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Are seasonal calving dynamics forced by buttressing from ice mélange or undercutting by melting? Outcomes from full-Stokes simulations of Store Glacier, West Greenland

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We use a full-Stokes 2-D model (Elmer/Ice) to investigate the flow and calving dynamics of Store Glacier, a fast-flowing outlet glacier in West Greenland. Based on a new, subgrid-scale implementation of the crevasse depth calving criterion, we perform two sets of simulations: one to identify the primary forcing mechanisms and another to constrain future stability. We find that the mixture of icebergs and sea ice, known as ice mélange or sikussak, is principally responsible for the observed seasonal advance of the ice front. On the other hand, the effect of submarine melting on the calving rate of Store Glacier appears to be limited. Sensitivity analysis demonstrates that the glacier's calving dynamics are sensitive to seasonal perturbation, but are stable on interannual timescales due to the strong topographic control on the flow regime. Our results shed light on the dynamics of calving glaciers and may help explain why neighbouring glaciers do not necessarily respond synchronously to changes in atmospheric and oceanic forcing.
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The Cryosphere, 8, 2353–2365, 2014
www.the-cryosphere.net/8/2353/2014/
doi:10.5194/tc-8-2353-2014
© Author(s) 2014. CC Attribution 3.0 License.
Are seasonal calving dynamics forced by buttressing from ice
mélange or undercutting by melting? Outcomes from full-Stokes
simulations of Store Glacier, West Greenland
J. Todd and P. Christoffersen
Scott Polar Research Institute, University of Cambridge, Cambridge, UK
Correspondence to: J. Todd (jat71@cam.ac.uk)
Received: 5 June 2014 – Published in The Cryosphere Discuss.: 2 July 2014
Revised: 30 September 2014 – Accepted: 13 November 2014 – Published: 16 December 2014
Abstract. We use a full-Stokes 2-D model (Elmer/Ice) to
investigate the flow and calving dynamics of Store Glacier,
a fast-flowing outlet glacier in West Greenland. Based on
a new, subgrid-scale implementation of the crevasse depth
calving criterion, we perform two sets of simulations: one to
identify the primary forcing mechanisms and another to con-
strain future stability. We find that the mixture of icebergs
and sea ice, known as ice mélange or sikussak, is princi-
pally responsible for the observed seasonal advance of the
ice front. On the other hand, the effect of submarine melt-
ing on the calving rate of Store Glacier appears to be limited.
Sensitivity analysis demonstrates that the glacier’s calving
dynamics are sensitive to seasonal perturbation, but are sta-
ble on interannual timescales due to the strong topographic
control on the flow regime. Our results shed light on the dy-
namics of calving glaciers and may help explain why neigh-
bouring glaciers do not necessarily respond synchronously to
changes in atmospheric and oceanic forcing.
1 Introduction
Recent studies show accelerating net mass loss from the
Greenland Ice Sheet (GrIS) (Rignot and Kanagaratnam,
2006; Howat et al., 2007; Khan et al., 2010), raising con-
cerns about its future response to changing global climate
and the impact this might have on global sea level. The two
factors which govern this loss are (1) an overall negative sur-
face mass balance stemming from intensified surface melt-
ing in the ice sheet’s ablation zone (Hanna, 2005; van den
Broeke et al., 2009; Enderlin et al., 2014) and (2) faster rates
of ice discharge through calving glaciers which terminate in
fjords (Luckman and Murray, 2005; Howat et al., 2005; Rig-
not and Kanagaratnam, 2006; Howat et al., 2007). The lat-
ter (dynamic) mechanism accounted for 67% of the total
net ice loss in 2005 (Rignot and Kanagaratnam, 2006), but
less in recent years (Enderlin et al., 2014), highlighting the
sensitivity of Greenland’s marine-terminating glaciers to the
transient pulse of warm Atlantic water flowing into many of
Greenland’s fjords over the last decade (Holland et al., 2008;
Straneo et al., 2010; Christoffersen et al., 2011).
Owing to the advancement of surface mass balance mod-
els over the last two decades (Hanna, 2005; Box et al., 2006;
van den Broeke et al., 2009; Enderlin et al., 2014), surface
mass balance is well represented in global sea level predic-
tions (IPCC, 2013). The rapid dynamics associated with sud-
den increases in the discharge of ice into fjords by marine-
terminating glaciers are, on the other hand, complex and
poorly understood, and their relationship with climate re-
mains elusive and is so far unconstrained (IPCC, 2013). The
main processes involved in rapid dynamics are fast glacier
flow and calving, i.e. the mechanism whereby pieces of ice
and bergs break off glaciers terminating in water. These pro-
cesses are complex because they interact with and respond to
atmospheric as well as oceanic forcing effects. As such, calv-
ing and its associated dynamics comprise one of the most sig-
nificant uncertainties in predictions of future ice sheet mass
balance and sea level change.
While atmospheric processes were previously thought to
be the main driver of rapid ice sheet dynamics (Zwally et al.,
2002), recent studies point to warm water in coastal cur-
rents as the main forcing of mass loss by discharge (Holland
et al., 2008). The rapid acceleration of Jakobshavn Isbræ,
Published by Copernicus Publications on behalf of the European Geosciences Union.
2354 J. Todd and P. Christoffersen: Calving dynamics of Store Glacier
from 4000ma1in 1995 to 17000ma1in 2012, is
clearly linked to the continuing retreat of the calving ice front
over this period (Joughin et al., 2012, 2014), and it has been
hypothesised that submarine melting plays a crucial role in
driving this retreat (Holland et al., 2008; Motyka et al., 2010).
Spaceborne tracking of calving fronts also shows that recent
glacier retreat along the East Greenland coastline has been
widespread and synchronous below 69N, but largely absent
at higher latitudes, where coastal water is much colder (Seale
et al., 2011). This suggests that these glaciers are retreating
in response to changes in the ocean system. Warmer fjord
water increases the rate of submarine melting of the calving
terminus. This effect is further amplified by atmospheric pro-
cesses; buoyant proglacial plumes, driven by the delivery of
surface meltwater to the terminus by the subglacial hydrolog-
ical system, are capable of significantly increasing melt rates
(Jenkins, 2011). Undercutting of calving ice fronts by sub-
marine melting should, in addition, amplify calving rate due
to the stress response (O’Leary and Christoffersen, 2013).
The formation of ice mélange, a rigid mixture of icebergs
and bergy bits, held together by sea ice, henceforth referred
to simply as mélange, may also play an important role with
regard to rapid ice sheet dynamics (Sohn et al., 1998; Joughin
et al., 2008). Data from Jakobshavn Isbræ indicate a com-
plete cessation of calving when the glacier is buttressed by
mélange, a response that may explain why the glacier ad-
vances by up to 5km in winter (Amundson et al., 2008)
and why the glacier retreats suddenly when the mélange dis-
integrates (Joughin et al., 2008). A similar correspondence
between mélange clearing date and increasing calving rate
has been found for a number of glaciers, including those
near Uummannaq in West Greenland (Howat et al., 2010).
Walter et al. (2012) used changes in velocity observations
and a force balance technique to infer a buttressing stress
of 30–60kPa exerted by mélange onto the terminus of Store
Glacier. This buttressing effect and the effect of submarine
melting (Xu et al., 2013) appear to be crucial for the calv-
ing dynamics of this glacier. However, temporal correlation
is insufficient evidence to confidently attribute seasonal calv-
ing retreat to either the collapse of ice mélange or submarine
melting. This highlights the need for numerical modelling to
attempt to partition these effects.
In this paper we present results from a numerical model
developed using the open-source finite-element (FEM) mod-
elling package, Elmer/Ice, with newly implemented calving
dynamics. Theoretical consideration of the calving process
indicates the importance of the near-terminus stress field in
controlling the propagation of crevasses and the detachment
of icebergs (Nye, 1957; van der Veen, 1998a, b; Benn et al.,
2007a, b). Linking calving to crevasse propagation and stress
in this way provides a useful and physically based framework
for investigating calving in numerical models of glacier dy-
namics. Here, we implement a calving model based on the
penetration of both surface and basal crevasses (Nick et al.,
2009, 2010), and incorporate the full stress solution into the
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |
Fig. 1. Store Gletscher in Ikerasak Fjord, Greenland. Colour scale shows summer velocity (m a-1) from
the MEaSUREs program (Joughin et al., 2011). Yellow line indicates the flowline used in this study,
and the green star indicates the location of the main proglacial plume forming where subglacial water is
discharged into the fjord.
28
Figure 1. Store Glacier in Ikerasak Fjord, Greenland. Colour
scale shows summer velocity (ma1) from the MEaSUREs project
(Joughin et al., 2011). The yellow line indicates the flow line used
in this study, and the green star indicates the location of the main
proglacial plume forming where subglacial water is discharged into
the fjord.
crevasse depth criterion, after Nye (1957). We use this model
to investigate the seasonal dynamics of Store Glacier, a fast-
flowing outlet glacier near Uummannaq in West Greenland,
which experiences a large seasonal variability in dynamics
and front position (Howat et al., 2010), but has been interan-
nually stable for at least four decades (Weidick et al., 1995;
Howat et al., 2010, p. C41). The stable, seasonal calving dy-
namics of Store, along with the recent discovery of a 28km
long trough behind the terminus, extending 900m below sea
level, make this glacier an ideal target for stability analysis
as well as process study.
To examine the calving process, we focus on the calv-
ing front’s position and seasonal fluctuation. We investigate
the effects of submarine melting, mélange buttressing and
glacier geometry on calving, with the aim of identifying the
role of each mechanism in driving the observed seasonal
variability at the front. We find that mélange is likely to be
the primary driver, and that submarine melting plays a sec-
ondary role. We also find that the topographic setting of Store
Glacier is responsible for its observed stability.
2 Store Glacier
Store Glacier, henceforth referred to as Store, is a fast-
flowing marine-terminating outlet glacier located in Ikerasak
Fjord, near Uummannaq in West Greenland (Fig. 1). The
glacier drains an area of 35000km2and is 5km wide at
the terminus, where surface velocity reaches 6600ma1
(Joughin et al., 2011). The location of the terminus coincides
with a bottleneck in fjord width (Fig. 1), as well as a pro-
nounced basal pinning point (Fig. 2), suggesting that fjord
topography may play an important role in calving dynamics.
The Cryosphere, 8, 2353–2365, 2014 www.the-cryosphere.net/8/2353/2014/
J. Todd and P. Christoffersen: Calving dynamics of Store Glacier 2355
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |
Fig. 2. a) Surface and basal geometry of central flowline used in this study. b) Model mesh of region
outlined by green box in (a). Blue line represents sea level.
29
Figure 2. (a) Surface and basal geometry of central flow line used
in this study. (b) Model mesh of region outlined by green box in (a).
The blue line represents sea level.
In terms of climate, data from the Regional Atmospheric
Climate Model (RACMO) suggest that 2km3of meltwater
forms on the surface of Store between June and August (Et-
tema et al., 2009). Recent modelling work (Xu et al., 2013)
shows that submarine melting at the terminus may occur at
rates of 8md1in summer because a large proportion of
runoff is discharged subglacially into Ikerasak Fjord. The lat-
ter is established from observations, which show upwelling
of dirty, subglacially derived meltwater near the centre of
the calving ice front during summer months (Chauché et al.,
2014) (Fig. 1). The high melt rates are caused by entrainment
of warm ambient fjord water into buoyant meltwater plumes,
which rise rapidly in front of the glacier, from a depth of
490m below sea level due to forced convection (Jenkins,
2011; Chauché et al., 2014). The glacier is buttressed by a
rigid proglacial mélange, which is typically present from late
January or early February to the end of May (Howat et al.,
2010). When present, this rigid ice mélange has been shown
to exert a significant backstress on the calving terminus of
Store (Walter et al., 2012).
Store exhibits characteristic seasonal variabilities in terms
of calving front position and velocity (Howat et al., 2010).
Estimates of the terminus velocity of Store differ depending
on where and when data were obtained. The most recently
collected TerraSAR-X data, obtained from NASA’s MEa-
SUREs project (Joughin et al., 2011), measure a peak veloc-
ity of 6600 m a1at the calving front with a seasonal variabil-
ity of 700ma1. Howat et al. (2010) measured velocities
a few kilometres behind the terminus and found values rang-
ing from 2500 to 4200 ma1between 2000 and 2010. Howat
et al. (2010) also tracked changes in front position through
time, finding a seasonal variability of at least 500m, when
averaged across the width of the terminus. This is consistent
with time-lapse photography showing seasonal advance of
1km near the central flow line (J. Box, personal commu-
nication, 2014).
3 Methods
In this work, we use Elmer/Ice in a 2-D configuration to
model the central flow line of Store. The modelled flow line
is 113km long and covers the region from the 100ma1
ice velocity contour to the calving front (Fig. 2a). The flow
line was produced using velocity vector data from the MEa-
SUREs project (Joughin et al., 2011).
We use a 2-D modelling framework in which both calving
front and grounding line evolve freely through time. Whereas
the representation of processes in 2-D requires parameterisa-
tion of key out-of-plane effects, as explained below, it is a
practical first step which will guide and help the future im-
plementation of calving processes in 3-D.
3.1 Elmer/Ice dynamics
Elmer/Ice is a finite-element model which solves the Stokes
equations and uses Glen’s flow law as a constitutive stress–
strain relation (see Gagliardini et al., 2013, for details). The
finite-element approach is a flexible solution which allows us
to vary the spatial resolution of the model and thereby focus
on the dynamics at the calving ice front (Fig. 2b). Because we
are principally interested in capturing processes at the calv-
ing terminus, we adopt a spatial resolution which varies from
250m in the upper region of the glacier to 20m near the ter-
minus (Fig. 2b). The model evolves through time with a time
step of 1 day.
Temperature is an important factor in the stress–strain
relationship of ice (Cuffey and Paterson, 2010). However,
near the terminus, which is our region of interest, extensive
crevassing makes the implementation of temperature diffi-
cult. The ability of subglacial meltwater to penetrate upwards
through basal crevasses, as well as the effect of air circulation
in surface crevasses, is likely to significantly affect the tem-
perature profile of the ice. Due to these complications, and
the lack of observations to constrain ice temperature, we as-
sume for the sake of simplicity that the glacier is isothermal
at 10C.
Because basal friction exerts a critical control on the dy-
namics of fast-flowing glaciers in general, we first use the ad-
joint inverse method (Gillet-Chaulet et al., 2012) to identify
the basal friction profile which results in surface velocity as
observed along the flow line. The result of the inverse method
is a profile for the basal friction parameter (β2) which is re-
lated to basal velocity (Ub) and basal shear stress (τb) by the
relation (MacAyeal, 1992)
τb=β2Ub.(1)
To integrate seasonal variation in ice flow in response to
seasonal change in basal friction, we run the inverse model
for both the summer and winter observed velocity profiles,
thereby obtaining two basal friction profiles. A seasonal vari-
ability in ice flow, very similar to what is observed in real-
www.the-cryosphere.net/8/2353/2014/ The Cryosphere, 8, 2353–2365, 2014
2356 J. Todd and P. Christoffersen: Calving dynamics of Store Glacier
ity, is imposed by varying the basal traction coefficient sinu-
soidally between summer and winter values.
3.2 Boundary conditions
Initial surface elevation along the modelled flow line is pre-
scribed from the GIMP DEM product (Howat et al., 2014).
The bed profile is obtained from airborne geophysical sur-
veys carried out by the Greenland Outlet Glacier Geophysics
(GrOGG) project and NASA’s Operation IceBridge (https:
//espo.nasa.gov/missions/oib/). We use a mass-conservation
algorithm similar to that of McNabb et al. (2012) to constrain
ice thickness and bed topography in the heavily crevassed
region of fast flow near the terminus, where radar data are
sparse.
Ice thickness evolves through time according to the mass
continuity equation (Cuffey and Paterson, 2010), and we add
and subtract mass according to RACMO surface mass bal-
ance data averaged between 1985 and 2008. The ice sur-
face is treated as a stress-free boundary, as we assume at-
mospheric pressure to be negligible. At the ice base, friction
is prescribed through inverse methods as described above,
except under the floating tongue, which, when it exists, is a
frictionless free surface. At the calving terminus, we apply an
external pressure equal to the hydrostatic pressure from sea-
water (see Eq. 5 below). Above sea level, atmospheric pres-
sure is neglected.
We simulate the seasonal advance and retreat of Store’s
floating tongue using an implementation of grounding line
dynamics developed by Favier et al. (2012). The grounding
line algorithm compares external water pressure and ice over-
burden pressure to detect where the glacier is floating, and
modifies basal friction accordingly.
3.3 New scheme for implementation of flow
convergence
Similar to most outlet glaciers, Store undergoes significant
lateral narrowing as ice flows from catchment to coast. As
such, it is important that dynamic effects from sidewall drag
(Raymond, 1996) and ice convergence (Thomas et al., 2003)
are accounted for.
Gagliardini et al. (2010) implemented a parameterisation
for sidewall friction in Elmer/Ice, and we use it here. The
issue of ice convergence in full-Stokes 2-D models, how-
ever, has thus far received little attention from the glacier
modelling community. Here, we have developed a routine
which adds flux sources to elements along the flow line, cor-
responding to the downstream narrowing of the glacier. We
derive a flux convergence term (see Supplement) and add it
to the Stokes incompressibility equation (Eq. S1 in the Sup-
plement), such that
∇ · u= dW
dxW1uxA, (2)
where uis the velocity vector, Wis glacier width, uxis the
along-flow component of velocity and Ais the area of the
element.
This convergence term represents an important 3-D ef-
fect, ensures that mass balance is maintained throughout the
model domain, and allows for realistic evolution of mass and
momentum near the terminus. We note that this prescribed
flux convergence differs from implementation of flow con-
vergence in earlier work with flow line models (e.g. Glad-
stone et al., 2012; Cook et al., 2014), where the additional
mass is added as an input to the surface mass balance. Al-
though the latter will result in correct flux, it neglects the di-
rect effect of the additional flux on the velocity field and may
consequently underestimate velocity change while overesti-
mating elevation change.
3.4 Numerics for implementing calving
We implement the crevasse-penetration calving criterion
(Benn et al., 2007a, b; Nick et al., 2010), based on the work
of Nye (1957) and van der Veen (1998a, b). This model is
based on the assumption that calving occurs when surface
and basal crevasses meet. Surface and basal crevasse depths
are calculated from the balance of forces:
σn=2τesgnxx )ρigd +Pw,(3)
where the result, σn, is the “net stress”, which is positive
in a crevasse field and negative in unfractured ice (van der
Veen, 1998a). The first term on the right-hand side of Eq. (3)
represents the opening force of longitudinal stretching, and
is adapted from Otero et al. (2010); τerepresents effective
stress, which is related to the second invariant of the devi-
atoric stress tensor and which, in 2-D, is defined by Cuffey
and Paterson (2010):
τ2
e=τ2
xx +τ2
zx ,(4)
where xis the direction of ice flow and zis the vertical. We
multiply τein Eq. (3) by the sign function of longitudinal
deviatoric stress (τxx ) to ensure crevasse opening is only pre-
dicted under longitudinal extension (τxx >0).
The second term on the right-hand side of Eq. (3) repre-
sents ice overburden pressure, which leads to creep closure,
where ρiis the density of glacier ice, gis the force of gravity
and dis depth through the ice.
The final term in Eq. (3) is water pressure (Pw), which
acts to open crevasses when present. In basal crevasses, Pw
is controlled by the subglacial hydrological system, and in
surface crevasses it is related to the depth of water in the
crevasse.
Crevasses will exist wherever σnis positive, and ice re-
mains intact elsewhere. Evaluating Eq. (3) for both surface
and basal crevasses at every node in our model allows us
to define “zero contours” which represent the base and top
of surface and basal crevasse fields, respectively. The modi-
fied crevasse-penetration calving criterion (Nick et al., 2010)
The Cryosphere, 8, 2353–2365, 2014 www.the-cryosphere.net/8/2353/2014/
J. Todd and P. Christoffersen: Calving dynamics of Store Glacier 2357
predicts that calving will occur where and when these zero
contours meet. By calculating the crevasse depth criterion as
an index at every node, and interpolating the nodal values
to find the zero contours (Fig. 3), we arrive at a calving im-
plementation which accounts for changes in stress between
surface and interior and which is reasonably insensitive to
the model’s mesh resolution.
The magnitudes of the force components of Eq. (3) vary
greatly between the surface and bed. Specifically, the cryo-
static pressure will be much higher at the bed. However,
when the terminus is near flotation, high basal water pressure
will almost completely counteract this closing force. High
basal water pressure is thus an essential condition for signif-
icant basal crevasse penetration (van der Veen, 1998a). Be-
cause our study focuses specifically on calving dynamics, we
make the simplifying assumption that an efficient subglacial
drainage system exists near the terminus and thus that there
is negligible difference in basal water pressure for any given
depth within the region where calving may occur. With this
assumption, basal water pressure is simply a function of sea
level and bed elevation (van der Veen, 1998a):
Pw= −ρwgz, (5)
where zis the zcoordinate, which is negative below sea level.
Water pressure is essential for basal crevasse penetration,
but it may also be significant in surface crevasses (Benn et al.,
2007b). The process of “hydrofracturing” by water in surface
crevasses is believed to have been a critical factor in the col-
lapse of the Larsen B Ice Shelf (Scambos et al., 2003). How-
ever, while water in surface crevasses may be important, it
is extremely difficult to quantify. The relationship between
surface melt rate and crevasse water depth depends on the
distribution, shape and depth of crevasses, and melting and
refreezing on crevasse walls, as well as potential drainage of
water from crevasses into englacial, subglacial or proglacial
water bodies. As such, it is currently impossible to estimate
even an order of magnitude for crevasse water depth at Store
in summer. However, outside the 3-month summer melt sea-
son, surface crevasses must be assumed to be dry.
Modelling calving in a 2-D continuum model involves
some implicit assumptions which may affect the accuracy of
the calving criterion presented above. Firstly, the implemen-
tation of valley sidewall friction assumes that the calving ter-
minus runs straight from one side of the valley to the other.
However, Store’s terminus is usually arcuate in shape, with
the centreline being further advanced in the fjord than the
sidewalls. Thus, our implementation will overestimate lat-
eral drag at the terminus. Secondly, by assuming a constant
temperature of 10C throughout the glacier, we neglect
temperature-dependent variations in viscosity and thus the
stress field. Finally, Eq. (3) slightly overestimates ice over-
burden pressure by assuming constant bulk density within
the glacier. In fact, the presence of a crevasse field may sig-
nificantly reduce bulk density; this represents a positive feed-
Figure 3. The terminus of the flow line mesh of Store Glacier. White
line indicates the net stress (σn) zero contour for both surface and
basal crevasses. Net stress (MPa) is >0 where crevasses exists and
<0 in solid and unfractured ice. Calving occurs in the model when
the surface and basal zero contours meet. The blue line indicates sea
level.
back whereby the growth of a crevasse field reduces ice over-
burden pressure, leading to further crevasse deepening.
For the reasons outlined above, we expect our model to
slightly underestimate the penetration of surface and basal
crevasses near the present terminus position. As such, we ap-
ply a constant scaling factor of 1.075 to the effective stress
term in Eq. (3). This scaling procedure is equivalent to the
assumed presence of water in crevasses throughout the year
in earlier work (Nick et al., 2010; Vieli and Nick, 2011). We
note, in this context, that for a typical value of effective stress
(τe=300 kPa), our 7.5 % scaling factor equates to just 2.3m
water depth added to crevasses. As there are several factors,
aside from water depth, which may explain why the calving
criterion does not predict full crevasse penetration exactly at
the observed terminus location, we consider the scaling fac-
tor to simply be a tunable parameter, encompassing the above
processes, and which we keep constant. A more robust treat-
ment of the issues outlined above will most likely require a
3-D model for calving.
3.5 Model forcing
We investigate the calving dynamics of Store in three stages.
First, we set up a baseline run in which flow is affected only
by a seasonal variation in basal traction. We then explore the
glacier’s response to (1) undercutting of ice front by subma-
rine melting in summer and (2) buttressing of the ice front
by rigid mélange in winter. The aim of these numerical ex-
periments (henceforth referred to as experiment 1) is to iden-
tify which forcing has the greatest influence on the glacier’s
flow, and the outcome represents a “present-day” simulation
in which the glacier’s frontal position varies seasonally as ob-
served under current climatic conditions. Finally, we perform
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2358 J. Todd and P. Christoffersen: Calving dynamics of Store Glacier
perturbation experiments by altering mélange and submarine
melt forcing in terms of their magnitude and duration. This
set of experiments (experiment 2) investigates the response
of Store to changes at its calving ice front in a warming cli-
mate.
3.5.1 Submarine melting
Time-lapse photography shows a meltwater plume at the
central section of the terminus of Store in summer months
(Chauché et al., 2014). Because the location of this plume
coincides with the terminus position in our model, we apply
summer melt rates at the calving front which vary linearly
from 8md1at the base to 0md1at sea level. This melt
distribution is a simplification of the one found by Xu et al.
(2013), who used MITgcm to investigate plume-induced ice
front melting at Store, based on previous estimates of fjord
water temperature (Rignot et al., 2010) and subglacial melt-
water discharge (van Angelen et al., 2012). Their results sug-
gest an average melt rate across the entire face of 3.6m d1
in summer, with a local maximum at the base of the plume
of 8md1. Because subglacial discharge is strongly influ-
enced by surface runoff in summer months, we assume, for
the sake of simplicity, that no submarine melting occurs in
winter. If and when the floating tongue exists during the melt
season, we apply a bottom melt rate of 1/10th of that applied
on the vertical face, based on the “geometrical scale factor”
proposed by Jenkins (2011).
In experiment 1, ice front melting is assumed to occur at a
constant rate from the start of June until the end of August,
as >90 % of all surface runoff in the Store catchment occurs
over this period. In experiment 2, we investigate the effects
of increasing summer melt rates by a factor of 1.5 and 2, and
increasing its duration by 33 and 66%.
3.5.2 Mélange backstress
We simulate the effect of mélange backstress by applying an
external pressure on the calving terminus in addition to that
exerted by the sea (Fig. 4). The applied pressure is similar to
that found by Walter et al. (2012) from a force-balance study
of Store, based on the observed speedup of the glacier fol-
lowing mélange collapse. Their results show that the mélange
yields a supporting pressure equivalent to a backstress of 30–
60 kPa acting on the entire face of the terminus. In reality, this
stress is applied only through the thickness of the mélange,
a property not measured by Walter et al. (2012). To obtain a
realistic forcing scenario at the calving front of our model,
we convert Walter et al.’s backstress (σfb) into an equivalent
mélange–glacier contact pressure:
σsik =σfb Hterm
Hsik ,(6)
where Hterm and Hsik are the thicknesses of the glacier ter-
minus and the mélange, respectively.
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |
Fig. 4. Schematic diagram showing proximal and distal processes affecting calving. a) Varying basal
friction (τb) affects the stress field in the glacier. b) Changing fjord water temperatures and subglacial
water flux affect the rate of submarine melting of the calving face and floating tongue (when present).
c) The seasonal formation of m´
elange provides a buttressing force which suppresses surface crevasse
depth and, thus, calving. d) Surface melt water in crevasses causes hydrofracturing, which acts to deepen
surface crevasses. e) Glacier geometry exerts a strong influence on crevasse field depth: compressional
forces on the stoss side of Store’s pinning point suppress the depth of crevasses, while rapid loss of basal
traction in the lee side deepen them.
31
Figure 4. Schematic diagram showing proximal and distal pro-
cesses affecting calving. (a) Varying basal friction (τb) affects the
stress field in the glacier. (b) Changing fjord water temperatures
and subglacial water flux affect the rate of submarine melting of
the calving face and floating tongue (when present). (c) The sea-
sonal formation of mélange provides a buttressing force which sup-
presses surface crevasse depth and thus calving. (d) Surface melt
water in crevasses causes hydrofracturing, which acts to deepen
surface crevasses. (e) Glacier geometry exerts a strong influence
on crevasse field depth: compressional forces on the stoss side of
Store’s pinning point suppress the depth of crevasses, while rapid
loss of basal traction on the lee side deepen them.
In experiment 1, we take the midpoint of the range esti-
mated by Walter et al. (2012) (45 kPa), acting over a mélange
thickness of 75m, as estimated from laser altimeter data
collected by NASA’s Operation IceBridge (https://espo.nasa.
gov/missions/oib/). Based on the work of Howat et al. (2010),
we assume mélange to be present and rigid from the start of
February until the end of May and absent from June to Jan-
uary. In experiment 2, we investigate the effect of reducing
mélange strength by 25 and 50% and its duration by 33 and
66%.
4 Results
4.1 Baseline run
The baseline configuration of our model includes only one
seasonal effect: the prescribed sinusoidal variation in the
basal friction parameter between winter and summer values.
The result is a slight increase in flow speed at the terminus,
from a minimum of 4700ma1in winter to a maximum of
4900ma1in summer (Fig. 5b). When the calving criterion
is implemented, calving activity is periodic and characterised
by 80–90m bergs breaking off with a frequency of one per
8.7 days (Fig. 5a). Terminus velocity increases when calving
occurs and is reduced afterwards as the front advances. The
amplitude of these velocity fluctuations is about 200m a1
(Fig. 5b), a similar magnitude to the seasonal effect of vary-
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J. Todd and P. Christoffersen: Calving dynamics of Store Glacier 2359
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |
Fig. 5. Plots showing variations in terminus position (a) and velocity (b), over the course of a year for
baseline model run (blue line) and run with submarine melting applied (red line). Red shading indicates
melt season. The saw-toothed pattern in both panels is a result of calving.
32
Figure 5. Plots showing variations in terminus position (a) and ve-
locity (b) over the course of a year for the baseline model run (blue
line) and a run with submarine melting applied (red line). Red shad-
ing indicates melt season. The saw-toothed pattern in both panels is
a result of calving.
ing basal friction, indicating that the position of the calving
front has a strong influence on terminus velocity. However,
the terminus position varies less than 100m through the en-
tire simulation and there is no discernible seasonality of the
glacier’s frontal position. This shows that the observed sea-
sonal advance and retreat of the calving front cannot be at-
tributed to seasonal variation in basal friction.
4.2 Experiment 1
To attain a realistic “present-day” simulation, we start by
adding submarine melting, as described above, with rates
up to 8md1from June to August. This forcing slightly in-
creases the frequency and reduces the magnitude of calving
events, though the overall terminus position varies only neg-
ligibly (Fig. 5a). Terminus velocity during the melt season
is slightly suppressed compared with the melt-free simula-
tion (Fig. 5b). This experiment suggests that neither seasonal
variability in basal dynamics nor submarine melting explains
the seasonal calving dynamics observed at Store. Only when
the stabilising effect of mélange buttressing is included does
our model respond with significant frontal advance and re-
treat. Figure 6 shows the evolution of calving terminus po-
sition through time for each of the two seasonal forcings as
well as the combined effect.
In our model, the formation of the mélange triggers the im-
mediate formation of a floating ice tongue which advances
into the fjord. The terminus advances by 1300m between
February and May, while the mélange is present, and begins
to retreat rapidly when the mélange disappears, irrespective
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |
Fig. 6. Plots showing changes in calving terminus position (a) and velocity (b) during a three year period
within a 40 year long stable simulation, with coloured solid lines illustrating the effect of four different
combinations of melting and ice m´
elange perturbation. Blue and red shade indicates m´
elange and melt
season respectively.
33
Figure 6. Plots showing changes in calving terminus position (a)
and velocity (b) during a 3-year period within a 40-year-long sta-
ble simulation, with coloured solid lines illustrating the effect of
four different combinations of melting and ice mélange perturba-
tion. Blue and red shading indicates mélange and melt season, re-
spectively.
of whether or not submarine melting is applied (Fig. 6). Fig-
ure 7 shows the evolution of the floating tongue through the
mélange season. As the floating tongue advances, both the
surface and basal crevasse fields are suppressed near the ter-
minus. Note that the surface elevation rises as the floating
tongue extends into the fjord, indicating that the dynamic
regime near the grounding line is forcing the terminus be-
low flotation level. This is only overcome once the floating
tongue is long enough to exert sufficient upward bending mo-
ment on the grounding line. Once significant upward bend-
ing is exerted, this is manifested as a suppression of surface
crevasse field, clearly visible in Fig. 7.
When the mélange effect is combined with submarine
melting, the collapse of the floating tongue is followed by
a further 250m retreat beyond the stable terminus position
at 113km. After this retreat, the terminus slowly readvances
through the melt season to 113 km, where it remains, calving
periodically, until the mélange forms during the following
winter.
Our simulations in this experiment demonstrate a strong
correlation between terminus position and velocity. Seasonal
dynamics imposed by changing basal friction (Fig. 5) are
dwarfed by the deceleration which occurs when the float-
ing ice tongue develops and advances (Fig. 6). The dynamic
effect of this slowdown is transmitted up to 30 km inland
(Fig. 8a). During the mélange season, surface velocity is
reduced and thickness increases slightly (Fig. 8b) between
90km and the terminus. Following mélange collapse, veloc-
ity immediately rebounds to values similar to those prior to
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2360 J. Todd and P. Christoffersen: Calving dynamics of Store Glacier
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |
Fig. 7. Sequential profiles of Store Gletscher during advance of its calving terminus due to m´
elange
backstress. As the floating tongue advances from the grounding line (marked GL), it rises upwards due
to buoyant forces, which also act to close surface crevasses near the grounding line. This indicates that
flow dynamics at the grounding line are forcing the terminus below flotation.
34
Figure 7. Sequential profiles of Store Glacier during advance of its
calving terminus due to mélange backstress. As the floating tongue
advances from the grounding line (marked GL), it rises upwards due
to buoyant forces, which also act to close surface crevasses near the
grounding line. This indicates that flow dynamics at the grounding
line are forcing the terminus below flotation.
the mélange formation, and this speedup is followed by a
gradual deceleration through the rest of the year. Interest-
ingly, surface velocity at 85 km is consistently faster through-
out the seasonal cycle than its 1 January value, peaking at
7.5% faster halfway through the year. Figure 8b also indi-
cates slight thickening upstream and thinning downstream of
this location, which coincides with a significant basal pin-
ning point and large surface slopes as the glacier flows into a
deep basal trough (Fig. 2).
The outcome of experiment 1 is a seasonally variable calv-
ing model of Store which is in overall good agreement with
observations (Howat et al., 2010; Walter et al., 2012). The
stable position adopted by the modelled terminus (113km)
following the summer melt season matches the observed
summer terminus position. As observed, the modelled ter-
minus retreats rapidly soon after mélange has collapsed in
the fjord. The total seasonal variability in modelled front po-
sition (1.3km) is in good agreement with that observed by
Howat et al. (2010), as well as time-lapse imagery collected
by the Extreme Ice Survey (www.eis.com), which shows that
the frontal position of Store can vary by more than 1km
between summer and winter (J. Box, personal communica-
tion, 2014).
4.3 Experiment 2
In this experiment, we perturb the stable “present-day” sim-
ulation obtained in experiment 1 in order to investigate the
response of Store to climate change. We specifically inves-
tigate the glacier’s response to changes in mélange buttress-
ing and submarine melting because these forcing factors are
poorly understood.
When mélange strength is reduced to 75% of its base-
line value (Fig. 9a–c, green lines), the floating tongue does
not begin to form until halfway through the mélange season.
As a result, the maximum length of the tongue is reduced
from 1.3 to 0.7km. When mélange strength is further re-
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |
Fig. 8. Plots showing velocity (a) and thickness (b) perturbations through a single calendar year. Line
colour indicates time of year. Velocity and thickness have been normalised against their Jan 1st values.
35
Figure 8. Plots showing velocity (a) and thickness (b) perturbations
through a single calendar year. Line colour indicates time of year.
Velocity and thickness have been normalised against their 1 January
values.
duced to 50 % (Fig. 9a–c, red lines), no floating tongue forms
in spring, though there remains a clear change in calving dy-
namics throughout the mélange season. These results suggest
that any future climate-related reduction in the strength of
mélange may significantly affect the calving dynamics and
seasonality of Store.
Reducing the duration of the mélange season to 66%
(Fig. 9b) limits the length of the floating tongue to 0.8km
for the 45kPa case. However, reduction to 33% (Fig. 9c)
has no further effect on calving dynamics, and the floating
tongue continues to advance for a month following mélange
break-up. This is a surprising result, which suggests that the
floating tongue is at least temporarily self-stabilising. In the
75 % mélange strength case, when season duration is reduced
to 66% (Fig. 9b, green line), the floating tongue begins to
advance slightly sooner and thus the final length is slightly
higher. However, no floating tongue forms when season du-
ration is further reduced to 33% (Fig. 9c, green line).
An increase in the duration of submarine melting, by 33
and 66% (Fig. 9e and f, respectively), leads to more rapid
collapse of the floating tongue, though in no case does the
tongue collapse while rigid mélange is still present. As in
experiment 1 (Fig. 6), submarine melting has an appreciable
effect on the calving dynamics of the grounded terminus in
late summer. As such, a longer submarine melt season leads
to a longer period of larger, less frequent calving events and
a retreat in average terminus position. The response of the
modelled terminus to increasing melt magnitude, on the other
hand, appears somewhat stochastic. It should be noted, how-
ever, that the positions shown in Figs. 5, 6 and 9 represent
the terminus at the surface, which is able to advance into the
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J. Todd and P. Christoffersen: Calving dynamics of Store Glacier 2361
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |
36
Figure 9. Plots showing terminus position through 1 year for vary-
ing mélange season duration (ac) and melt season duration (df).
Durations of mélange and melt season are indicated by blue and red
shading, respectively. Line colour indicates varying magnitude of
melt rate and mélange backstress. The blue line in panels (a) and (d)
represent the baseline model from experiment 1 (Fig. 6). Changing
panels and line colours indicate perturbations under progressively
warmer climate scenarios.
fjord when undercutting takes place, due to the fact that the
glacier’s topography exerts a control on the position of the
grounding line. Broadly speaking, the calving dynamics are,
according to this model, relatively unaffected by increasing
melt magnitude. In even the most severe “warming climate”
scenario, with melt rate double that of present-day values and
duration increased from 3 to 5 months, the modelled termi-
nus remains stable.
5 Discussion
The results of our modelling experiments shed new light on
marine-terminating glacier dynamics and the calving mech-
anism. The calving dynamics of the modelled glacier vary
significantly through the year (experiment 1, Fig. 6), from
high-frequency (8.7 days), low-magnitude (80m) calving
events when no seasonal forcing is applied to complete ces-
sation of calving during the mélange season, with rapid re-
treat following mélange collapse, and seemingly stochastic
calving behaviour during the melt season. This behaviour
is in good overall agreement with year-round observation
of Store (N. Chauché, personal communication, 2014). Our
model captures two important aspects of Store’s behaviour.
Seasonally, Store’s terminus position is highly sensitive to
external perturbation. However, on interannual timescales,
Store’s calving dynamics are stable, and the terminus posi-
tion remains fairly constant (Howat et al., 2010).
In our model, the seasonal advance and retreat is specifi-
cally related to a floating tongue, which forms during win-
ter in response to the buttressing effect of rigid mélange
(Figs. 6, 7) and breaks apart once the buttressing effect
of the mélange disappears. This finding provides theoreti-
cal understanding for the observed temporal correlation be-
tween mélange break-up and frontal retreat at Store and other
glaciers in the Uummannaq region (Howat et al., 2010), as
well as Jakobshavn Isbræ (Amundson et al., 2010) farther
south, and glaciers such as Kangerdlugssuaq and Daugaard-
Jensen on the east coast (Seale et al., 2011). Our results from
experiment 2 suggest that the estimate of Walter et al. (2012)
of a mélange strength of 30–60 kPa is most likely correct, and
that any future climate-driven reduction in mélange strength
or thickness could significantly impact the seasonal dynam-
ics of Store (Fig. 9).
When we isolated the effect of submarine melting of the
ice front (experiment 1, Fig. 5), we found a slight increase in
calving frequency, an associated decrease in calving event
size, and a slight dampening of the glacier’s velocity re-
sponse to calving events. However, the overall effect of sub-
marine melting alone was minimal. Only when combined
with mélange forcing was submarine melting capable of sig-
nificantly affecting calving dynamics (Fig. 6). This suggests
that some process during the mélange season preconditions
the glacier for slight instability later in the season. Poten-
tially, the upward bending associated with the formation of
the floating tongue (Fig. 7) changes the glacier geometry near
the grounding line such that it is more susceptible to the ef-
fect of undercutting by submarine melting.
Despite doubling melt rates and increasing melt duration
by 66% in experiment 2 (Fig. 9), the terminus of Store re-
mained stable at 113km, suggesting that there is no direct
link between submarine undercutting and longer-term calv-
ing stability of the grounded terminus at present. This re-
sult contradicts previous work suggesting that undercutting
of the terminus promotes calving (Motyka et al., 2003; Rig-
not et al., 2010) by intensifying extensional stresses near the
terminus (O’Leary and Christoffersen, 2013). We propose,
however, that this apparent contradiction is a feature specific
to Store, due to the strong stabilising influence of topography.
The location of the terminus of Store coincides with a sig-
nificant basal pinning point (Fig. 2), as well as a “bottleneck”
in the fjord width (Fig. 1). The combined effect of these topo-
graphical features is to significantly affect the stress field and
crevasse depth (Fig. 4). The suppression of crevasses pene-
tration depth at the stoss side of the basal pinning point at the
terminus exceeds the deepening of crevasses in response to
undercutting of the ice front by submarine melting. As such,
the latter alone cannot cause the front to retreat in this case.
This suggests that, as long as the melt rate is less than the rate
of ice delivery to the front, the terminus position of Store will
be relatively insensitive to the rate of ice front melting. Thus,
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2362 J. Todd and P. Christoffersen: Calving dynamics of Store Glacier
the rate of iceberg production will be solely controlled by the
velocity at the terminus. The topographic setting of Store ex-
plains why this glacier remained stable during a period when
others in the same region experienced rapid retreats (Howat
et al., 2010) and, more generally, why neighbouring glaciers
are often observed to respond asynchronously to similar cli-
mate forcing (Moon et al., 2012).
Inland of Store’s stable frontal pinning point is a 28km
long overdeepening reaching 950m below sea level (Fig. 2),
which could make Store susceptible to sudden retreat, i.e.
if the terminus becomes ungrounded from its current pinning
point at 113 km. We found that, by forcing the model with un-
physically large values for submarine melt rate (not shown),
we were able to force the terminus back off its pinning point,
which led to rapid retreat through this trough. However, none
of our climate forcing scenarios were able to trigger such a
retreat, which suggests that the current configuration of Store
is stable and will most likely remain so in the near future.
As laid out above, our model is capable of reproducing
the flow and seasonal calving dynamics of Store simply by
perturbing the backstress exerted by mélange and the rate of
submarine melting. Our model excludes the effect of water in
surface crevasses, which may conceivably affect calving due
to hydrofracture if water levels are high (Benn et al., 2007a).
Although recent work included this effect (Nick et al., 2010),
we ignore it because high-resolution images captured in re-
peat surveys of Store with an unmanned aerial vehicle in
July 2013 detected water in only a small number of surface
crevasses near the terminus (Ryan et al., 2014). Although we
cannot exclude the possibility that undetected water is con-
tributing to crevasse penetration, it is not necessary to invoke
this process to explain the observed behaviour of Store. This
exclusion of hydrofracturing is a useful model simplification,
as it is difficult and potentially impossible to accurately es-
timate the depth of water in crevasses. The latter would re-
quire knowledge of surface meltwater production as well as
the number and size of surface crevasses, which is infeasible
with the type of model used here.
Although our model captures the flow and seasonal calv-
ing dynamics of Store in a realistic manner, it is important
to note that the outcome of our study is specifically limited
to this glacier and that multiyear dynamics remain to be fully
investigated. We use inverse methods to determine basal trac-
tion, rather than a hydrological model; this ensures that the
flow field matches observations, allowing us to focus on pro-
cesses at the terminus. However, prescribing basal traction
means we are unable to investigate its interannual evolution
in response to dynamic thinning, rising sea level or hydrolog-
ical processes. The difficulty of implementing realistic hy-
drological routing in a flow line model suggests that only a
3-D model will be fully capable of representing these pro-
cesses.
It is useful, at this point, to compare the development of
time-evolving models for calving with recent developments
in the implementation of grounding line dynamics. The lack
of consistency of grounding line treatment in ice flow models
was raised by Vieli and Payne (2005), and this issue has since
received a great deal of attention from the ice sheet modelling
community. A comprehensive intercomparison study, MIS-
MIP (Pattyn et al., 2012), compared the ability of various 2-D
ice flow models to simulate grounding line dynamics, before
MISMIP3d (Pattyn et al., 2013) did the same for 3-D mod-
els. Similarly, we hope that the 2-D model presented here
will guide the future development of full 3-D time-evolving
models for calving.
Finally, we note that, in terms of accounting for the feed-
back between crevasse formation and bulk density and flow
characteristics, a damage mechanics approach may prove
useful (Pralong and Funk, 2005; Borstad et al., 2012). A
counterpart study to this one by Krug et al. (2014) attempts
to couple a damage model with a calving model for Helheim
Glacier using Elmer/Ice.
6 Conclusions
Here we have presented results from a seasonally transient
but interannually stable calving model of Store Glacier in
West Greenland. The calving numerics in our model differ
from previous implementations of the crevasse depth crite-
rion (Nick et al., 2010; Vieli and Nick, 2011; Cook et al.,
2014) in that the balance of crevasse opening and clos-
ing forces is calculated through the entire thickness, not
just at the boundaries, meaning that changes through depth
are taken into account. In agreement with recent related
work (Nick et al., 2010), we find that the inclusion of basal
crevasses in the calving criterion is important. We propose
the addition of a new divergence term to the Stokes equa-
tions, which is not only practical but most likely essential
for accurate simulation of glaciers in 2-D flow line models.
We also find that the frequently assumed presence of water
in surface crevasses is not necessary for seasonal calving dy-
namics at Store.
We find that basal traction varies very little between winter
and summer; basal lubrication by surface meltwater is there-
fore unlikely to play an important role in the seasonal ad-
vance and retreat of the ice front. This does not imply, how-
ever, that calving and flow dynamics are not strongly cou-
pled. Our results indicate a strong correlation between ter-
minus position and velocity (Figs. 5, 6). The deceleration
which results from advance of the floating tongue is trans-
mitted up to 30 km inland (Fig. 8). This finding supports pre-
vious studies which found that dynamic change at Helheim
Glacier (Nick et al., 2009) and Jakobshavn Isbræ (Joughin
et al., 2012) were triggered at the terminus.
A key outcome from this study is that the buttressing pres-
sure from rigid mélange is principally responsible for ob-
served seasonal advance and retreat. However, sensitivity
analysis revealed that, in a warming climate, reduction in
mélange strength or duration could prevent Store from ad-
The Cryosphere, 8, 2353–2365, 2014 www.the-cryosphere.net/8/2353/2014/
J. Todd and P. Christoffersen: Calving dynamics of Store Glacier 2363
vancing a floating tongue in winter. The model also indicates
that submarine melting has only a limited effect on calv-
ing dynamics and that even large changes to melt rates in
the future are unlikely to destabilise the terminus of Store.
We propose that Store’s highly stable terminus configura-
tion is due to its topographic setting, being located at both
a basal pinning point and a “bottleneck” in fjord width. We
also find, however, that behind this basal pinning point, Store
flows across a very large trough, reaching 950m below sea
level and extending 28km inland from the current ground-
ing line. This suggests that, were the terminus to be forced to
retreat from its current pinning point, further retreat may be
rapid and sudden, of a similar magnitude to that experienced
by Jakobshavn Isbræ, which resulted in a sustained increase
of ice flux and contribution to sea level rise (Joughin et al.,
2012).
The Supplement related to this article is available online
at doi:10.5194/tc-8-2353-2014-supplement.
Acknowledgements. This study was funded by the Natural En-
vironment Research Council through a PhD studentship (grant
no. NE/K500884/1) to J. Todd and research grant (NE/K005871/1)
to P. Christoffersen. We thank Thomas Zwinger, Peter Råback
and Olivier Gagliardini for help with the Elmer/Ice model,
Michiel van den Broeke for providing RACMO climate data,
Alun Hubbard and Jason Box for useful discussions related to
Store Glacier. We are grateful to Chris Borstad and an anonymous
reviewer for useful feedback during the review process.
Edited by: E. Larour
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... Increasingly, these models are also used to examine the stress field within glaciers to better understand factors that control crevasse formation and the onset of calving events ( Here we show that a common method used to implement the ice-ocean boundary condition in Stokes models can result in solutions that are unphysically sensitive to the choice of simulation time step size. This behavior manifests in applications that allow for rapid changes in the model domain -a type of change associated with models that allow for instantaneous calving events or crevasses (Todd and Christoffersen, 2014;Todd and others, 2018;Yu and others, 2017). The time step dependence arises because for glaciers outside of hydrostatic equilibrium, the acceleration is not small as assumed in Stokes flow. ...
... The Nye zero stress criterion provides a simple method to introduce a physically motivated parameterization of calving into glacier models (Nye, 1957;Nick and others, 2010;Todd and Christoffersen, 2014;Ma and others, 2017;Ma and Bassis, 2019). This criterion assumes crevasses penetrate to the depth in the glacier where the largest effective principle stress vanishes. ...
... After every time step, we analyze the stress field using a method similar to that used by (Todd and Christoffersen, 2014). We generate a contour of crevassed ice based on the binary scalar crevasse field stored on the tracers. ...
Thesis
Modeling of the climate system including, but not limited to, atmospheric dynamics, ocean dynamics, and ice dynamics, is one of the crucial scientific problems of the 21st century. This work focuses on modeling of iceberg calving, the process by which high stresses within ice cause fractures and eventual detachment of icebergs from glaciers. Iceberg calving causes approximately half of mass loss of the world's glaciers, but remains poorly understood and implemented in climate models. Existing model implementations of calving exist at a wide range of spatial and temporal scales, ranging from models that seek to resolve crevasse propagation on the scale of seconds or minutes to broad parameterizations of calving implemented in ice sheet scale climate models. In this work, we seek to develop an intermediate model that can run for year to decade timescales and determines calving based on the internal stresses within the ice. We first focus on crevasse advection, the memory of previously formed crevasses within the ice and their impact on calving behavior. Second, we implement the possibility for mixed mode calving, where ice can fail either via high tensile stresses, high shear stresses, or a combination of the two. Lastly, we include submarine melt to see the combined effect of melt and mixed mode calving on glacier stability. This model is developed using the highly flexible Python finite element library FEniCS and LEoPart, a particle tracking library developed for use with FEniCS. Our novel use of particles to track previously crevassed ice provides a computationally efficient method to track glacier parameters that does not diffuse over time. In initial model development, we identified a key numerical consideration related to the buoyant boundary condition on ice that can create unphysical results if not careful managed in a calving model. Once this issue was documented and addressed through a simple addition to the glacier momentum balance, including crevasse advection in the model, which should increase calving rates, reduces overall rate of glacier advance but is incapable of causing retreat in most cases. This indicates that other mass loss mechanisms, such as shear calving and submarine melt, are crucial to mass balance and should be included in future models whenever possible. When mixed-mode calving is included, we find that glacier behavior is highly unstable with regards to the shear strength of ice. Sharp transitions exists at shear strengths where the ice transitions from slow advance or retreat to rapid, catastrophic collapse. Including submarine melt causes further steady mass loss, but does not significantly alter the shear strength threshold necessary for rapid collapse. This shows that if models seek to model potential catastrophic glacier collapse, which is a point of current scientific contention, mixed mode failure including shear localization should be a key focus of modeling efforts.
... Earlier work has shown that climate-related processes including terminus ablation and undercutting (driven by ocean warming) (Holland et al., 2008;Motyka et al., 2011;Slater et al., 2019;Rignot et al., 2012;Wood et al., 2021), mélange rigidity (driven by changes in ocean temperature, sea-ice concentration, and/or runoff) Moon et al., 2015;Sohn et al., 1998;Todd and Christoffersen, 2014), and enhanced hydrofracture (driven by changes in runoff and surface mass balance) (Benn et al., 2007;Nick et al., 2010) may affect terminus position. Hence, we acquired several ice-sheet and oceanographic datasets in order to compare our glacier terminus position changes with climatic factors. ...
... Models indicate that the absence of a rigid mélange may be more important than ocean-driven melting of the terminus in enhancing glacier retreat (Todd and Christoffersen, 2014). Observations show a correlation between terminus change 818 T. E. Black and I. Joughin: Multi-decadal retreat of marine-terminating outlet glaciers Figure 9. Annual duration of sea-ice season (when sea-ice concentration is greater than 15 %) from Hadley-OI (orange) and NOAA (purple). ...
Article
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The retreat and acceleration of marine-terminating outlet glaciers in Greenland over the past 2 decades have been widely attributed to climate change. Here we present a comprehensive annual record of glacier terminus positions in northwest and central-west Greenland and compare it against local and regional climatology to assess the regional sensitivity of glacier termini to different climatic factors. This record is derived from optical and radar satellite imagery and spans 87 marine-terminating outlet glaciers from 1972 through 2021. We find that in this region, most glaciers have retreated over the observation period and widespread regional retreat accelerated from around 1996. The acceleration of glacier retreat coincides with the timing of sharp shifts in ocean surface temperatures, the duration of the sea-ice season, ice-sheet surface mass balance, and meltwater and runoff production. Regression analysis indicates that terminus retreat is most sensitive to increases in runoff and ocean temperatures, while the effect of offshore sea ice is weak. Because runoff and ocean temperatures can influence terminus positions through several mechanisms, our findings suggest that a variety of processes – such as ocean-interface melting, mélange presence and rigidity, and hydrofracture-induced calving – may contribute to, but do not conclusively dominate, the observed regional retreat.
... We suggest that by the 2020/21 winter, the retreat of the terminus ~6 km from fjord edge enabled mélange to form fast to the fjord margins, resulting in great enough backstress on the glacier front to influence calving rate (e.g. Todd and Christoffersen, 2014). This was aided by the glacier thinning sufficiently that the terminus 1 km from the front was at or near floatation ( fig. ...
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Discharge from Greenland’s marine-terminating glaciers contribute to half of all mass loss from the ice sheet, but the factors forcing their retreat are complex and contested. Here, we examine K.I.V Steenstrups Nordre Bræ (‘Steenstrup’), which, between 2018—2021, retreated ~7 km, thinned by ~20%, doubled in ice discharge, and quadrupled in flow speed. This rate of acceleration is unprecedented amongst Greenland’s glaciers, and now places Steenstrup in the top 10% of glaciers by contribution to Greenland’s discharge. In contrast to expected behaviour from a shallow, grounded tidewater glacier, Steenstrup was insensitive to high surface temperatures that destabilised many regional glaciers in 2016, responding instead to an extreme anomaly in deeper Atlantic Water (AW) in 2018. Steenstrup’s behaviour highlights that, as AW intrusions occur at increasingly shallow depths, even apparently long-term stable glaciers with high sills are vulnerable to sudden and rapid retreat.
... Jenkins, 2011;Motyka and others, 2013), can affect calving rates by shaping glacier termini and continuously modifying near-terminus stresses (e.g. O'Leary and Christoffersen, 2013;Cook and others, 2014;Todd and Christoffersen, 2014;Krug and others, 2015;Cowton and others, 2019;Ma and Bassis, 2019). Calving events also shape glacier termini and modify near-terminus stresses. ...
Article
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Tidewater glaciers have been observed to experience instantaneous, stepwise increases in velocity during iceberg-calving events due to a loss of resistive stresses. These changes in stress can potentially impact tidewater glacier stability by promoting additional calving and affecting the viscous delivery of ice to the terminus. Using flow models and perturbation theory, we demonstrate that calving events and subsequent terminus readvance produce quasi-periodic, sawtooth oscillations in stress that originate at the terminus and propagate upstream. The stress perturbations travel at speeds much greater than the glacier velocities and, for laterally resisted glaciers, rapidly decay within a few ice thickness of the terminus. Consequently, because terminus fluctuations due to individual calving events tend to be much higher frequency than climate variations, individual calving events have little direct impact on the viscous delivery of ice to the terminus. This suggests that the primary mechanism by which calving events can trigger instability is by causing fluctuations in stress that weaken the ice and lead to additional calving and sustained terminus retreat. Our results further demonstrate a stronger response to calving events in simulations that include the full stress tensor, highlighting the importance of accounting for higher order stresses when developing calving parameterizations.
... Overall, these changes to the water column temperature can cause non-negligible (up to several tens of percent) changes in terminus submarine melt rates across the large areas of the calving front that are not directly affected by plume-inducing subglacial discharge. The vertical pattern of changes to terminus submarine melt rates (reduced near the surface and increased at intermediate depths) induced by iceberg melting is expected to exacerbate undercutting of glacier termini, with potentially important impacts on calving rates (Benn et al., 2017;Ma and Bassis, 2019;O'Leary and Christoffersen, 2013;Todd and Christoffersen, 2014). Although fjords hosting icebergs this large and numerous are relatively few in number, it is these fjords (and the glaciers hosted by them) that contribute the most to dynamic mass loss from the Greenland Ice Sheet (Enderlin et al., 2014;Khan et al., 2020). ...
Article
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The rate of ocean-driven retreat of Greenland's tidewater glaciers remains highly uncertain in predictions of future sea level rise, in part due to poorly constrained glacier-adjacent water properties. Icebergs and their meltwater contributions are likely important modifiers of fjord water properties, yet their effect is poorly understood. Here, we use a 3-D ocean circulation model, coupled to a submarine iceberg melt module, to investigate the effect of submarine iceberg melting on glacier-adjacent water properties in a range of idealised settings. Submarine iceberg melting can modify glacier-adjacent water properties in three principal ways: (1) substantial cooling and modest freshening in the upper ∼50 m of the water column; (2) warming of Polar Water at intermediate depths due to iceberg melt-induced upwelling of warm Atlantic Water and; (3) warming of the deeper Atlantic Water layer when vertical temperature gradients through this layer are steep (due to vertical mixing of warm water at depth) but cooling of the Atlantic Water layer when vertical temperature gradients are shallow. The overall effect of iceberg melt is to make glacier-adjacent water properties more uniform with depth. When icebergs extend to, or below, the depth of a sill at the fjord mouth, they can cause cooling throughout the entire water column. All of these effects are more pronounced in fjords with higher iceberg concentrations and deeper iceberg keel depths. These iceberg melt-induced changes to glacier-adjacent water properties will reduce rates of glacier submarine melting near the surface, increase them in the Polar Water layer, and cause typically modest impacts in the Atlantic Water layer. These results characterise the important role of submarine iceberg melting in modifying ice sheet-ocean interaction and highlight the need to improve representations of fjord processes in ice sheet scale models.
... The effect of ice melange, a consolidated agglomeration of icebergs and fast-ice, on glacier margins has been studied more broadly for Greenland. Ice melange is strongly correlated with glacier retreat and advance (Howat et al., 2010;Moon et al., 2015), can alter the buttressing effect of ice tongues affecting marine terminating glacier dynamics (Krug et al., 2015;Todd & Christoffersen, 2014), and can stabilize the ice margin impeding calving (Amundson et al., 2010;Cassotto et al., 2021;Robel, 2017). ...
Article
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Plain Language Summary Ice tongues are iconic glaciological landmarks along the Victoria Land Coast in the western Ross Sea of Antarctica. Forming at the seaward margin of marine‐terminating glaciers, they extend many kilometers from the coast out to the sea. Over much of the year, ice tongues are embedded and protected by land‐fast sea ice (fast‐ice), which is sea ice attached to land. Fast‐ice recession can trigger ice tongue calving, which is an indicator of environmental change. We document a unique event in observational history, which is the complete loss of Parker Ice Tongue after exceptional seasons of repeated and complete fast‐ice break‐out. An event of this magnitude could have only occurred previously around the 1850s or more likely much longer back in time. We show that ice tongue integrity, evolution, and motion are affected by fast ice variability. Using satellite image time series, we observed that fast‐ice delayed Parker Ice Tongue collapse by protecting it from oceanic processes like currents and waves. It demonstrates the importance of coastal sea ice for these floating ice masses. It also confronts us with the question about the fate of such ice tongues if coastal sea ice coverage decreases as a result of future climate change.
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The disintegration of the eastern Antarctic Peninsula’s Larsen A and B ice shelves has been attributed to atmosphere and ocean warming, and increased mass losses from the glaciers once restrained by these ice shelves have increased Antarctica’s total contribution to sea-level rise. Abrupt recessions in ice-shelf frontal position presaged the break-up of Larsen A and B, yet, in the ~20 years since these events, documented knowledge of frontal change along the entire ~1,400-km-long eastern Antarctic Peninsula is limited. Here, we show that 85% of the seaward ice-shelf perimeter fringing this coastline underwent uninterrupted advance between the early 2000s and 2019, in contrast to the two previous decades. We attribute this advance to enhanced ocean-wave dampening, ice-shelf buttressing and the absence of sea-surface slope-induced gravitational ice-shelf flow. These phenomena were, in turn, enabled by increased near-shore sea ice driven by a Weddell Sea-wide intensification of cyclonic surface winds around 2002. Collectively, our observations demonstrate that sea-ice change can either safeguard from, or set in motion, the final rifting and calving of even large Antarctic ice shelves. Most of the eastern Antarctic Peninsula’s coastline has undergone uninterrupted advance since the early 2000s due to enhanced near-shore sea ice, according to satellite observations and reanalysis data.
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Exact information on calving front positions of marine- or lake-terminating glaciers is a fundamental glacier variable for analyzing ongoing glacier change processes and assessing other variables like frontal ablation rates. In recent years, researchers started implementing algorithms that could automatically detect the calving fronts on satellite imagery. Most studies use optical images, as in these images, calving fronts are often easy to distinguish due to sufficient spatial resolution and the presence of different spectral bands, allowing the separation of ice features. However, detecting calving fronts on SAR images is highly desirable, as SAR images can also be acquired during the polar night and are independent of weather conditions, e.g., cloud cover, facilitating all-year monitoring worldwide. In this paper, we present a benchmark dataset of SAR images from multiple regions of the globe with corresponding manually defined labels to train and test approaches for the detection of glacier calving fronts. The dataset is the first to provide long-term glacier calving front information from multi-mission data. As the dataset includes glaciers from Antarctica, Greenland and Alaska, the wide applicability of models trained and tested on this dataset is ensured. The test set is independent of the training set so that the generalization capabilities of the models can be evaluated. We provide two sets of labels: one binary segmentation label to discern the calving front from the background and one for multi-class segmentation of different landscape classes. Unlike other calving front datasets, the presented dataset contains not only the labels but also the corresponding preprocessed and geo-referenced SAR images as PNG files. The ease of access to the dataset will allow scientists from other fields, such as data science, to contribute their expertise. With this benchmark dataset, we enable comparability between different front detection algorithms and improve the reproducibility of front detection studies. Moreover, we present one baseline model for each kind of label type. Both models are based on the U-Net, one of the most popular deep learning segmentation architectures. Additionally, we introduce Atrous Spatial Pyramid Pooling to the bottleneck layer. In the following two post-processing procedures, the segmentation results are converted into one-pixel-wide front delineations. By providing both types of labels, both approaches can be used to address the problem. To assess the performance of the models, we first review the segmentation results using the recall, precision, F1-score, and the Jaccard Index. Second, we evaluate the front delineation by calculating the mean distance error to the labeled front. The presented vanilla models provide a baseline of 150 m ± 24 m mean distance error for the Mapple Glacier in Antarctica and 840 m ± 84 m for the Columbia Glacier in Alaska, which has a more complex calving front, consisting of multiple sections, as compared to a laterally well constrained, single calving front of Mapple Glacier.
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The rate of ocean-driven retreat of Greenland’s tidewater glaciers remains highly uncertain in predictions of future sea level rise, in part due to poorly constrained glacier-adjacent water properties. Icebergs and their meltwater contributions are likely important modifiers of fjord water properties, yet their effect is poorly understood. Here, we use a 3-D ocean circulation model, coupled to a submarine iceberg melt module, to investigate the effect of submarine iceberg melting on glacier-adjacent water properties in a range of idealised settings. Submarine iceberg melting can modify glacier-adjacent water properties in three principle ways: (1) substantial cooling and modest freshening in the upper ~50 m of the water column; (2) warming of Polar Water at intermediate depths due to iceberg melt-induced upwelling of warm Atlantic Water, and; (3) warming of the deeper Atlantic Water layer when vertical temperature gradients through this layer are steep (due to vertical mixing of warm water at depth), but cooling of the Atlantic Water layer when vertical temperature gradients are shallow. The overall effect of iceberg melt is to make glacier-adjacent water properties more uniform with depth. When icebergs extend to, or below, the depth of a sill at the fjord mouth, they can cause cooling throughout the entire water column. All of these effects are more pronounced in fjords with higher iceberg concentrations and deeper iceberg keel depths. These iceberg melt-induced changes to glacier-adjacent water properties will reduce rates of glacier submarine melting near the surface, but increase them in the Polar Water layer, and cause typically modest impacts in the Atlantic Water layer. These results characterise the important role of submarine iceberg melting in modifying ice sheet-ocean interaction, and highlight the need to improve representations of fjord processes in ice sheet-scale models.
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Calving is an important mass-loss process for many glaciers worldwide, and has been assumed to respond to a variety of environmental influences. We present a grounded, flowline tidewater glacier model using a physically-based calving mechanism, applied to Helheim Glacier, eastern Greenland. By qualitatively examining both modelled size and frequency of calving events, and the subsequent dynamic response, the model is found to realistically reproduce key aspects of observed calving behaviour. Experiments explore four environmental variables which have been suggested to affect calving rates: water depth in crevasses, basal water pressure, undercutting of the calving face by submarine melt and backstress from ice mélange. Of the four variables, only crevasse water depth and basal water pressure were found to have a significant effect on terminus behaviour when applied at a realistic magnitude. These results are in contrast to previous modelling studies, which have suggested that ocean temperatures could strongly influence the calving front. The results raise the possibility that Greenland outlet glaciers could respond to the recent trend of increased surface melt observed in Greenland more strongly than previously thought, as surface ablation can strongly affect water depth in crevasses and water pressure at the glacier bed.
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Calving of icebergs is a major negative component of polar ice-sheet mass balance. We present a new calving modeling framework relying on both continuum damage mechanics and linear elastic fracture mechanics. This combination accounts for both the slow sub-critical surface crevassing and fast propagation of crevasses when calving occurs. First, damage of the ice occurs over long timescales and enhances the viscous flow of ice. Then brittle fracture propagation happens downward, over very short timescales, in ice considered as an elastic medium. The model is validated on Helheim Glacier, South-West Greenland, one of the most monitored fast-flowing outlet glacier. This allows to identify sets of model parameters giving a consistent response of the model and producing a dynamic equilibrium in agreement with observed stable position of the Helheim ice front between 1930 and today.
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To quantify the ice-ocean processes which drive dynamic and geometric change at calving outlet glaciers, detailed measurements beyond the capability of present satellites are required. This study presents the application of a cost-effective (< USD 2000), unmanned aerial vehicle (UAV) to investigate frontal dynamics at a major outlet draining the western sector of the Greenland Ice Sheet. The UAV was flown over Store Glacier on three sorties during summer 2013 and acquired over 2000 overlapping, geo-tagged images of the calving front at ∼40 cm resolution. Stereo-photogrammetry applied to these images enabled the extraction of high-resolution digital elevation models with an accuracy of ±1.9 m which we used to quantify glaciological processes from early July to August 2013. The central zone of the calving front advanced by ~500 m whilst the lateral margins remained stable. In addition, the ice surface thinned by 3.5 m m-1during the melt-season in association with dynamic thinning. Ice flux through the calving front is calculated at 2.96 × 107 m3 d-1, equivalent to 11 Gt a-1, which is comparable to flux-gate estimates of Store Glacier's annual discharge. Water-filled crevasses were observed throughout the observation period, but covered a limited area (1200 to 12 000 m2 of the ∼5 × 106 m2 surveyed area) and did not appear to exert any significant control over calving. We conclude that the use of repeat UAV surveys coupled with the processing techniques outlined in this paper have a number of important potential applications to tidewater outlet glaciers.
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Analytical and numerical techniques are used to examine the flow response of a sloped slab of power-law fluid (power n) subjected to basal boundary conditions that vary spatially across the flow direction, as for example near an ice-stream margin with planar basal topography. The primary assumption is that basal shear stress is proportional to the basal speed times a spatially variable slip resistance. The ratio of mean basal speed to the speed originating from shearing through the thickness. denoted as r, gives a measure of how slippery the bed is. The principal conclusion is that a localized disturbance in slip resistance affects the basal stress and speed in a zone spread over a greater width of the flow. In units of ice thickness H, the spatial scale of spreading is proportional to a single dimensionless number R n ≡ (r/n+ 1)1/n+1 derived from n and r. The consequence for a shear zone above a sharp jump in slip resistance is that the shearing is spread out over a boundary layer with a width proportional to R n. For an ice stream caused by a band of low slip resistance with a half-width of w H, the margins influence velocity and stress in the central part of the band depending on Rn in comparison to w. Three regimes can be identified, which for n = 3 are quantified as follows: low r defined as R 3 w, for which the central flow is essentially unaffected by the margins and the driving stress is supported entire by by basal drag; high r defined as R 3 > 1w, for which the boundary layers from both sides bridge across the full flow width and the driving stress in the center is supported almost entirely by side drag; intermediate r, for which the driving stress in the center is supported by a combination of basal and side drag. Shear zones that are narrower than predicted on the basis of this theory (≈ R 3) would require localized softening of the ice to explain the concentration of deformation at a shorter scale.