Content uploaded by Timo Riikilä

Author content

All content in this area was uploaded by Timo Riikilä on Nov 16, 2016

Content may be subject to copyright.

Available via license: CC BY 4.0

Content may be subject to copyright.

The Cryosphere, 7, 1591–1602, 2013

www.the-cryosphere.net/7/1591/2013/

doi:10.5194/tc-7-1591-2013

© Author(s) 2013. CC Attribution 3.0 License.

The Cryosphere

Open Access

A particle based simulation model for glacier dynamics

J. A. Åström1, T. I. Riikilä2, T. Tallinen2, T. Zwinger1, D. Benn3,4, J. C. Moore5,6,7, and J. Timonen2

1CSC – IT Center for Science, P.O. Box 405, 02101, Esbo, Finland

2Department of Physics, University of Jyväskylä, P.O. Box 35 (YFL), 40014, Jyväskylä, Finland

3Department of Geology, University Centre in Svalbard, 9171 Longyearbyen, Norway

4School of Geography and Geosciences, University of St Andrews, Fife, KY16 8ST, UK

5State Key Laboratory of Earth Surface Processes and Resource Ecology,

College of Global Change and Earth System Science, Beijing Normal University, Beijing, China

6Arctic Centre, University of Lapland, PL122, 96100 Rovaniemi, Finland

7Department of Earth Sciences, Uppsala University, Villavägen 16, Uppsala, 75236, Sweden

Correspondence to: J. A. Åström (jan.astrom@csc.ﬁ)

Received: 29 January 2013 – Published in The Cryosphere Discuss.: 6 March 2013

Revised: 15 August 2013 – Accepted: 19 August 2013 – Published: 8 October 2013

Abstract. A particle-based computer simulation model was

developed for investigating the dynamics of glaciers. In the

model, large ice bodies are made of discrete elastic parti-

cles which are bound together by massless elastic beams.

These beams can break, which induces brittle behaviour. At

loads below fracture, beams may also break and reform with

small probabilities to incorporate slowly deforming viscous

behaviour in the model. This model has the advantage that it

can simulate important physical processes such as ice calv-

ing and fracturing in a more realistic way than traditional

continuum models. For benchmarking purposes the deforma-

tion of an ice block on a slip-free surface was compared to

that of a similar block simulated with a Finite Element full-

Stokes continuum model. Two simulations were performed:

(1) calving of an ice block partially supported in water, simi-

lar to a grounded marine glacier terminus, and (2) fracturing

of an ice block on an inclined plane of varying basal fric-

tion, which could represent transition to fast ﬂow or surging.

Despite several approximations, including restriction to two-

dimensions and simpliﬁed water-ice interaction, the model

was able to reproduce the size distributions of the debris ob-

served in calving, which may be approximated by univer-

sal scaling laws. On a moderate slope, a large ice block was

stable and quiescent as long as there was enough of friction

against the substrate. For a critical length of frictional con-

tact, global sliding began, and the model block disintegrated

in a manner suggestive of a surging glacier. In this case the

fragment size distribution produced was typical of a grinding

process.

1 Introduction

The formation and propagation of fractures underpins a

wide range of important glaciological processes including

crevasse formation, iceberg calving and rheological weaken-

ing of ice in shear margins and icefalls. Numerical simula-

tions of glaciers, however, almost exclusively employ con-

tinuum methods, which treat ice as a continuous medium

with uniform or smoothly varying properties. The difﬁculty

of dealing with discontinuities in continuum models means

that the effects of fracturing are routinely represented by sim-

ple parameters, such as depth of fracture penetration (Benn

et al., 2007a, b; Nick et al., 2010), bulk “damage” (Jouvet

et al., 2011; Borstad et al., 2012), or ice softness (Vieli et al.,

2006). While useful for many purposes, these approaches im-

pose major limitations on the kinds of glacier behaviour that

can be represented in prognostic models.

Iceberg calving and the fracture of ice remain intensively

active topics of interest, which is a testament both to the

difﬁculty of the work, and the long term monitoring re-

quired to quantify its statistical nature (Weertman, 1974,

1980; Schulson, 2001). Calving constitutes up to 40–50%

of mass loss on marine terminating ice fronts (Burgess et al.,

2005; Dowdeswell et al., 2008; Walter et al., 2010; Thomas

et al., 2004) in the regions where it has been documented.

Marine terminating glaciers and ice shelves account for al-

most all mass loss through calving in the case of Antarctica

and about 50% for Greenland (Rignot et al., 2011; Jacob

et al., 2012). Calving glaciers are very variable, but two end

Published by Copernicus Publications on behalf of the European Geosciences Union.

1592 J. A. Åström et al.: A particle based simulation model for glacier dynamics

member types can be recognised: (i) glaciers with grounded

termini, and (ii) ﬂoating ice shelves that are constrained only

at their lateral margins. The two scenarios produce radically

different types of calving: (i) small ice blocks that fall off

the calving cliff in typically warm tidewater glacier settings,

and (ii) large ﬂat-topped bergs that can be tens of kilometres

across from the colder ice shelves that fringe the polar ice

sheets. At present ice sheet models do not typically incorpo-

rate calving as a function of atmospheric and oceanic forc-

ing. Indeed, no formulation for calving has yet been agreed

as suitable for models, though several have been proposed

(Benn et al., 2007a; Nick et al., 2010; Bassis, 2011; Lev-

ermann et al., 2012), and indeed different ones may be suit-

able for different applications such as large scale models (e.g.

Levermann et al., 2012) or basin-scale studies (Benn et al.,

2007a) with ﬂoating ice tongues (e.g. Nick et al., 2010).

Benn et al. (2007a) proposed a physically based model

with the position of the calving front depending on the depth

of penetration of surface crevasses, which in turn depends on

the longitudinal strain rate. A modiﬁcation suggested was to

increase crevasse depth by the ﬁlling of crevasses by surface

melt water, which is common occurrence in summer even

on ice sheets, and certainly typical of many marine terminat-

ing smaller glaciers. Nick et al. (2010) introduced a further

modiﬁcation by including basal crevasses with a calving cri-

terion when surface crevasses reach basal crevasses. Basal

crevasses can penetrate much farther than surface air-ﬁlled

crevasses, hence potentially triggering calving at a greater

distance from the terminus. The existence of huge tabular

icebergs, originating from ﬂoating ice shelves, provides am-

ple motivation for incorporating this effect. An upper bound

for the height of calving ice cliffs was suggested by Bassis

and Walker (2012).

There is a long history of using particle models to simu-

late geophysical phenomena (Cundall and Strack, 1979; Jing,

1992; Gethin et al., 2001; Potyondy and Cundall, 2004),

but usually the material behaviour studied with these mod-

els have been restricted to elastic and brittle properties. In

this paper, we introduce a new, particle-based method for

modelling ice ﬂow, which allows elastic, viscous and brit-

tle behaviour to be represented within a single framework.

Although based on simple rules, a very wide range of glacio-

logical phenomena emerge from the model, allowing de-

tailed investigation of processes that are difﬁcult or impos-

sible to represent in continuum models. We ﬁrst describe the

model, then illustrate some of its potential applications, in-

cluding calving events, the effects of variable basal friction,

and threshold behaviour in sliding rates (“surging”). Further,

we present an ice-deformation calculation comparing the re-

sults obtained with the particle model to the ones obtained

with the FEM code Elmer/Ice (http://elmerice.elmerfem.org)

for benchmarking.

Fig. 1. The particles connected with a beam can (a) stretch when a

force Fis applied and (b) bend when a torque Tis applied. Parti-

cles that overlap, i.e. come into contact (c) will experience repulsive

forces. The amount of stretching and bending required for beam

breaking is highly exaggerated as is the amount of particle overlap

in the simulations.

2 Model

In our simulation model, a large ice-body is divided into

discrete particles. A detailed theoretical description of the

model is given in Appendix A. The typical diameter of the

particles is in the present simulations of the order of 1m.

Initially, at the start of a simulation, particles are densely

packed (close-packed) or deposited to form a large body, and

the particles are assumed to be frozen together. The parti-

cles can either be arranged more or less randomly as in an

amorphous solid, or as in a regular lattice. The frozen con-

tacts between the particles are modelled by elastic massless

beams which can break if stretched, sheared or bent beyond

elastic threshold limits. In such a case the beam vanishes (see

Fig. 1). Choosing a proper fracture criterion is far from triv-

ial. A general elliptical criterion (Zhang and Eckert, 2005)

that includes the ‘classical’ fracture criterion of Tresca, von

Mises, Mohr-Coulomb and the maximum normal, i.e. hydro-

static pressure, stress criterion is rather useful. This criterion

states that a material under tension fails at locations where

σ2

I

σ2

0

+σ2

I I

τ2

0

≥1,(1)

in which σIis the normal stress, or pressure determined by

the ﬁrst invariant of the stress tensor and σII is shear stress

determined by the second invariant. σ0and τ0are material

dependent constants. Instead of fracture stress thresholds,

it is also possible to use thresholds for elastic strain. It is

trivial to change between stress and strain via the relation

σ=K, where σis the stress tensor, Kis stiffness tensor

and is strain tensor. Also, energy based criteria can easily

be formed. Then fracture takes place if the total elastic en-

ergy of a beam grows beyond a threshold. It is still an open

The Cryosphere, 7, 1591–1602, 2013 www.the-cryosphere.net/7/1591/2013/

J. A. Åström et al.: A particle based simulation model for glacier dynamics 1593

question which criteria are best for glacier simulations. No

mass is lost when beams break. If particles detach, they are

able to ﬂow past each other and thereby collide with other

particles. The shape deformations of the particles are not cal-

culated exactly. The contact forces in a collision are calcu-

lated as a function of overlapping of particles. The collisions

are inelastic and kinetic energy is lost in every collision. This

means that once all contacts are broken in an ice-block un-

der local compression, it will display granular ﬂow. In parts

of an ice-block with extant connections the ice will continue

to behave as an elastic solid. Under tension the ice is able

to fracture via crack formation and propagation. The model

should thus contain the necessary ingredients for simulating

a visco-elastic material, like ice, that fractures, at least on a

qualitative level. The equations of motion may vary slightly

with the exact implementation for the interaction of the par-

ticles (cf. Appendix A), but can typically be written in the

form:

M¨

ri+C˙

ri+X

j

γij C0˙

rij +X

j

γ0

ij Krij =Fi,(2)

where Mis the diagonal mass-matrix containing the masses

(i.e. volume times density Vρ ) and moments of inertia

(ρRr2dV) of the particles. Vectors riand ˙

ridenote the po-

sition and velocity of particle iand rij and ˙

rij are the cor-

responding relative vectors for particles iand j. The diag-

onal damping matrix Ccontains damping coefﬁcients for

drag, i.e. drag force = (1/2)ρv2cDS, where vis velocity, S

is cross-sectional area of the object to which the drag is ap-

plied, and cDis the Reynolds number dependent drag coef-

ﬁcient. The other damping matrix, C0, contains coefﬁcients

for the inelastic collisions. The parameter γij is zero for par-

ticles not in contact and unity for particles in contact, and

γ0

ij is unity for connected particles and zero otherwise. The

stiffness matrix is denoted by Kand Fiis the sum of other

forces acting on particle i. These forces may include gravita-

tion (gρV ), buoyancy (gδρV ), where δρ is density difference

between ice and ﬂuid, atmospheric and hydrodynamic/static

forces, etc. In order to simulate ice we use the Young’s mod-

ulus E=109Nm−2, density ρ=910kgm−3and fracture

strain c=(1–5)×10−4(Schulson, 1999). The damping co-

efﬁcient for collisions is C0=105Nsm−1.Crepresents the

drag force on ice falling into water in a calving event. A typ-

ical value for this parameter is 103kgs−1. If the contacts be-

tween the particles are broken, the material consisting of only

the particles behaves as a nearly incompressible ﬂuid. If the

contacts are not broken, the material consisting of particles

and beams, deforms elastically under small deformations.

In granular ﬂow, the viscosity depends on various factors

such as the packing density and the cohesion between parti-

cles in contact. These are two important parameters that af-

fect the diffusion and the momentum transfer between collid-

ing particles, which is the microscopic origin of viscosity in

any material, including polycrystalline ice. One of the many

contributions to viscosity of ice comes from grain-boundary

sliding (Johari et al., 1995). In general, diffusion increases

with temperature, which means more “liquid-like” for higher

temperatures and more “solid-like” for lower temperatures.

For ice, it therefore seems reasonable to model the viscous

cohesion as a “melting-refreezing probability”. This means

that once the elastic beam that models the frozen contact be-

tween adjacent particles is stretched or bent, the probability

for that beam to break becomes non-zero. In contrast, if the

tensile strain of the beam reaches the fracture strain the con-

tact always breaks. Also, when particles without a connecting

beam are close to each other, a beam can be created with a

small probability allowing the material to “refreeze”. When

combined, the two effects allow the material to undergo con-

stant liquid-like deformation (or regelation in the case of ice

bodies) as well as fracture. Notice that this method differs

from ﬂuid-like particle models such as smoothed particle hy-

drodynamics (Monghan, 1992, 2005).

Furthermore, the melting-freezing probability can be ad-

justed to produce stress-dependent viscous ﬂow obeying

Glen’s law for ﬂow rate, D=A(T )σ n−1

etD, where A(T ) is

a temperature dependent Arrhenius factor, σeis the second

invariant of the deviatoric stress tensor, Dis the strain-rate

tensor, tDis the deviatoric stress tensor, and n≈3. Corre-

spondingly, dynamic viscosity is µ≈1

Aσ 2. Details of this

derivation are given in Appendix A. The probabilities can be

adjusted such that the desired viscosity can be acquired, that

is the pre-factor, A, can be changed by adjusting the probabil-

ities. Usefully this allows the temperature dependence of A

in Glen’s ﬂow law to be incorporated in the model. Compu-

tational problems arise, however, from the fact that the time

step length is limited by the rapid timescale of the brittle fail-

ure events to approximately 10−4s, while the relevant time

scale for viscous ﬂow of ice is much longer. To cover both

relevant time scales in a single simulation is impractical. It is

however possible to use lower viscosities and thereby higher

strain rates and re-scale the simulation time to match ice be-

haviour as long as the viscous ﬂow timescale remains slow

compared with that for fracture events (Riikilä et al., 2013).

This approach is somewhat similar to the plasticity model

used by Timar et al. (2010). Another, simpler, approach to

imitate viscous behaviour is to use a weak and short-range

attraction force for particles that are close to being in contact,

similar to cohesion models of wet granular materials (Huang

et al., 2005). Both approaches seem to give fairly realistic

results. The former approach is benchmarked against a con-

tinuum model below.

It is also possible to introduce friction between the parti-

cles. This would add another, tangential, interaction potential

between grains in contact, although that would be in addition

to existing interactions modelled which also include tangen-

tial forces. Moreover, as long as particles are connected to

form larger blocks rather than just being individual particles,

(which is almost always the case in the simulations), granu-

lar friction appears as a natural consequence of the surface

roughness of the blocks. Because of these effects, and for

www.the-cryosphere.net/7/1591/2013/ The Cryosphere, 7, 1591–1602, 2013

1594 J. A. Åström et al.: A particle based simulation model for glacier dynamics

Fig. 2. A ﬂow chart showing the algorithm in a schematic fashion. Details may vary.

simplicity, we have not introduced an explicit friction force

between the particles. A ﬂow chart showing a representative

algorithm of a simulation is presented in Fig. 2.

3 Results

Figure 3 shows snapshots of a calving 30m×30m ice-block.

In this simulation the material is purely elastic, i.e. no vis-

cous component is present. The Young’s modulus and frac-

ture strain are set rather low, 108Pa and 10−4, respectively.

This example is not intended to exactly mimic any partic-

ular real glacier. There is also a signiﬁcant fraction of the

beams missing to mimic damaged ice and there is an artiﬁ-

cial crevasse at the top left of the ice block to initialise calv-

ing on the left side. The model block rests on a soft substrate

that hinders the block from sliding. This “muddy sea ﬂoor” is

modelled as a linear spring force prohibiting the block from

sinking too deep and from moving sideways once it is stuck

in the mud. The block is immersed in 20 m of water. The wa-

ter is modelled here only as a buoyancy force. The simulation

times of the snapshots are indicated. The time resolution in

the simulation is 10−4s. In this case the initial conﬁguration

is unstable and as soon as the simulation starts at time t=0,

cracks appear and the ice-block calves. The duration of this

single calving event is 10–20s, which is realistic in compar-

ison with similar events in nature.

Figure 4 shows the fragment size distribution, n(s), from

the simulations displayed in Fig. 3. n(s)dsis the number of

fragments found in the size interval [s−ds/2, s +ds/2], and

sis the number of particles in a fragment. This is, obviously,

proportional to the volume of the fragment, i.e. roughly the

same amount in m3. Note that the distribution in Fig. 4 is

a relative probability distribution, which means that values

of n(s) below unity can occur. Larger bin sizes have been

used for larger fragments to avoid empty bins and the relative

probabilities are calculated as fragments in one bin over the

bin width. Results for three different simulations are shown

and we distinguish between the size distributions early dur-

ing the calving event and late during this process when the

fragments have come to rest ﬂoating on the water. The frag-

ment size distributions are compared with that of the uni-

versal crack-branching-merging model for fragmentation of

brittle materials. This distribution is given by (Åström et al.,

2004; Åström, 2006; Kekäläinen et al., 2007):

n(s) ∝s−(2D−1)/D exp(−s/s0), (3)

where sis again the fragment size or volume, Dis the di-

mension (D=2 for the simulations here and D=3 for real

The Cryosphere, 7, 1591–1602, 2013 www.the-cryosphere.net/7/1591/2013/

J. A. Åström et al.: A particle based simulation model for glacier dynamics 1595

Fig. 3. Snapshots of a calving ice-block. The size of the block is 30m×30m. The block rests on a soft substrate that efﬁciently hinders the

block from sliding, thus resembling, e.g. a muddy sea ﬂoor. The block is immersed in 20m of water. The simulation times of the snapshots

are indicated.

glaciers), s0is a parameter which depends on, e.g. the ma-

terial and the fracture energy. The result shown in Fig. 3 is

consistent with ﬁeld data by Crocker (1993) and Savage et al.

(2000) from Bonavista Bay on Newfoundland, and with the

data of Dowdeswell and Forsberg (1992) from Kongsfjor-

den on Svalbard. The consistency shown means that the ﬁeld

data presented in these papers have, approximately, the same

shape for the fragment size distribution as Eq. (3).

Next we turn to verifying the viscous behaviour without

fracture. In order to verify the ﬂow behaviour of the model a

comparison with Elmer/Ice (Gagliardini and Zwinger, 2008;

Zwinger and Moore, 2009) was made. In both simulations an

ice block was placed on a ﬂat surface with little/no friction

and gravity as the only driving force. The result is displayed

in Fig. 5. The deformation of the ice blocks are obviously

quite similar. The main difference appears during the early

times of the simulations. This is probably due to a partial

jamming effect of the granular material. The particle model

parameters are set such that the resulting viscosity is 105

times lower than in the Elmer model, leading to 105times

faster strain rate, i.e. strain rate is proportional to the inverse

of viscosity. In a set of similar tests with varying shear rates,

we also veriﬁed that the particle model can reproduce Glen’s

ﬂow law (cf. Appendix A).

In order to further investigate the behaviour of our model,

we simulated the dynamics of an ice-block on a slope. We

chose a block of size 200m×50 m on a 18◦slope. Again, the

viscosity was roughly 105lower than realistic values for ice,

i.e. an Arrhenius factor ≈5×10−19 s−1Pa−3. We would thus

expect that the strain rates will be roughly 105higher than

realistic rates for ice deformation. It is thus, in a crude sense,

possible to re-scale the simulation time, which is calculated

in seconds, to approximately days (24h=0.864×105s).

In order to mimic natural variation in bed “stickyness”,

we also introduced the possibility to locally switch between

a high friction, i.e. a no-slip condition, and zero friction for

the contact between the substrate and the ice-block. We an-

chored the base of the block, indicated by a red line in Figs. 6

and 7, by high friction against the substrate. We also included

a pressure, corresponding to the over-burden pressure on the

upper vertical edge of the ice-block (Figs. 6 and 7). This sim-

ulates roughly the pressure induced by a slab of ice with same

thickness upstream. Finally, we investigate how the ice-block

slides down-slope as a function of the fraction of the rest of

the glacier being anchored to the substrate.

Figure 6 displays the case when only the top part of the

ice-block is anchored. The rest is free to slide down-slope

without friction. It is evident from this ﬁgure that the an-

chored part is not enough to keep the ice-block in place. It

breaks near the substrate and the entire block slides down-

slope. If the time is re-scaled as explained above, the velocity

reaches roughly 5 mday−1, which is comparable to observed

rates (about 10–100md−1, Cuffey and Paterson, 2010) for

surging glaciers.

In the opposite limit, when there are additional frictional

anchoring points on the lower part of the slope, the ice-block

cannot move, but remains stuck. In this case only a smaller

part of the block, near the lower edge, calves, fragments

and ﬂows a limited distance down the slope. As this layer

of highly fragmented ice ﬂows, it gets thinner and the force

driving it down-hill decreases and it slows down. This is dis-

played in Fig. 7.

www.the-cryosphere.net/7/1591/2013/ The Cryosphere, 7, 1591–1602, 2013

1596 J. A. Åström et al.: A particle based simulation model for glacier dynamics

Fig. 4. (A) The fragment size distribution for simulations like the

one in Fig. 3. The ﬁgure displays the results for three different sim-

ulations and both the distribution early during the calving event and

later when the fragments have come to rest ﬂoating on the water.

The line is the distribution predicted from Eq. (3). (B) The fragment

size distribution n(s) for a surging glacier. In this case n(s) ∝s−2.3.

In order to quantify the above, Fig. 8 shows kinetic energy

(Ekin) as a function of time for varying amounts of frictional

contact on the lower slope as described in Figs. 6 and 7. The

various lines in Fig. 8 correspond to two different phases,

a surging phase, when the entire ice-block slides down the

slope, and a quiescent phase, when only part of the front of

the ice-block fractures and ﬂows down-hill. For some of the

simulated cases there appears to be a phase-transition dur-

ing the simulation run time. In these cases the kinetic energy

is initially in the quiescent phase and at some point in time

the kinetic energy suddenly increases and rapidly approaches

that of the surging phase. Sometimes surging does not appear

for the entire block, and the kinetic energy only increases

part-way towards the surging phase before stabilising.

Finally, to highlight the difference between the single calv-

ing event represented by Fig. 3, and the surging glaciers in

Fig. 6, the fragment size distribution was calculated for the

surging glacier simulation (Fig. 4). In this case the fragment

size distribution was equivalent to that usually found for a

grinding process (Åström, 2006).

Fig. 5. Snapshots of a deforming ice block simulated with Elmer/Ice

and our particle model. In the Elmer/Ice simulation (red markers)

the snapshots are from time steps 0, 3, 5 and 7yr and the particle

model snapshots (red area) are from corresponding time steps. The

size of the block is 30m×30m and the time span of the Elmer/Ice

simulation is roughly 108s but only 103s in the particle simulation.

4 Discussion

The new model we introduced in this paper is certainly not

feasible to incorporate into ice sheet models given the ex-

tensive computing power required. It may however be used

to investigate details of calving processes and relationships

such as dependence of crevassing rate and fragment size on

the water depth at the calving front, or the presence and

inﬂuence of water in crevassses on fracture processes. The

model also has considerable potential to test and improve pa-

rameterizations of fracturing and calving used in continuum

models. The resolution of many models simply does not in-

clude small ice-cliff failure, and, since calving and fractur-

ing are essentially discontinuous processes, introducing them

into continuum models is problematic. Cuffey and Paterson

(2010) summarised the situation as: most models either let

ice shelves advance to the edges of the model grid, or as-

sume that ice shelves terminate at a prescribed water depth

(400m typically). For marine-terminating glaciers that are

not fully ﬂoating, most models either assume that calving rate

increases with water depth, or constrain the ice front thick-

ness instead of the calving rate. However, recent progress has

been considerable in the ﬁeld of parameterizing crevassing

by weakening the ice rheology in a damage model.

Our discrete particle formulation may be seen as a comple-

mentary method to statistical continuum damage approaches

that have been applied to ice shelves (e.g. Borstad et al.,

2012) or to mountain glacier calving (e.g. Jouvet et al.,

2011). This can be illustrated by, for example, Levermann

et al. (2012) who formulated the vertically averaged ice strain

rate tensor, which can be determined from the spatial deriva-

tive of the remotely sensed velocity pattern. His model can

The Cryosphere, 7, 1591–1602, 2013 www.the-cryosphere.net/7/1591/2013/

J. A. Åström et al.: A particle based simulation model for glacier dynamics 1597

Fig. 6. Snapshots of a 200m×50m ice-block on a 18◦slope. The

red line marks a high friction contact. The colours of the ice-block

represent elastic areas where the elastic beams between particles

have not been broken (green) and areas where beams are no longer

intact (gray). The pressure of an upstream ice slab on the slope

above the ice-block is not shown in the ﬁgure.

then be tuned to observations of speciﬁc ice shelves, and

no other observations are needed for the model to “carry it-

self forward” into the future. That is, it can “predict” calv-

ing without any other observation inputs (the necessary in-

puts would all come from the dynamic ice sheet model).

Over broad areas of an ice shelf, the viscosity is reduced

by crevasses (e.g. along the ﬂow units coming from different

tributary ice streams and glaciers), or the ice may be strength-

ened by the presence of sub ice shelf freeze-on of ocean wa-

ter or weakened by melting processes. The crevasses can be

readily seen in imagery, and these images can be used to tune

models for ice viscosity and fracture initiation stress in spe-

ciﬁc ice shelves or tributary ice streams to give similar pat-

terns of both crevassing and velocities as observations (Al-

brecht and Levermann, 2012a, b).

Above we demonstrated the importance of basal friction

to the behaviour of the particle model. Fast outlet ice streams

and surging glaciers are governed by the physics of basal

sliding. In temperate glaciers (i.e. glaciers with temperatures

Fig. 7. Similar as in Fig. 6, but with several “frictional anchors”,

indicated by red markers, also on the lower part of the slope.

at the pressure melting point) sliding behaviour is often

tightly interlinked with basal hydrology. On the continental

ice sheets, the fast ﬂowing ice streams, and outlet glaciers

owe their speed to basal sliding in addition to internal ice

deformation. Schoof (2009) showed that a variety of friction

laws converged on the Coulomb friction law in appropriate

parametric limits which can usefully describe a plastic till

rheology. The motion of ice streams appears to depend crit-

ically on the distribution and nature of regions of high drag

(“sticky spots”, Alley, 1993). It is not known what controls

the present conﬁguration of these features, though presum-

ably they are related to the bed roughness and geometry ei-

ther directly as a bedrock bump, or by routing water sup-

ply and till properties. Inverse methods can be used to deter-

mine the spatial variability of basal friction (Raymond and

Gudmundsson, 2009; Morlighem et al., 2010; Petra et al.,

2012; Arthern and Gudmundsson, 2010; Jay-Allemand et al.,

2011; Schäfer et al., 2012), in an analogous way to the dam-

age weakening of shear margins and crevasse damage on ice

shelves.

www.the-cryosphere.net/7/1591/2013/ The Cryosphere, 7, 1591–1602, 2013

1598 J. A. Åström et al.: A particle based simulation model for glacier dynamics

Fig. 8. The kinetic energy (Ekin) as function of time for “surging”

simulations exempliﬁed by Figs. 6 and 7 for 7 different simulations.

The cases when the ice-block is the entire time in either the surging

or the quiescent phase the energy is indicated by discrete markers.

For the blocks for which a phase transition appear, the kinetic en-

ergy is represented by continuous lines.

The discrete particle model we use here clearly suf-

fers from lack of three dimensional geometry; hence it is

presently limited to the testing or veriﬁcation of the parame-

terization used in continuum models. It is plausible to incor-

porate basal friction laws that could mimic more accurately

a plastic basal till with sticky spots. Our model prediction for

the particle size distributions can be readily tested in obser-

vational data on fragmentation and calving.

Appendix A

NUMFRAC – a particle based code for fracture

mechanics in nonlinear materials

A1 Elastic model

NUMFRAC is a particle based code which has been designed

to simulate fracture in nonlinear, e.g. visco-elastic or plas-

tic, materials. The code has been used for modelling brittle

fracture (Åström, 2006), cytoskeleton dynamics (Åström et

al., 2010), and mechanics of ﬁber networks (Åström et al.,

2012). A material is modelled as a set of particles that are

connected by interaction potentials. The code allows for arbi-

trary arranged “packings” of particles of different size, kind

and shape. For example, a rather loose random packing of

spheres of different size and stiffness can be used as a model

of an isotropic porous material with structure ﬂuctuations. In

contrast, a close packed hexagonal arrangement of identical

spheres models a dense uniform anisotropic material. Geom-

etry, interaction, and all other relevant parameters can be set

separately for all particles and particle-particle interactions,

Fig. 9. Model test result for Glen’s ﬂow law. Five constant shear

stresses were applied to a test block and strain rate was measured

and viscosity extracted. The melting-refreezing probabilities are

set so that the pre-factor, A, in Glen’s ﬂow law is of the order

10−10, rather than 10−24, which would be the realistic range for

ice: the simulated material in its shear dependent behaviour resem-

bles Glen’s ﬂow law but has an overall viscosity that is about 14

orders of magnitude lower than that of ice. This corresponds to a

viscosity roughly at the same order of magnitude as that of bitumen

or asphalt. The simulation data is represented by red markers and

ﬁtting the function, xB/A gave A≈1.5×10−10,B≈ −2.0 as it

should.

making the code an excellent tool for investigating mechani-

cal behaviour of strongly disordered materials.

To initialise a simulation, the particles are arranged and

packed to form the desired material. Thereafter the connec-

tions between particles are determined, e.g. by elastic beams,

which then determine the potentials.

The equations of motion may vary from case to case. A

simple example is given by:

M¨

ri+C˙

ri+X

j

Krij =Fi,(A1)

where Mis a mass-matrix containing the masses and mo-

ments of inertia of the particles, ri,˙

ri,¨

riare the position,

velocity and acceleration vectors of particle i, including ro-

tations. rij are the corresponding position vectors for all par-

ticles jthat are connected to particle i.Cis a damping matrix

containing damping coefﬁcient. Kis the stiffness matrix and

Fiis the sum of other forces acting on particle i. These forces

may include gravitation, buoyancy, atmospheric and hydro-

dynamic/static forces, etc.

An example of an interaction potential between two parti-

cles is an Euler–Bernoulli (E–B) beam. The elastic energy of

an E–B beam can be written as (1/2)kx2, where xis the dis-

placement vector containing translational and rotational dis-

placements of two connected mass points. If the components

of the displacement vector, x1and x4are the displacements

of the two connected mass points along the axis of the beam

The Cryosphere, 7, 1591–1602, 2013 www.the-cryosphere.net/7/1591/2013/

J. A. Åström et al.: A particle based simulation model for glacier dynamics 1599

that connects them, x2and x5the displacements in the per-

pendicular direction and x3and x6the rotations of the mass

points, the stiffness matrix kfor small deformations of a lin-

ear elastic E–B beam in two dimensions is:

k=

EA

L0 0 −EA

L0 0

012EI

L36EI

L20−12EI

L36EI

L2

06EI

L24EI

L0−6EI

L22EI

L

−EA

L0 0 EA

L0 0

0−12EI

L3−6EI

L2012EI

L3−6EI

L2

06EI

L22EI

L0−6EI

L24EI

L

,(A2)

where Iis the moment of inertia of the beam cross-section, L

the length of the beam, EYoung’s modulus and Athe cross-

section area of the beam.

In 3-D the matrix must be extended to a size of 12×12 en-

tries. Euler-Bernoulli beams overestimate the bending/shear

stiffness for short and fat beams. That is, beams with an as-

pect ratio smaller than or roughly equal to 20. For smaller

aspect ratios, Timoshenko beams may be used.

Another efﬁcient and easy potential which we have also

used for the ice application below, is to deﬁne a tension stiff-

ness which is a harmonic potential with respect to the dis-

tance between two particles, and another harmonic potential

with respect to node rotations away from the axis that con-

nects the mid-points of the two particles. These two stiffness

moduli can be directly linked to the macroscopic Young’s

modulus and Poisson ratio of the material to be modelled.

This potential is a type of shear-beam. Calving behaviour

seems to be rather insensitive to which beam model is used.

No signiﬁcant differences in results could be detected. Timo-

shenko beams are, from a theoretical point of view, the most

exact, but demand slightly more computations than Euler-

Bernoulli beams or shear beams. For the results shown here

we used the latter ones.

Once the interactions for the particle-particle connections

are determined and the appropriate stiffness matrices are as-

sembled, the global stiffness matrix, K, for the entire mate-

rial body to be simulated is assembled by expanding, with

zeros, the matrices for the individual particle-particle inter-

action matrices, and adding them together. Since the orienta-

tion is different for each interaction pair and the orientations

change over time, the last term in Eq. (A1) on the left side,

is typically nonlinear and can be obtained, for the present

example, via:

Krij =TT

t

Z

0

kT drij ,(A3)

where drij is the time dependent incremental displacement

vector for connected particles iand j,Tis a time dependent

rotation matrix for converting a global coordinate system to

the orientation of the beam axis connecting iand j.

In order to implement Eq. (A1) on a computer, the dif-

ferential equation must be rewritten as a difference equation

via:

r(t +1t) =

2M

1t2−Kr(t ) −M

1t2−C

21t r(t −1t)

M

1t2+C

21t −1

,(A4)

where ris now the global position vector containing all par-

ticle translations and rotations. tis time and 1t is the time

incremental, i.e. time-step.

This discrete form is easily implemented on a computer

and given time and space-dependent boundary conditions,

and r(t =0), the time development r(t) can unambiguously

be calculated.

For most fragmentation simulations there is further a need

to determine a fracture criterion. If “softening” potentials are

used, for example, the Lennard–Jones potential, there is no

such need since the attractive force between mass points van-

ishes continuously. For harmonic potentials the fracture cri-

terion must be deﬁned explicitly.

Choosing a proper fracture criterion is far from trivial. A

general elliptical criterion (Zhang and Eckert, 2005) that in-

cludes the “classical” fracture criterion of Tresca, von Mises,

Mohr-Coulomb and the maximum normal, i.e. hydrostatic

pressure, stress criterion is rather useful. This criterion states

that a material under tension fails at locations where

σ2

I

σ2

0

+σ2

I I

τ2

0

≥1,(A5)

in which σIis the normal stress, or pressure determined by

the ﬁrst invariant of the stress tensor and σII is shear stress

determined by the second invariant. σ0and τ0are material

dependent constants. Because both shear and normal stresses

are easily deﬁned for beams it is straightforward to imple-

ment this fracture criterion in the beam model. A fracture

limit can either be deﬁned for every beam separately or as

a limit on the average stress over several beams. For a sin-

gle beam, σIcan simply be set to the stress along the axis of

the beam and shear σI I the off-axis stress. For several beams

(e.g. for all beams connected to a single mass point) the aver-

age stress tensor can be divided in a trace-less, isotropic and

a diagonal part which then deﬁnes local shear and normal

stress, respectively.

A2 Plastic and visco-elastic models

In order for the particle model to be able to model not only

fracture of elastic material, but also plastic and visco-elastic

materials more elaborate interaction potentials, as compared

to purely elastic ones, must be used. The microscopic mech-

anism behind plasticity and visco-elasticity is that local in-

teractions cannot just be broken but also reform in conﬁgura-

tions different from the original one. This is the general prin-

ciple behind irreversible material deformations like viscous

www.the-cryosphere.net/7/1591/2013/ The Cryosphere, 7, 1591–1602, 2013

1600 J. A. Åström et al.: A particle based simulation model for glacier dynamics

Table A1. Typical parameters used in the simulations.

Ice density Water density Young’s modulus Fracture strain Drag coefﬁcient Time step

910 kg m−31000kg m−30.1–1.0 GPa 10−4–10−3103kgs−1(10−5–10−3)s

ﬂow and plastic deformation. In the particle model, plasticity

can be introduced via an ordinary yield/fracture stress cri-

terion combined with an additional criterion that allow new

contacts between particles to be formed. These new contacts

may be formed between any two particles that come close

enough to each other as the material deforms. Close enough

means here that the particles touch, or almost touch, each

other. With this mechanism, the material stress will not only

be dependent on the load as in an elastic material before any

fracture has taken place, but also on the load history.

A visco-elastic material differs from a plastic material in

that the material stress depends not only on load and load his-

tory but also on time. This means that for example a constant,

time independent, external displacement, stress will slowly

relax to zero in a visco-elastic material.

It is rather straightforward to incorporate this behaviour in

particle models. To demonstrate the procedure, Glen’s ﬂow

law can be obtained as a result of the adopted shear-beams

rheology via, so called, “melting-freezing probabilities”. If

these probabilities are set right, they can produce a stress-

dependent viscous ﬂow obeying Glen’s law for ﬂow rate,

D=A(T )σ n−1

etD, where A(T ) is a temperature dependent

Arrhenius factor, σeis the second invariant of the devia-

toric stress tensor, Dis the strain-rate tensor, tDis the de-

viatoric stress tensor, and n≈3. To derive Glen’s law for our

model in an approximate scalar form, we assume that melt-

ing events are random and uncorrelated which means that

the probability for an event to appear with a time interval

1t obeys an exponential probability function P=1−e−λ1t .

This function gives the probability of melting a beam dur-

ing a timestep 1t, where λis the rate of melting (melting

events per second). Each time a beam is broken all elastic

stress and strain, =σ/E , will be relaxed. Here , σ and

Eare strain, stress and Young’s modulus of the beam. As

long as the model material is under compression and the

particles are close-packed, the model material will be prac-

tically incompressible. This implies that the deformation of

the simulated material will be by shear ﬂow only and of the

form D≈tD

Eλ. By choosing λ=2AU

a2E2≈Aσ 2Ewe ob-

tain Glen’s law D=A(T )σ 2tD. Here Uis the elastic energy

of the beam and ais the diameter of the discrete blocks (and

length of a beam). The elastic energy of a beam Uis roughly

equal to the elastic energy density σ2

2E, multiplied by the ef-

fective volume of a beam a2. Correspondingly, dynamic vis-

cosity is µ≈E

λ≈1

Aσ 2. Figure 9 shows viscosity as a func-

tion of shear stress in a simulation series where a varying

shear stress was applied to a test square.

A3 Application to glacier mechanics

The application to real world ice problems is still under de-

velopment. Some general issues that are important input to

the model are still poorly understood for glacier ice. The

most important is probably the correlation and distribution of

structural and mechanical disorder. Disorder, in this context,

means the impurities, weaknesses, cracks that are abundant

in natural ice. Such impurities have an affect on local stiff-

ness moduli and fracture thresholds on the “particle scale”,

which is presently cubic-metre.

Since some of the model key parameters are only known

with accuracies of an order of magnitude, there is no point

to model other parameters to higher accuracy. Therefore we

simply use, e.g. 9103kg m−3for ice density, a 10% density

difference between ice and water for buoyancy. The Young’s

modulus of ice is a particularly difﬁcult parameter. For pure

crystalline ice the value of Young’s modulus Eis roughly

1010 Pa (Schulson, 1999), while the effective stiffness Sof

polycrystalline glacier ice commonly drops to about 109Pa

(Vaughan, 1995). It is quite obvious that when the “quality”

of ice weakens, stiffness, in particular when measured as ten-

sion, will decrease drastically. Eventually, as ice begins to

melt or is highly damaged, it practically vanishes. The same

kind of uncertainty is relevant for fracture strain of ice. For

the present simulations we have used values [108−109]Pa

for Young’s modulus and [10−4−10−3]fracture strain. The

damping coefﬁcient for colliding ice-blocks is another pa-

rameter that is not known to us. It is, however, reasonable

to expect that colliding square-metre ice blocks would not

bounce back very efﬁciently, and therefore we use values

similar to, or slightly less than that for critical damping of the

harmonic potential. The water drag coefﬁcient for a square-

metre object is roughly 103kgs−1. Typical simulation pa-

rameters are displayed in Table A1.

From a computational point of view the most problematic

aspect of the simulations are the short time-spans that can

be simulated. In practice, the only reasonable way to slightly

reduce this problem is to use a viscosity that is lower than

that of ice, but which still resembles Glen’s ﬂow law. The

relaxation time near ice cliffs does not have a simple deﬁ-

nition but is likely in the range of weeks or months. Since

the calving time scale is of the order of a few seconds we can

drastically reduce the viscous relaxation time without getting

much interference between the two, i.e. the Deborah number

deﬁned as viscous relaxation time over typical calving time

is of the order of 106(Reiner, 1964). In other words, reducing

The Cryosphere, 7, 1591–1602, 2013 www.the-cryosphere.net/7/1591/2013/

J. A. Åström et al.: A particle based simulation model for glacier dynamics 1601

the viscous timescale by 105from days to seconds still leaves

the viscous and the fracture timescales well separated.

A4 Computational implementation

The code has been constructed by the authors TR,JA,TT and

does not use any commercial sub-routines or libraries. The

code is written in Fortran and C++ and utilises MPI for par-

allel computing. The scalability to large glaciers, i.e. many

particles, is close to perfect. The most severe limiting factor

is the time-step, which is of the order 10−4−10−3s. There

is no useful way to formulate the time discretization in par-

allel and, at present, the code may, in the best case, simulate

about one hour of glacier dynamics within 24h of comput-

ing. The only reasonable way to speed up the computations

is to calibrate the particle model such that the macroscopic

viscosity of the model is considerably lower than that of ice

as explained above.

Since computation of time cannot be made parallel, it

is important, for computational efﬁciency to maximise the

time-step without violating the stability of the computation.

Since there are no explicit temperature ﬂuctuations in the

simulations and no external heat sources it is easy to check

for stability by recording the total energy and check for en-

ergy conservation during simulations.

The particle size sets a very intuitive resolution limit for

the simulation. At present the particles are of the order of

one cubic metre, but the model is inherently scale invariant.

As long as the equations of motion, Eq. (A1) remain un-

changed, all parameters can be re-scaled without changing

the simulation results. This means, the same simulation can

be viewed as, e.g. a cubic-metre or cubic-millimetre simu-

lation if the length unit in r,M,C,Kand Fare all changed

accordingly. For example, if the length unit is decreased, M

and F, which is dominated by gravity, are reduced in propor-

tion to the volume of the particles, while Kis reduced only

in proportion to the surface area of the particles. This means

that small glaciers are much more stable, under gravity, than

large ones.

Acknowledgements. This publication is contribution number 22 of

the Nordic Centre of Excellence SVALI, “Stability and Variations

of Arctic Land Ice”, funded by the Nordic Top-level Research

Initiative (TRI). The work has been supported by the SVALI project

through the University of Lapland, Arctic Centre, and through

the University Centre in Svalbard. Funding was also provided by

the Conoco-Phillips and Lunding High North Research Program

(CRIOS: Calving Rates and Impact on Society). The funding is

gratefully acknowledged.

Edited by: O. Gagliardini

References

Albrecht, T. and Levermann, A.: Fracture-induced softening for

large-scale ice dynamics, The Cryosphere Discuss., 7, 4501–

4544, doi:10.5194/tcd-7-4501-2013, 2013.

Albrecht, T. and Levermann, A.: Fracture ﬁeld for

large-scale ice dynamics, J. Glaciol., 58, 165–176,

doi:10.3189/2012JoG11J191, 2012b.

Alley, R. B.: In search of ice-stream sticky spots, J. Glaciol., 39,

447–454, 1993.

Arthern, R. J. and Gudmundsson, G. H.: Initialization of ice-sheet

forecasts viewed as an inverse Robin problem, J. Glaciol., 56,

527–533, 2010.

Åström, J. A.: Statistical models of brittle fragmentation, Adv.

Phys., 55, 247–278, 2006.

Åström, J. A., Ouchterlony, F., Linna, R. P., and Timonen, J.: Uni-

versal dynamic fragmentation in D dimensions, Phys. Rev. Lett.,

92, 245506, doi:10.1103/PhysRevLett.92.245506, 2004.

Åström, J. A., von Alfthan, S., Sunil Kumar, P. B., and Karttunen,

M.: Myosin motor mediated contraction is enough to produce cy-

tokinesis in the absence of polymerisation, Soft Matter, 6, 5375,

doi:10.1039/C0SM00134A, 2010.

Åström, J. A., Sunil Kumar, P. B., and Karttunen, M.: Stiffness

transition in anisotropic ﬁber nets, Phys. Rev. E, 86, 021922,

doi:10.1103/PhysRevE.86.021922, 2012.

Bassis, J. N.: The statistical physics of iceberg calving and the emer-

gence of universal calving laws, J. Glaciol, 57, 3–16, 2011.

Bassis, J. N. and Walker, C. C.: Upper and lower limits on the

stability of calving glaciers from the yield strength envelope of

ice, Proc. R. Soc. A, 468, 913–931, doi:10.1098/rspa.2011.0422,

2012.

Benn, D. I., Hulton, N. R. J., and Mottram, R. H.: “Calving

laws”, “sliding laws” and the stability of tidewater glaciers, Ann.

Glaciol., 46, 123–130, 2007a.

Benn, D. I., Warren, C. R., and Mottram, R. H.: Calving processes

and the dynamics of calving glaciers, Earth Sci. Rev., 82, 143–

179, 2007b.

Borstad, C. P., Khazendar, A., Larour, E., Morlighem, M., Rig-

not, E., Schodlok, M. P., and Seroussi, H.: A damage mechanics

assessment of the Larsen B ice shelf prior to collapse: towards a

physically based calving law, Geophys. Res. Lett., 39, L18502,

doi:10.1029/2012GL053317, 2012.

Burgess, D. O., Sharp, M. J., Mair, D. W. F., Dowdeswell, J. A.,

and Benham, T. J.: Flow dynamics and iceberg calving rates of

Devon Ice Cap, Nunavut, Canada, J. Glaciol., 51, 219–230, 2005.

Crocker, G. B.: Size distributions of bergy bits and growlers calved

from deteriorating icebergs, Cold. Reg. Sci. Technol., 22, 113–

119, 1993.

Cuffey, K. M. and Paterson, W. S. B.: The Physics of Glaciers,

4th Edn., Academic Press, 704 pp., 2010.

Cundall, P. A. and Strack, O.: A discrete numerical model for gran-

ular assemblies, Geotechnique, 29, 47–65, 1979.

Dowdeswell, J. A. and Forsberg, J. A.: The size and frequency of

icebergs and bergy bits derived from tidewater glaciers in Kongs-

fjorden, northwest Spitsbergen, Polar Res., 11, 81–91, 1992.

Dowdeswell, J. A., Benham, T. J., Strozzi, T., and Hagen, J. O.:

Iceberg calving ﬂux and mass balance of the Austfonna ice cap

on Nordaustlandet, Svalbard, J. Geophys. Res., 113, F03022,

doi:10.1029/2007JF000905, 2008.

www.the-cryosphere.net/7/1591/2013/ The Cryosphere, 7, 1591–1602, 2013

1602 J. A. Åström et al.: A particle based simulation model for glacier dynamics

Gagliardini, O. and Zwinger, T.: The ISMIP-HOM benchmark ex-

periments performed using the Finite-Element code Elmer, The

Cryosphere, 2, 67–76, doi:10.5194/tc-2-67-2008, 2008.

Gethin, D. T., Ransing, R. S., Lewis, R. W., and Dutko, M.: Nu-

merical comparison of a deformable discrete element model and

an equivalent continuum analysis for the compaction of ductile

porous material, Comput. Struct., 79, 1287–1294, 2001.

Huang, N., Ovarlez, G., Bertrand, F., Rodts, S., Coussot, P., and

Bonn, D.: Flow of Wet Granular Materials, Phys. Rev. Lett. 94,

028301, doi:10.1103/PhysRevLett.94.028301, 2005.

Jacob, T., Wahr, J., Pfeffer, W. T., and Swenson, S.: Recent con-

tributions of glaciers and ice caps to sea level rise, Nature, 482,

514–518, doi:10.1038/nature10847, 2012.

Jay-Allemand, M., Gillet-Chaulet, F., Gagliardini, O., and

Nodet, M.: Investigating changes in basal conditions of Var-

iegated Glacier prior to and during its 1982–1983 surge, The

Cryosphere, 5, 659–672, doi:10.5194/tc-5-659-2011, 2011.

Jing, L.: Formulation of discontinuous deformation analysis

(DDA)- an implicit discrete element model for block systems,

Eng. Geol., 49, 371–381, 1998.

Johari, G. P., Pascheto, W., and Jones, S. J.: Anelasticity and grain

boundary relaxation of ice at high temperatures, J. Phys. D Appl.

Phys., 28, 112, doi:10.1088/0022-3727/28/1/018, 1995.

Jouvet, G. and Rappaz, J.: Analysis and ﬁnite element ap-

proximation of a nonlinear stationary Stokes problem

arising in glaciology, Adv. Numer. Anal., 2011, 164581,

doi:10.1155/2011/164581, 2011.

Jouvet, G., Picasso, M., Rappaz, J., Huss, M., and Funk, M.: Mod-

elling and numerical simulation of the dynamics of glaciers in-

cluding local damage effects, Math. Model. Nat. Phenom., 6,

263–280, doi:10.1051/mmnp/20116510, 2011.

Kekäläinen, P., Åström, J. A., and Timonen, J.: Solution

for the fragment-size distribution in crack-branching

model of fragmentation, Phys. Rev. E, 76, 026112,

doi:10.1103/PhysRevE.76.026112, 2007.

Levermann, A., Albrecht, T., Winkelmann, R., Martin, M. A.,

Haseloff, M., and Joughin, I.: Kinematic ﬁrst-order calving law

implies potential for abrupt ice-shelf retreat, The Cryosphere, 6,

273–286, doi:10.5194/tc-6-273-2012, 2012.

Monaghan, J. J.: Smoothed particle hydrodynamics, Ann. Rev. As-

tro. Astrophys., 30, 543–574, 1992.

Monaghan, J. J.: Smoothed particle hydrodynamics, Rep. Prog.

Phys., 68, 1703–1759, 2005.

Morlighem, M., Rignot, E., Seroussi, H., Larour, E., Dhia, H. B.,

and Aubry, D.: Spatial patterns of basal drag inferred using con-

trol methods from a full-Stokes and simpler models for Pine Is-

land Glacier, West Antarctica, Geophys. Res. Lett., 37, L14502,

doi:10.1029/2006JF000576, 2010.

Nick, F. M., van der Veen, C. J., Vieli, A., and Benn, D. I.: A phys-

ically based calving model applied to marine outlet glaciers and

implications for their dynamics, J. Glaciol., 56, 781–794, 2010.

Petra, N., Zhu H., Stadler, G., Hughes, T. J. R., and Ghattas, O.:

An inexact Gauss–Newton method for inversion of basal sliding

and rheology parameters in a nonlinear Stokes ice sheet model,

J. Glaciol., 58, 889–903, 2012.

Potyondy, D. O. and Cundall, P. A.: A bonded-particle model for

rock, Int. J. Rock Mech. Min., 41, 1329–1364, 2004.

Raymond, M. J. and Gudmundsson, G. H.: Estimating basal prop-

erties of ice streams from surface measurements: a nonlin-

ear Bayesian inverse approach applied to synthetic data, The

Cryosphere, 3, 265–278, doi:10.5194/tc-3-265-2009, 2009.

Reiner, M.: “The Deborah Number”, Physics Today, 17, 62,

doi:10.1063/1.3051374, 1964.

Riikilä, T., Åström, J., Tallinen, T., Zwinger, T., Benn, D., Moore,

J., and Timonen, J.: Discrete element model for viscoelastic ma-

terials with fracture, in preparation, 2013.

Rignot, E., Mouginot, J., and Scheuchl, B.: Ice ﬂow

of the Antarctic ice sheet, Science, 333, 1427–1430,

doi:10.1126/science.1208336, 2011.

Savage, S. B., Crocker, G. B., Sayed, M., and Carriers, T.: Cold.

Reg. Sci. Technol., 31, 163–172, 2000.

Schäfer, M., Zwinger, T., Christoffersen, P., Gillet-Chaulet, F.,

Laakso, K., Pettersson, R., Pohjola, V. A., Strozzi, T., and

Moore, J. C.: Sensitivity of basal conditions in an inverse model:

Vestfonna ice cap, Nordaustlandet/Svalbard, The Cryosphere, 6,

771–783, doi:10.5194/tc-6-771-2012, 2012.

Schoof, C.: Coulomb friction and other sliding laws in a higher or-

der glacier ﬂow model, Math. Model. Meth. Appl. Sci., 20, 157–

189, 2009.

Schulson, E. M.: The structure and mechanical behaviour of ice,

JOM, 51, 21–27, 1999.

Schulson, E. M.: Brittle failure of ice, Eng. Fract. Mech., 68, 1839–

1887, doi:10.1016/S0013-7944(01)00037-6, 2001.

Thomas, R. H., Rignot, E. J., Kanagaratnam, K., Krabill, W. B., and

Casassa, G.: Force-perturbation analysis of Pine Island Glacier,

Antarctica, suggests cause for recent acceleration, Ann. Glaciol.,

39, 133–138, doi:10.3189/172756404781814429, 2004.

Timar, G., Blomer, J., Kun, F., and Herrmann, H. J.: New univer-

sality class for the fragmentation of plastic materials, Phys. Rev.

Lett., 104, 095502, doi:10.1103/PhysRevLett.104.095502, 2010.

Vaughan, D. G.: Tidal ﬂexure at ice margins, J. Geophys. Res., 100,

6213–6224, doi:10.1029/94JB02467, 1995.

Vieli, A., Payne, A. J., Du, A., and Shepherd, A.: Numerical mod-

elling and data assimilation of Larsen B ice shelf, Antarctic

Peninsula, Philos. T. Roy. Soc. A, 364, 1815–1839, 2006.

Walter, F., O’Neel, S., McNamara, D., Pfeffer, W. T., Bassis, J. N.,

and Fricker, H. A.: Iceberg calving during transition from

grounded to ﬂoating ice: Columbia Glacier, Alaska, Geophys.

Res. Lett., 37, L15501, doi:10.1029/2010GL043201, 2010.

Weertman, J.: Depth of water-ﬁlled crevasses that are closely

spaced, J. Glaciol., 13, 544–544, 1974.

Weertman, J.: Bottom crevasses, J. Glaciol., 25, 185–188,

doi:10.1029/JB089iB03p01925, 1980,

Zhang, Z. F. and Eckert, J.: Uniﬁed Tensile Fracture Criterion, Phys.

Rev. Lett. 94, 094301, doi:10.1103/PhysRevLett.94.094301,

2005.

Zwinger, T. and Moore, J. C.: Diagnostic and prognostic simula-

tions with a full Stokes model accounting for superimposed ice

of Midtre Lovénbreen, Svalbard, The Cryosphere, 3, 217–229,

doi:10.5194/tc-3-217-2009, 2009.

The Cryosphere, 7, 1591–1602, 2013 www.the-cryosphere.net/7/1591/2013/