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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 particles 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 calving and fracturing in a more realistic way than traditional continuum models. For benchmarking purposes the deformation 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, similar to a grounded marine glacier terminus, and (2) fracturing of an ice block on an inclined plane of varying basal friction, 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 observed in calving, which may be approximated by universal 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 contact, 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.
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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
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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 ) 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=109Nm2, density ρ=910kgm3and fracture
strain c=(1–5)×104(Schulson, 1999). The damping co-
efficient for collisions is C0=105Nsm1.Crepresents the
drag force on ice falling into water in a calving event. A typ-
ical value for this parameter is 103kgs1. 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 n1
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 n3. Corre-
spondingly, dynamic viscosity is µ1
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 104s, 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
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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 104, 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 104s. 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 [sds/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(2D1)/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
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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 18slope. Again, the
viscosity was roughly 105lower than realistic values for ice,
i.e. an Arrhenius factor 5×1019 s1Pa3. 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 mday1, which is comparable to observed
rates (about 10–100md1, 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.
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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) s2.3.
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
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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 18slope. 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.
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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
1010, rather than 1024, 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 A1.5×1010,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
L2012EI
L36EI
L2
06EI
L24EI
L06EI
L22EI
L
EA
L0 0 EA
L0 0
012EI
L36EI
L2012EI
L36EI
L2
06EI
L22EI
L06EI
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
1t2Kr(t ) M
1t2C
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
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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 m31000kg m30.1–1.0 GPa 104–103103kgs1(105–103)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 n1
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 n3. 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=1eλ1t .
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 DtD
Eλ. By choosing λ=2AU
a2E22Ewe 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 µE
λ1
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 m3for 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 [108109]Pa
for Young’s modulus and [104103]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 103kgs1. 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 104103s. 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
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... The main objective of this investigation is to use the Bonded Particle Model (BPM) HiDEM [16][17][18] to reproduce the experiments of Prasanna et al. [15]. BPMs are often used to model the failure of complex materials, as a wide range of failure phenomena can be modeled via the interaction rules for the particles. ...
... Figures 2(a) (I) and (II) show illustrations of a beam element failing due to force applied or moment acting on particles. The equation of motion for particle i is [17] ...
... (II) Stress-train response of a beam with cohesive crack. (Panels (a) and (b) reproduced from Åström et al.[17] and Paavilainen et al.[39], respectively.) ...
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A major challenge within material science is the proper modeling of force transmission through fragmenting materials under compression. A particularly demanding material is sea ice, which on small scales is an anisotropic material with quasibrittle characteristics under failure. Here we use the particle-based model HiDEM and laboratory-scale experiments on saline ice to develop a material model for fragmenting ice. The material behavior of the HiDEM model-ice, and the experiments are compatible on force transmission and fragmentation if: (i) the typical HiDEM glacier-scale particle size of meters is brought down to millimeters corresponding to the grain size of the laboratory ice, (ii) the often used HiDEM lattice structure is replaced by a planar random structure with an anisotropy in the direction normal to the randomized plane, and (iii) the instant tensile and bending failure criterion, used in HiDEM on glacier scale, is replaced by a cohesive softening failure potential for energy dissipation. The main outcomes of this exercise is that many of the, more or less, traditional ice modeling schemes are proven to be incomplete. In particular, local crushing of ice is not valid as a generic failure mode for fragmented ice under compression. Rather, shear failure, as described by Mohr-Coulomb theory is demonstrated to be the dominant failure mode.
... Specifically, we progressively remove basal support from the glacier, rather than pushing it. In addition, in order to add some randomness to the system, we added a Voronoi-based spatial distribution of weak joints (zero strength) within the glacier in order to represent pre-existing crevasses or cracks (similarly to Aström et al. [2013Aström et al. [ , 2014). As you can see in Figure 5.3, these modifications led to a highly-detailed very large-scale glacier-ocean simulation that not only illustrates the high visual fidelity of our method, but that we have also shown to be numerically accurate. ...
Thesis
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Material fracture surrounds us every day from tearing off a piece of fresh bread to dropping a glass on the floor. Modeling this complex physical process has a near limitless breadth of applications in everything from computer graphics and VFX to virtual surgery and geomechanical modeling. Despite the ubiquity of material failure, it stands as a notoriously difficult phenomenon to simulate and has inspired numerous efforts from computer graphics researchers and mechanical engineers alike, resulting in a diverse set of approaches to modeling the underlying physics as well as discretizing the branching crack topology. However, most existing approaches focus on meshed methods such as FEM or BEM that require computationally intensive crack tracking and re-meshing procedures. Conversely, the Material Point Method (MPM) is a hybrid meshless approach that is ideal for modeling fracture due to its automatic support for arbitrarily large topological deformations, natural collision handling, and numerous successfully simulated continuum materials. In this work, we present a toolkit of augmented Material Point Methods for robustly and efficiently simulating material fracture both through damage modeling and through plastic softening/hardening. Our approaches are robust to a multitude of materials including those of varying structures (isotropic, transversely isotropic, orthotropic), fracture types (ductile, brittle), plastic yield surfaces, and constitutive models. The methods herein are applicable not only to the needs of computer graphics (efficiency and visual fidelity), but also to the engineering community where physical accuracy is key. Most notably, each approach has a unique set of parametric knobs available to artists and engineers alike that make them directly deployable in applications ranging from animated movie production to large-scale glacial calving simulation.
... Also, remarkable works [13][14][15][16][17][18][19][20][21] should be cited, which concern the problem under consideration. ...
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We have presented in this analytical research the revisiting of approach for mathematical modeling the Glacier dynamics in terms of viscous-plastic theory of 2-dimensional movements within (x, y)-plane in cartesian coordinates. The stationary creeping approximation for the plane-parallel flow of slowly moving glacial ice on absolutely flat surface without any inclination has been considered. Even in such simple formulation, equations of motion that governs by the dynamics of viscous-plastic flow of glacial ice is hard to be solved analytically. We have succeeded in obtaining analytical expression for the components of velocity in Ox-direction of motion for slowly moving glacial ice (Ox-axis coincides to the initial main direction of slowly moving glacial ice). Restrictions on the form of flow stem from the continuity equation as well as from the special condition for non-Newtonian (viscous-plastic) flow have been used insofar.
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Preprint
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Ice shelves play a key role in the dynamics of marine ice sheets, by buttressing grounded ice and limiting rates of ice flux to the oceans. In response to recent climatic and oceanic change, ice shelves fringing the West Antarctic Ice Sheet (WAIS) have begun to fragment and retreat, with major implications for ice sheet stability. Here, we focus on the Thwaites Eastern Ice Shelf (TEIS), the remaining pinned floating extension of Thwaites Glacier. We show that TEIS has undergone a process of fragmentation in the last five years, including brittle failure along a major shear zone, formation of tensile cracks on the main body of the shelf, and release of tabular bergs on both eastern and western flanks. Simulations with the Helsinki Discrete Element Model (HiDEM) show that this pattern of failure is associated with high backstress from a submarine pinning point at the distal edge of the shelf. We show that a significant zone of shear upstream of the main pinning point developed in response to the rapid acceleration of the shelf between 2002 and 2006, seeding damage on the shelf. Subsequently, basal melting and positive feedbacks between damage and strain rates weakened TEIS, allowing damage to accumulate. Thus, although backstress on TEIS has likely diminished through time as the pinning point has shrunk, accumulation of damage has ensured that the ice in the shear zone has remained the weakest link in the system. Experiments with the BISICLES ice sheet model indicate that additional damage to or unpinning of TEIS are unlikely to trigger significantly increased ice loss from WAIS, but the calving response to loss of TEIS remains highly uncertain. It is widely recognised that ice-shelf fragmentation and collapse can be triggered by hydrofracturing and/or unpinning from ice shelf margins or grounding points. Our results indicate a third mechanism, backstress-triggered failure, that can occur when ice ffractures in response to stresses associated with pinning points. In most circumstances, pinning points are essential for ice shelf stability, but as ice shelves thin and weaken the concentration of backstress in damaged ice upstream of a pinning point may provide the seeds of their demise.
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Mass loss caused by glacier calving is one of the direct contributors to global sea level rise. Reliable calving laws are required for accurate modelling of ice sheet mass balance. Both continuous and discontinuous methods have been used for glacial calving simulations. In this study, the discrete element method (DEM) based on dilated polyhedral elements is introduced to simulate the calving process of a tidewater glacier. Dilated polyhedrons can be obtained from the Minkowski sum of a sphere and a core polyhedron. These elements can be utilized to generate a continuum ice material, where the interaction force between adjacent elements is modeled by constructing bonds at the joints of the common faces. A hybrid fracture model considering fracture energy is introduced. The viscous creep behavior of glaciers on long-term scales is not considered. By applying buoyancy and gravity to the modelled glacier, DEM results show that the calving process is caused by cracks which are initialized at the top of the glacier and spread to the bottom. The results demonstrate the feasibility of using the dilated polyhedral DEM method in glacier simulations, additionally allowing the fragment size of the breaking fragments to be counted. The relationship between crack propagation and internal stress in the glacier is analyzed during calving process. Through the analysis of the Mises stress and the normal stress between the elements, it is found that geometric changes caused by the glacier calving lead to the redistribution of the stress. The tensile stress between the elements is the main influencing factor of glacier ice failure. In addition, the element shape, glacier base friction and buoyancy are studied, the results show that the glacier model based on the dilated polyhedral DEM is sensitive to the above conditions.
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We present repeated radio-echo sounding (RES, 5 MHz) on a profile grid over the eastern Skaftá cauldron (ESC) in Vatnajökull ice cap, Iceland. The ESC is a ∼ 3 km wide and 50–150 m deep ice cauldron created and maintained by subglacial geothermal activity of ∼ 1 GW. Beneath the cauldron and 200–400 m thick ice, water accumulates in a subglacial lake and is released semi-regularly in jökulhlaups. The RES record consists of annual surveys conducted at the beginning of every summer during the period 2014–2020. Comparison of the RES surveys reveals variable lake area (0.5–4.1 km2) and enables traced reflections from the lake roof to be distinguished from bedrock reflections. This allows construction of a digital elevation model (DEM) of the bedrock in the area, further constrained by two borehole measurements at the cauldron centre. It also allows creation of lake thickness maps and an estimate of lake volume at the time of each survey, which we compare with lowering patterns and released water volumes obtained from pre- and post-jökulhlaup surface DEMs. The estimated lake volume was 250 GL (gigalitres = 106 m3) in June 2015, but 320 ± 20 GL drained from the ESC in October 2015. In June 2018, RES profiles revealed a lake volume of 185 GL, while 220 ± 30 GL were released in a jökulhlaup in August 2018. Considering the water accumulation over the periods between RES surveys and jökulhlaups, this indicates 10 %–20 % uncertainty in the RES-derived volumes at times when significant jökulhlaups may be expected.
Thesis
Numerical models currently in use for projections of future ice sheet mass balance lack a mechanistic description of iceberg calving, introducing uncertainty in the future glaciological contribution to global sea level. Constraining dynamic mass loss associated with particular future scenarios can help us parse that uncertainty. We have modified the plastic approximation of Nye (1952) to apply to ocean- terminating glaciers (published derivation: Ultee & Bassis, 2016) and generate physically consistent constraints on dynamic mass loss. Our approach accounts for the interaction of multiple glacier tributary branches (published methods: Ul- tee & Bassis, 2017) and their contribution to sea level. For four large Greenland outlet glacier catchments—Sermeq Kujalleq (Jakobshavn Isbræ), Koge Bugt, Hel- heim, and Kangerlussuaq Glaciers—we find an upper bound of 29 mm on dy- namic contribution to sea level after 100 years of warming. This bound accounts for dynamic loss only and can be summed with surface mass balance projections to bound the total glaciological contribution to sea level from those catchments. The convergence of upper bounds derived from our two strongest forcing scenar- ios agrees with studies that suggest surface mass balance will dominate future mass loss from Greenland. Although our work is motivated by coastal communities’ exposure to rising seas, the constraints we produce here are unlikely to be immediately usable for coastal adaptation. Intermediaries such as extension agents, climate consultants, or re- gional science-policy boundary organizations may be able to tailor our results for use in local adaptation contexts (published commentary: Ultee, Arnott, Bassis, & Lemos, 2018). Understanding the landscape of science intermediation, as well as working directly with stakeholders, can help researchers produce more usable sea level information.
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Floating ice shelves can exert a retentive and hence stabilizing force onto the inland ice sheet of Antarctica. However, this effect has been observed to diminish by fracture-coupled dynamic processes within the protective ice shelves leading to accelerated ice flow and hence to a sea-level contribution. In order to better understand the role of fractures in ice dynamics we apply a large-scale continuum representation of fractures and related fracture growth into the prognostic Parallel Ice Sheet Model (PISM). To this end we introduce a higher-order accuracy advection scheme for the transport of the two-dimensional fracture density across the regular computational grid. Dynamic coupling of fractures and ice flow is attained by a reduction of effective ice viscosity proportional to the inferred fracture density. This formulation implies the possibility of a non-linear threshold behavior due to self-amplified fracturing in shear regions triggered by small variations in damage threshold. As a result of prognostic flow simulations, flow patterns with realistically large across-flow velocity gradients in fracture-weakened regions as seen in observations are reproduced. This model framework is expandable to grounded ice streams and accounts for climate-induced effects on fracturing and hence on the ice-flow dynamics. It further allows for an enhanced fracture-based calving parameterization.
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Recently observed large-scale disintegration of Antarctic ice shelves has moved their fronts closer towards grounded ice. In response, ice-sheet discharge into the ocean has accelerated, contributing to global sea-level rise and emphasizing the importance of calving-front dynamics. The position of the ice front strongly influences the stress field within the entire sheet-shelf-system and thereby the mass flow across the grounding line. While theories for an advance of the ice-front are readily available, no general rule exists for its retreat, making it difficult to incorporate the retreat in predictive models. Here we extract the first-order large-scale kinematic contribution to calving which is consistent with large-scale observation. We emphasize that the proposed equation does not constitute a comprehensive calving law but represents the first-order kinematic contribution which can and should be complemented by higher order contributions as well as the influence of potentially heterogeneous material properties of the ice. When applied as a calving law, the equation naturally incorporates the stabilizing effect of pinning points and inhibits ice shelf growth outside of embayments. It depends only on local ice properties which are, however, determined by the full topography of the ice shelf. In numerical simulations the parameterization reproduces multiple stable fronts as observed for the Larsen A and B Ice Shelves including abrupt transitions between them which may be caused by localized ice weaknesses. We also find multiple stable states of the Ross Ice Shelf at the gateway of the West Antarctic Ice Sheet with back stresses onto the sheet reduced by up to 90 % compared to the present state.
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Glaciers and ice caps (GIC) are important contributors to present-day global mean sea-level rise (SLR). Most previous global mass balance estimates for GIC rely on interpolation of sparse mass balance measurements, representing a small fraction of the GIC area. Instead, we here perform a global, simultaneous inversion of mass change over all ice-covered regions larger than 100 km2 using monthly GRACE-derived satellite gravity fields spanning January 2003 to December 2010. This is the first GRACE-based study where every such GIC region is considered and the results quantified. We conclude that GIC, excluding the Greenland and Antarctic peripheral glaciers and ice caps (PGIC), lost mass at a rate of 148 ± 30 giga tonnes per year (Gt/yr) during this period, contributing 0.41 ± 0.08 mm/yr to SLR. This rate is significantly smaller than previous estimates that rely on extrapolation of mass balance measurements.
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The basal shear stress of an ice stream may be supported disproportionately on localized regions or “sticky spots”. The drag induced by large bedrock bumps sticking into the base of an ice stream is the most likely cause of sticky spots. Discontinuity of lubricating till can cause sticky spots, but they will collect lubricating water and therefore are unlikely to support a shear stress of more than a few tenths of a bar unless they contain abundant large bumps. Raised regions on the ice-air surface can also cause moderate increases in the shear stress supported on the bed beneath. Surveys of large-scale bed roughness would identify sticky spots caused by bedrock bumps, water-pressure measurements in regions of thin or zero till might reveal whether they were sticky spots, and strain grids across the margins of ice-surface highs would show whether the highs were causing sticky spots. Sticky spots probably are not dominant in controlling Ice Stream Β near the Upstream Β camp, West Antarctica.
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Depth of water-filled crevasses that are closely spaced - Volume 13 Issue 69 - J. Weertman
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The aim of this paper is to describe in detail how the benchmark tests ISMIP-HOM (Ice Sheet Model Intercomparison Project – Higher-Order ice-sheet Model) has been performed using the open-Source finite element code Elmer (http://www.csc.fi/elmer). The ISMIP-HOM setup consists of five diagnostic and one prognostic experiments, for both 2-D and 3-D geometries. For all the tests, the full-Stokes equations are solved. Some FE technical points, such as mesh characteristics, stabilisation methods, numerical methods used to solve the linear system and parallel performance are discussed. For all these setups, the CPU time consumption is analysed in comparison to the accuracy of the solution. Some general rules are then inferred that optimise the computing time versus the accuracy of the results.
Conference Paper
Ice exhibits either ductile or brittle behavior, depending upon the conditions under which it is loaded. Glaciers, for instance, are loaded by gravity, under deviatoric stresses of ~0.1 MPa or lower. At temperatures of interest, they flow through dislocation creep at strain rates of the order of 10−9 s−1 or lower (Patterson 1994). Sheets of sea ice—another terrestrially important ice feature—are loaded predominantly by wind, under global compressive stresses similar in magnitude to the shear stresses within glaciers (Richter-Menge and Elder 1998). These bodies deform through a combination of creep and fracture, the latter process manifested by oriented leads or open cracks and by compressive shear faults which often form as conjugate sets that traverse a large fraction of the Arctic Basin (Kwok 1999; Schulson 2002). The cold, icy crust of Europa, an extraterrestrial feature which may shield an ocean beneath within which a form of life may exist or may once have existed (Reynolds et al. 1987; Hoppa et al. 1999; Pappalardo et al. 1999; Greenberg et al. 2000; Kargel et al. 2000), is loaded by the motion of diurnal tides and by non-synchronous rotation (Greenberg and Weidenschilling 1984). The crust deforms in a brittle manner, as evident from the networks of cracks that lace through it (Greeley et al. 2000). Of the two kinds of ice deformation, creep is probably the better known by readers of the geological and geophysical literature, and is certainly the more fully explored and better understood. The interested reader may wish to consult a number of excellent reviews of the subject (e.g., see Duval et al. 1983; Weertman 1983; Durham and Stern 2001). Fracture, in comparison, has only recently been systematically examined. The motivation with respect to terrestrial mechanics (Schulson 2001) comes largely …
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In 1928 I came from Palestine to Easton, Pa., to assist Eugene Cook Bingham at the birth of Rheology. I felt strangely at home. There was Bethlehem quite near, there was a river Jordan and a village called little Egypt. The situation was, however, also slightly confusing. To go from Bethlehem to Egypt, one had to cross the river Jordan, a topological feature which did not conform to the original. Then there were, here, places such as Allen town to which there was no analogy. And this could lead to strange situations, such as when a girl at school was asked where Christ was born and replied, “In Allentown”. When corrected by “No, in Bethlehem,” she remarked, “I knew it was somewhere around here.” The following lines are from an after‐dinner talk presented at the Fourth International Congress on Rheology, which took place last August in Providence, R. I. Marcus Reiner, research professor at the Israel Institute of Technology, is currently in the United States as a visiting professor at the Polytechnic Institute of Brooklyn.
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The fundamentals of the smoothed particle hydrodynamics (SPH) method and its applications in astrophysics are reviewed. The discussion covers equations of motion, viscosity amd thermal conduction, spatially varying resolution, kernels, magnetic fields, special relativity, and implementation. Applications of the SPH method are discussed with reference to gas dynamics, binary stars and stellar collisions, formation of the moon and impact problems, fragmentation and cloud collisions, and cosmological and galactic problems. Other applications discussed include disks and rings, radio jets, motion near black holes, supernovae, magnetic phenomena, and nearly incompressible flow.