bed. Consequently the driving stress and hence
velocities and stresses far upstream from the calving front are close
to zero. In reality glaciers have a sloping surface. Therefore, we
repeated some of the above experiments on a simple glacier geometry
with a sloped bed and surface, a fixed cliff height and no sliding.
Bed and surface slopes were chosen as

To summarize, the stress and velocity fields in the vicinity of the calving front are only slightly altered for sloping bed and surface. It is, however, noteworthy that the reclining surface slope induces higher stresses near the surface, which could potentially induce crevassing and thus advect pre-damaged ice to the calving front.

Combinations of five stress tensor invariants at the
surface of an idealized glacier with a vertical calving front
without water pressure and zero basal motion. Each black line
represents a linear combination of five stress invariants. The
blue envelope contains the maxima of all stress invariant
combinations. The green triangle, red square and purple circle
represent the maximum of the scaled Hayhurst stress

The Hayhurst stress, typically used as the driving force for damage
evolution

Envelopes of stress invariant combinations at the surface of
the idealized glacier with zero basal motion
for varying relative water level

The magnitudes and positions of the maximum stress invariant
combinations for different relative water levels

The Hayhurst, maximum principal and von Mises stress distributions are
shown in Figs.

The similarity of stress distribution curves along the glacier surface
for varying relative water levels (Fig.

Modeled (dashed lines) and corresponding parametrized (solid
lines) maximum extensive stresses

The distribution along the glacier surface of scaled maximum principal
stress

Using the parametrizations of magnitude and position of the maximum
extensional stress at the surface (Eqs.

One major assumption is that a large crevasse forms at the location of
the maximum tensile surface stress where the ice is weakened until
failure. Such crevassing seems realistic as both observations and
model results show the formation of huge crevasses. When failure of
the surface ice is complete, we assume that all ice in front of the
crevasse is removed and a new calving front forms at the location of
the crevasse. Here, we do not consider explicitly which processes are
responsible for downward propagation of the crevasse. Several
processes could be considered, such as bottom crevassing,
hydro-fracturing by ponding water in surface crevasses, rapid elastic
crevasse propagation

The stress intensity and therefore ice deformation rates are
decreasing as the relative water level increases due to the pressure
exerted by the water at the calving front. This feature is already
captured by the depth-integrated extensional stress at the front
(Eq.

In addition, the detailed modeling shows that the stress peak at the
glacier surface moves upstream for lowering relative water level
(Figs.

A higher relative water level results in a more stable calving front

Using freshwater instead of seawater at the calving front yields
slightly higher stresses and velocities
(Fig.

The model results demonstrate that reclining calving fronts lead to
lower velocities and stresses and thereby implicitly confirm that
inclined calving fronts should reach larger stable heights than
vertical cliffs, as observed for example at Eqip Sermia (200

Basal sliding leads to increased stresses at the surface throughout
the computational domain. Thus, basal sliding may cause an onset of
ice damaging and crevasse opening in a greater distance from the
calving front (Fig.

For a sloping glacier surface the location and magnitude of the stress
maximum in the vicinity of the calving front remain almost identical, as
shown in Fig.

The proposed calving rate parametrization
(Eq.

The calving rate parametrization (Eq.

for constant relative water level

for constant absolute water depth

for constant ice thickness the calving rate decreases with
increasing relative water level (Fig.

Calving rates predicted by the parametrization in relation to
ice thickness. Calving rates increase with increasing total ice
thickness for a given water depth

Calving rates predicted by the parametrization as a function
of relative water level. Calving rate decreases under increase of
the relative water level

Calving rates (

Comparison of measured calving rates with predictions from
the calving parametrization. The glacier names are abbreviated
according to Table

The calving rate parametrization (Eq.

Values of calving front height, water depth and calving rate for different glaciers.

To calibrate these parameter choices, data on calving rate, ice
thickness and water depth for a wide variety of tidewater glaciers in
the Arctic were collected. The data set covers the full range of water
levels (relative and absolute), velocities and ice thicknesses that
are found in Arctic tidewater glaciers. Unfortunately, many studies
report only width-averaged data on calving front geometry and calving
rate, which are not suitable for our proposed relation which relies on
local stresses on a flowline. Only a limited set of point data on
calving front geometries are available from the published literature
from which total ice cliff thickness, water depth and calving rate can
be obtained. For the calibration, we used the values shown in
Table

Contours of calving rates calculated with
Eq. (

Note that the derivation of the parametrization is independent of the specific geometry or location of a tidewater glacier and thus the calibration is expected to be “global” and valid for any tidewater glacier.

This study improves our knowledge on the influence of geometry and water depth on the stress and flow regimes in the vicinity of the calving front and proposes a novel calving rate parametrization.

The magnitude of the stresses and flow speeds near a grounded vertical calving front are dominantly dependent on water depth and increase with decreasing water depth. Thus, the presence of water at the calving front has a strong stabilizing effect. Importantly, the extensional stress at the surface can be parametrized as a function of relative water level only. Further, we find that grounded tidewater glaciers with reclining calving faces have the potential to reach larger maximum stable heights than those with vertical calving fronts. Spatially uniform variations in basal sliding likely have a weaker effect than water depth and calving front slope on the stability, as the magnitude and location of the stress maximum show a small sensitivity to variations in bed slipperiness.

A simple calving rate parametrization was derived that was calibrated with calving rate data of a set of tidewater glaciers in the Arctic. This approach can be used to compute calving rates for grounded tidewater glaciers with relatively simple geometries when front thickness and water depth are known. The application of this parametrization in flow models of different complexity should be straightforward.

The present study lays the foundation for future, more detailed, studies of the calving process on more realistic geometries. Detailed analyses including time evolution, further processes such as frontal melt and water-filled crevasses, and data validation will be necessary for the implementation of improved calving parametrizations.

The libMesh library is a C++ framework for the numerical
simulation of partial differential equations on serial and parallel platforms
available at

The distribution of longitudinal tensile stress at the surface

The authors wish to thank Jaime Otero and Jeremy Bassis for their reviews and Douglas Benn, Joe Todd and Olivier Gagliardini for their comments that helped considerably to improve this paper. This work was funded by the Swiss National Science Foundation Grant 200021_156098. Edited by: Olivier Gagliardini Reviewed by: Jeremy Bassis and Jaime Otero