Over large coastal regions in Greenland and Antarctica the ice sheet calves directly into the ocean. In contrast to ice-shelf calving, an increase in calving from grounded glaciers contributes directly to sea-level rise. Ice cliffs with a glacier freeboard larger than

To this end we compute the stress field for a glacier with a simplified two-dimensional geometry from the two-dimensional Stokes equation. First we assume a constant yield stress to derive the failure region at the glacier front from the stress field within the glacier. Secondly, we assume a constant response time of ice failure due to exceedance of the yield stress. With this strongly constraining but very simple set of assumptions we propose a cliff-calving law where the calving rate follows a power-law dependence on the freeboard of the ice with exponents between 2 and 3, depending on the relative water depth at the calving front. The critical freeboard below which the ice front is stable decreases with increasing relative water depth of the calving front. For a dry water front it is, for example,

Ice loss from Greenland and Antarctica is increasingly contributing to global sea-level rise

Tidewater glaciers calve vigorously when they are near floatation thickness, producing icebergs with a horizontal extent smaller than the ice thickness. This has been expressed in semiempirical height-above-floatation calving laws

In order to model calving not just for single glaciers but for whole ice sheets, a calving parametrization is needed. Models describing the nucleation and spreading of crevasses in ice

First, calving can be described as a function of strain rate and crevasse depth.

Second, a number of approaches have been taken to analyze calving processes via the stress balance.

Finally,

All these approaches agree on the basic physics of glacier calving: thicker ice at the terminus leads to higher stresses and larger calving rates. Glaciers terminating in water are stabilized by the water's back-pressure and have smaller calving rates.

The stability limit derived by

In this study, we analyze stresses at the calving front by solving the 2-D Stokes equation with a finite element model in order to propose a simple cliff-calving law. The purpose of this study is not to provide a comprehensive analysis. By contrast, we seek a minimalistic set of assumptions that paths the way to a simple stress-based cliff-calving law.

Geometrical setup of the stress computation: two-dimensional plane, flat glacier frozen to the bedrock with a calving front at its terminus. The glacier length

In this study we consider a plane, flat glacier of constant thickness

In order to compute the stress field near the calving front we set the glacier to be grounded (relative water depth

The ice flow and the stresses within the ice are governed by the Stokes equations,

The surface boundary is assumed to be traction-free. At the calving front boundary, we assume traction continuity to the water pressure and no traction above the water line. At the glacier bed, a no-slip boundary condition is assumed, which corresponds to a glacier frozen to its bed. No inflow is assumed at the upstream boundary.

The boundary value problem was solved with the Finite Element package FEniCS

Since the Stokes equations are linear in the stresses and the terminus boundary condition is linear in the ice thickness, the equations can be solved on a dimensionless domain and the stresses scaled to arbitrary ice thickness. Velocities do not scale linearly but can be obtained from the scaled stresses through the ice rheology equation. The water depth at the calving front was incorporated via the relative (dimensionless) water depth

In order to determine a suitable stress criterion for cliff calving we consider a number of commonly used stresses that have a clear physical role (Fig.

The deviatoric normal stress,

The different components of the deviatoric stress tensor are not invariants of the stress tensor, i.e., they depend on the coordinate system in which they are computed, and therefore they are not suitable as failure criteria.
The largest principal stress,

The von Mises stress is the second invariant

Stress configurations at the calving front for different relative water depths (

As a first step we select a failure criterion, which then yields a failure region based on the computed stress fields. As a second step we decide on a timescale for the failure in order to derive a simple calving law.

Crevasses are a natural candidate for ice front failure. In the case of glaciers that are frozen to the ground, crevasses, generally, do not form from the base upward

Surface meltwater filling surface crevasses can increase their depth (hydrofracturing)

Instead, we assume shear faulting to be the dominant process in ice-cliff failure. We could use the von Mises stress as a failure criterion instead and reach qualitatively the same result because they differ only by a factor of

The failure region is defined as the region close to the calving front where the maximum shear stress exceeds a critical shear stress of

Assuming Coulomb failure, the required cohesion,

In general, brittle compressive failure happens through shear faulting

Outline of the failure region for different ice thicknesses on a dimensionless domain and without water stabilizing the front (ice thickness

We define the failure region as the region close to the calving front where the maximum shear stress exceeds the critical shear stress

For a given water depth, the failure distance

Above a critical freeboard of about

In Fig.

Figure

Size of shear failure region

There is a theory for damage evolution in ice for tensile damage

However, Eq. (

There is plenty of literature about compressive creep and failure in rocks

This leaves us with a dilemma: there have been no studies that determined the material properties of ice under time-dependent brittle compressive failure. Also, we cannot determine those material properties ourselves by fitting the resulting calving law to observations because, so far, cliff calving has not been observed as the major calving process in any glacier. That makes it impossible to estimate the time to failure using Eq. (

Nevertheless, we will use it as a starting point for our further analysis.
For the stresses above the shear failure threshold,

Time to failure given by Eq. (

With a constant failure time, the calving rate is proportional to the size of the failure region

Table of parameters in the cliff-calving relation (Eq.

Cliff-calving rates

How do cliff-calving rates given by Eq. (

Jakobshavn glacier in Greenland is one of the few glaciers that are currently in a cliff-calving mode. Jakobshavn glacier terminates in water with a depth of

It is difficult to determine calving rates directly. The ice flow velocity to the front of Jakobshavn is up to

Inserting values of glacier freeboard and water depth given above into Eq. (

We solved the 2-D Stokes equation numerically for a flat glacier frozen to its bed in a flow-line model and investigated the stresses at the calving front.

The following four simplifications were made.

The model was solved in one horizontal direction, neglecting lateral shear effects. Without lateral shear effects, the result is independent of the topography of individual glaciers.

We assumed a basal boundary condition corresponding to a glacier frozen to its bed. Sliding was not considered.

The main failure mechanism was assumed to be shear faulting. We assumed brittle compressive failure according to the Coulomb law without friction stabilizing the ice cliff. Friction would allow glaciers with larger freeboards than observed to be stable.

A constant time to failure has been assumed.

Under these assumptions, crevasses cannot penetrate the whole glacier depth and shear failure was chosen as the main failure mechanism. The region where shear stresses exceed a critical shear stress of

The cliff-calving rate was derived using an idealized setup, given by the first two of the four assumptions described above. Realistic glaciers that might experience cliff calving sit in valleys where they experience lateral drag and may be sliding. The calving front may have a slope rather than a vertical cliff and there might be an undercut caused by frontal melt.

First consider sliding with a constant velocity

In general, sliding velocities increase towards the glacier terminus. The steepest possible velocity gradient can be obtained with a free-slip basal boundary condition: we assume no influx at the upstream boundary,

To summarize, the derived cliff-calving law is valid for glaciers that are frozen to the bed or sliding with a constant velocity and vanishing strain rate. It serves as a lower bound on the calving rate for glaciers in which velocities increase towards the calving front.

Stress configurations at the calving front for different relative water depths (

In order to investigate how lateral drag influences cliff calving, we will assume ice flow in a channel with a flow-line in the

Assuming

Hence, lateral shear increases the maximum shear, therefore increasing the size of the failure region and the cliff-calving rate. The derived cliff-calving rate can serve as a lower bound if lateral drag is present.

Maximum shear stress

Other studies have shown that a calving front with a slope has significantly reduced stresses compared to a calving front with a vertical cliff

We have not analyzed this effect here because once cliff calving has been initiated, the full thickness calving probably prevents calving front slopes from forming. We aim to find a parametrization that can be implemented in ice sheet models capable of simulating the Antarctic ice sheet. These simulations are done on resolutions of several kilometers and cannot resolve calving front slopes on length scales of several tens or hundreds of meters.

Undercut from melt would increase the stresses near the calving front

Cliff calving is still a rather hypothetical process with a very limited scope of observations. Since there are currently no glaciers that are clearly in a cliff-calving regime, the calving rate cannot be fitted to observed calving rates. There is uncertainty in the maximum shear stress used to determine the failure distance as well as the time to failure.

Laboratory studies give a range of values between

The scaling parameter

It is difficult to say at which glacier freeboard the tensile failure regime ends and the shear failure regime begins, not only due to uncertainty in the scaling parameter

Comparison of the cliff-calving law given by Eq. (

The calving law proposed here was derived under a number of constraining assumptions. First, it was assumed that friction plays no role in shear failure. Second, it was assumed that once the critical shear stress is exceeded, ice fails after a constant time to failure. An improved cliff-calving model might include friction and allow a stress-dependent time to failure.

If the Coulomb law with a friction component is used, the immediate failure region is smaller than in the no-friction case. Time to failure relations for compressive failure, as given by Eqs. (

Another problem is that there are no laboratory studies on the parameters in the time to failure relations for ice. It is also not possible to calibrate the calving relation using observed calving rates because there are no glaciers currently available where cliff calving is the primary failure mechanism. Paleorecords might provide some means to calibrate cliff-calving rates as attempted in

Paleorecords might not be constraining enough to provide a useful limit for the Antarctic sea-level contribution of the next 85 years. But even if it is difficult to constrain the rate of cliff-calving there are important qualitative consequences of a monotonously increasing cliff-calving dependence on ice thickness. The most important is the potential of a self-amplifying ice loss mechanism, which is not constrained by the reduction in calving but must be constrained by other processes. Without some kind of cliff-calving mechanism it is likely that ice sheet models are lacking an important ice loss mechanism.

FeniCS can be downloaded from the project website

It is possible to solve the stress balance at the calving front analytically in a depth-averaged model with a simplifying assumption for the isotropic pressure. This has been used by

Together with incompressibility, which means that the trace of the strain rate disappears (

The resulting stresses are smaller than the stresses obtained in Sect.

The biggest difference between the two approaches lies in the largest principal stress: in this simplified problem, the largest principal stress is negative in the whole ice volume; there is no region of tensile stresses, so no crevasses form. This is due to the assumption that the isotropic pressure is equal to the gravitational pressure, which is not actually the case in the vicinity of the glacier terminus.

Stress configurations at the calving front for different relative water depths (

AL conceived the study, TS designed and carried out the numerical experiments, both authors analysed the data, and TS wrote the manuscript with input from AL.

The authors declare that they have no conflict of interest.

Tanja Schlemm would like to thank Yue Ma and Christian Helanow for their valuable help with FeniCS. We would like to thank Andy Aschwanden and two anonymous reviewers for their very useful comments on earlier versions of the manuscript.

As a doctoral student, Tanja Schlemm is funded by a doctoral stipend granted by the Heinrich Böll Foundation. The publication of this article was funded by the Open Access Fund of the Leibniz Association.

This paper was edited by Eric Larour and reviewed by Andy Aschwanden and two anonymous referees.