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A physically based calving model applied to marine outlet glaciers and implications for the glacier dynamics

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We present results from numerical ice-flow models that include calving criteria based on penetration of surface and basal crevasses, which in turn is a function of longitudinal strain rates near the glacier front. The position of the calving front is defined as the point where either (1) surface crevasses reach the waterline (model CDw), or (2) surface and basal crevasses penetrate the full thickness of the glacier (model CD). For comparison with previous studies, results are also presented for a height-above-buoyancy calving model. Qualitatively, both models CDw and CD produce similar behaviour. Unlike previous models for calving, the new calving criteria are applicable to both grounded termini and floating ice shelves and tongues. The numerical ice-flow model is applied to an idealized geometry characteristic of marine outlet glaciers. Results indicate that grounding-line dynamics are less sensitive to basal topography than previously suggested. Stable grounding-line positions can be obtained even on a reverse bed slope with or without floating termini. The proposed calving criteria also allow calving losses to be linked to surface melt and therefore climate. In contrast to previous studies in which calving rate or position of the terminus is linked to local water depth, the new calving criterion is able to produce seasonal cycles of retreat and advance as observed for Greenland marine outlet glaciers. The contrasting dynamical behaviour and stability found for different calving models suggests that a realistic parameterization for the process of calving is crucial for any predictions of marine outlet glacier change.
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A physically based calving model applied to marine outlet glaciers
and implications for the glacier dynamics
F.M. NICK,
1,2,*
C.J. VAN DER VEEN,
3
A. VIELI,
4
D.I. BENN
5,6
1
Geological Survey of Denmark and Greenland, Øster Voldgade 10, DK-1350 Copenhagen, Denmark
E-mail: fmnick@ulb.ac.be
2
Institute for Marine and Atmospheric Research Utrecht, Utrecht University, Princetonplein 5, 3584 CC Utrecht,
The Netherlands
3
Department of Geography and Center for Remote Sensing of Ice Sheets, University of Kansas, 2335 Irving Hill Road,
Lawrence, Kansas 66045-7612, USA
4
Department of Geography, Durham University, South Road, Durham DH1 3LE, UK
5
The University Centre in Svalbard (UNIS), PO Box 156, NO-9171 Longyearbyen, Norway
6
School of Geography and Geosciences, University of St Andrews, St Andrews, Fife KY16 9AL, UK
ABSTRACT. We present results from numerical ice-flow models that include calving criteria based on
penetration of surface and basal crevasses, which in turn is a function of longitudinal strain rates near
the glacier front. The position of the calving front is defined as the point where either (1) surface
crevasses reach the waterline (model CDw), or (2) surface and basal crevasses penetrate the full
thickness of the glacier (model CD). For comparison with previous studies, results are also presented for
a height-above-buoyancy calving model. Qualitatively, both models CDw and CD produce similar
behaviour. Unlike previous models for calving, the new calving criteria are applicable to both grounded
termini and floating ice shelves and tongues. The numerical ice-flow model is applied to an idealized
geometry characteristic of marine outlet glaciers. Results indicate that grounding-line dynamics are less
sensitive to basal topography than previously suggested. Stable grounding-line positions can be obtained
even on a reverse bed slope with or without floating termini. The proposed calving criteria also allow
calving losses to be linked to surface melt and therefore climate. In contrast to previous studies in which
calving rate or position of the terminus is linked to local water depth, the new calving criterion is able to
produce seasonal cycles of retreat and advance as observed for Greenland marine outlet glaciers. The
contrasting dynamical behaviour and stability found for different calving models suggests that a realistic
parameterization for the process of calving is crucial for any predictions of marine outlet glacier change.
1. INTRODUCTION
Calving of icebergs is an important mechanism for rapidly
transferring mass from the polar ice sheets into the
surrounding ocean. Recent observations have shown that
changes in calving rate can greatly reduce the extent of
floating ice shelves and ice tongues, potentially resulting in
increased discharge from the interior (Joughin and others,
2004; Rignot and others, 2004). While the break-up of
floating ice tongues has negligible direct effect on global sea
level, the resulting speed-up of grounded ice can have major
consequences for global sea level. Indeed, a wide range of
observations applying to both current ice masses and
palaeo-ice sheets point to iceberg calving as a major factor
in rapid ice-sheet changes (Van der Veen, 2002). Rapid
changes in ice dynamics of Greenland outlet glaciers have
been documented in a number of recent studies (Thomas
and others, 2003; Joughin and others, 2004, 2008a,b,c;
Howat and others, 2005, 2007, 2008b; Luckman and others,
2006; Rignot and Kanagaratnam, 2006; Csatho and others,
2008; Moon and Joughin, 2008). In particular, recent
changes on Jakobshavn Isbræ on the west coast, and
Helheim and Kangerdlugssuaq glaciers on the east coast
have been detailed on seasonal timescales. Determining the
most likely causes for recent behaviour of these outlet
glaciers is crucial for assessing the future contribution of the
Greenland ice sheet to global sea level. It is therefore
important to formulate a calving model that can be readily
incorporated into time-evolving numerical ice-flow models
and, at the same time, is able to reproduce observed glacier
behaviour. As a first step, the objective of the present study is
to introduce a physically based calving model and to
evaluate its impact on flow and dynamics of calving glaciers.
Calving criteria used in previous model studies fall into
one of two main categories. The first type parameterizes the
calving rate as a function of some independent variable such
as water depth (Brown and others, 1982; Meier and Post,
1987; Hanson and Hooke, 2000), ice-front thickness (Pfeffer
and others, 1997) or stretching rate (Alley and others, 2007,
2008). The second approach specifies the position of the
calving front, based on a height-above-buoyancy criterion,
and the calving rate is then given by the rate at which ice is
delivered to the ice front (Van der Veen,1996, 2002). Data
collected on Columbia Glacier, Alaska, USA, prior to and
during its rapid retreat and speed-up suggest that the
position of the calving front is controlled by local geometry
such that at the terminus the thickness in excess of flotation
cannot become less than a certain threshold value (~50 m
for Columbia Glacier). Thus, for such height-above-
buoyancy or flotation criteria the calving rate is not explicitly
parameterized but is a result of flow and dynamics at the
terminus. Inclusion of a flotation criterion in a numerical
Journal of Glaciology, Vol. 56, No. 199, 2010
*Present address: Laboratoire de Glaciologie Polaire, De´partement des
Sciences de la Terre et de l’Environnement, CP 160/03, Universite´ Libre de
Bruxelles, Avenue F.D. Roosevelt 50, B-1050 Brussels, Belgium.
781
model reproduced unstable retreat through basal over-
deepenings, and showed a more complex relationship
between calving rate and water depth than assumed in
earlier models (Vieli and others, 2001; Nick and others,
2007). A model incorporating the height-above-buoyancy
criterion for calving was to some extent successful in
explaining the recent retreat of Helheimgletscher, but could
not reproduce the seasonal cycle of retreat and advance
(Nick and others, 2009).
A major difficulty with both the water-depth model and
the height-above-buoyancy model is that these apply to
grounded termini only. Moreover, these models produce
inherently unstable glacier behaviour where the bed slopes
downward towards the interior. A modest retreat of the
calving front into deeper waters leads to further retreat that is
halted only at the head of the fjord where the bed rises above
sea level or where the bed slope reverses. Neither model can
produce an advancing ice front without invoking other
factors such as sedimentation at the terminus to reduce local
water depth (Nick and others, 2007, 2009). Most of the
Antarctic marine-based glaciers and ice streams and many
Greenland calving glaciers are buttressed by floating ice
tongues or ice shelves from which icebergs are discharged.
Thus, there is a need for calving models that can be applied
to both floating and grounded ice fronts, and allow the
calving front to advance on seasonal timescales, either by
forming a short-lived floating ice tongue or by advancing a
grounded terminus.
The model of Pfeffer and others (1997) calculated calving
losses from a floating ice shelf as a nonlinear function of ice-
front thickness, based on empirically determined fracture
propagation rates. While having some physical basis, this
model does not take account of key factors governing
fracture location and extent (e.g. strain rate) and is only
applicable to floating ice. Alley and others (2008) also
formulated a law for ice-shelf calving based on regression
between stretching rate and velocity near the ice front,
assumed to be closely approximating the calving flux. The
implication of this model is that it tends to make ice shelves
inherently unstable, whereby one calving event and associ-
ated terminus retreat leads to higher calving fronts, and
consequently enhanced stretching, and could result in
complete collapse. Because their regression is valid for
near-steady termini, it is not immediately clear whether the
model proposed by Alley and others (2008) can be extended
to rapidly retreating calving fronts. As is the case for the
water-depth and height-above-buoyancy models, the phys-
ical underpinnings of the stretching-rate model remain
tenuous at best.
The calving process is highly stochastic in nature, and
may involve frequent detachment of smaller pieces from
above or below the waterline, as well as more infrequent
breaking of larger icebergs. Each calving event involves
propagation of fractures, but considering local character-
istics (e.g. shape of the snout, pre-existing planes of
structural weakness, wave impacts) it is unreasonable to
expect any model to be able to predict with any confidence
when and where the next iceberg will break off. Never-
theless, it is possible to formulate a ‘bulk’ calving model
that captures the main features of the average calving
process that can be included in prognostic numerical ice-
sheet models.
In an attempt to overcome limitations of existing calving
models, Benn and others (2007a,b) introduced a calving
criterion based on the depth of penetration of surface
crevasses, which is in turn a function of longitudinal strain
rates on the glacier tongue. The position of the calving front
was defined as the point where crevasse depth equals the ice
height above sea or lake level, based on the observation that
many glaciers calve when crevasses reach the waterline,
with failure of the subaerial part of the calving face followed
after some interval by calving of the submerged super-
buoyant ice toe (Motyka, 1997). The waterline crevasse-
depth criterion of Benn and others (2007a,b) has been
incorporated into a three-dimensional (3-D), full-Stokes
glacier model by Otero and others (2010). Their model
could successfully predict ice-margin position for a specified
glacier geometry, although glacier evolution through time
was not investigated. Here we implement a modified
crevasse-depth model in which the calving front is defined
as the point where water-filled surface crevasses and basal
crevasses penetrate the full thickness of the glacier. The
waterline crevasse-depth model may be applicable to small,
relatively slow tidewater glaciers such as those on Svalbard,
whereas the modified model may be more representative of
large, fast-flowing Greenlandic outlet glaciers. It must be
emphasized that these models are not intended as literal
representations of how individual calving events occur, but
rather as a means of relating terminus position to ice
dynamics in a simple but physically based way.
For tensile stresses of a few hundred kPa, air-filled
crevasses extend to a depth of several tens of meters.
However, where extensive surface melting takes place,
existing surface crevasses can become water-filled, allowing
further downward growth (Weertman, 1973; Van der Veen,
1998b, 2007). Scambos and others (2000) proposed that this
mechanism explains how relatively minor changes in local
climate conditions can lead to rapid disintegration of ice
shelves. Similarly, Sohn and others (1998) found seasonal
variations in calving rate on Jakobshavn Isbræ associated
with surface melting. Consequently, as suggested by Benn
and others (2007b), the calving model presented here
includes the effects of water on crevasse depth.
Prior studies on crevasse penetration (Smith, 1976, 1978;
Rist and others, 1996; Van der Veen, 1998a,b) applied linear
elastic fracture mechanics (LEFM) to estimate penetration
depth of crevasses on glaciers. In that approach, the stress
intensity factor is used to describe elastic stresses near the
crevasse tip, thereby accounting for local stress concen-
trations promoting fracture growth. When compared to the
fracture toughness of ice, the stress intensity factor provides
a measure for how deep a crevasse can penetrate into the
ice, if stresses acting on the crevasse are known. This
approach, however, is not readily incorporated into numer-
ical ice-flow models. Therefore, following Benn and others
(2007b) the simplifying assumption is made here that
crevasses propagate to the depth at which the tensile stress
equals the lithostatic stress and the net longitudinal stress is
zero (Nye, 1955, 1957). This simplification is appropriate
because where crevasses are closely spaced, as is the case
for most calving termini, stress concentrations at crevasse
tips are small. Crevasse depths predicted by the zero-stress
model are very close to those obtained by the LEFM
approach for a field of crevasses (Van der Veen, 1998b;
Mottram and Benn, 2009). The zero-stress condition allows
crevasse penetration depth to be readily estimated anywhere
in the terminus region and a simple calving criterion to be
implemented into a time-evolving numerical model.
Nick and others: Calving model applied to marine outlet glaciers782
Our primary objective is to investigate whether the new
calving model can generate realistic patterns of glacier
advance and retreat, both long-term and on seasonal
timescales, and how the bed topography influences flow
dynamics and terminus stability. In order not to obfuscate
interpretation of model results, processes that may force
terminus migration are, in this study, limited to variations in
water level in surface crevasses, in accumulation and in
back pressure at the calving front. We are aware that other
processes (e.g. basal melting under floating ice tongues) may
play an important role in the control of the position of
marine glacier termini, as highlighted by Holland and others
(2008) for Jakobshavn Isbræ, and that they should be
included in attempts to model observed behaviour of actual
outlet glaciers.
In the following sections, we first describe the crevasse-
depth calving model, and the ice-flow model used to evolve
the glacier through time. The numerical model is then
applied to an idealized geometry, to evaluate glacier
response and stability to various imposed forcings. For
comparative purposes, similar model runs were conducted
using the waterline crevasse-depth model and a height-
above-buoyancy calving criterion.
2. CALVING MODEL
In a field of closely spaced crevasses, little tensile stress can
exist within the thin slabs of ice separating adjacent
crevasses, so there are no large stress concentrations near
the tips of crevasses (Weertman, 1973). This suggests that
under these conditions the depth of surface crevasses may
be estimated following the model introduced by Nye (1955,
1957). That is, crevasses will penetrate to the depth at which
the net longitudinal stress becomes zero. In the absence of
water in the crevasses, at this depth the longitudinal tensile
stress equals the compressive ice overburden pressure. The
normal stress responsible for crevasse opening is the resistive
stress, R
xx
, defined as the full stress minus the lithostatic
stress and related to the longitudinal stretching rate through
Glen’s flow law (Van der Veen, 1999, p.38),
Rxx ¼2_
"xx
A

1
n
,ð1Þ
where Ais the temperature-dependent rate factor, n= 3 is the
flow parameter, and the contribution of other strain rates to
the effective strain rate has been neglected. The simplifying
assumption is made that this stress is constant with depth
(Rist and others, 1996; Van der Veen, 1998b). Allowance
can be made for depth variation resulting from non-uniform
temperatures throughout an ice column if, for example,
stretching rate is considered independent of depth (Van der
Veen, 1998a). Such refinement will not, however, signifi-
cantly alter the behaviour of the model glacier.
The lithostatic stress, or ice overburden pressure, in-
creases with depth according to
L¼igðHzÞ,ð2Þ
where
i
is the ice density, gis the gravitational acceleration,
His the ice thickness and zis the vertical coordinate with
z= 0 at the glacier or ice-shelf base. As noted by Rist and
others (1996), the density of near-surface firn is considerably
lower than that of solid ice, thereby reducing the crevasse
closing stress and allowing the crevasse deeper penetration
than if constant density is assumed (cf. Van der Veen,
1998b). We assume a constant value of
i
(920 kg m
–3
)
corresponding to solid ice, which is probably realistic in
glacier ablation zones (e.g. the termini of Greenlandic
glaciers). We note, however, that density variations can be
readily incorporated into the model.
Equating the tensile stress (Equation (1)) with the ice
overburden pressure (Equation (2)) yields the penetration
depth, d
s
, of surface crevasses (Nye, 1955, 1957),
ds¼Rxx
ig:ð3Þ
The Nye model does not take into account the strength of ice
and allows crevasses to exist for all values of the tensile
stress. In reality, if the tensile stress is less than some
threshold value, no crevasses will form. This condition is
important in determining where crevasses will first form, but
for the heavily crevassed terminus region just upstream of
the calving front, where the tensile stress is likely to be
greater than the threshold stress, this issue may be ignored
(Van der Veen, 1998b).
For a surface crevasse containing water, an additional
opening stress allows the crevasse to penetrate deeper
(Weertman, 1973; Van der Veen, 1998b). If d
w
is the water
height in the crevasse, this additional stress equals
w
gd
w,
where
w
represents the density of meltwater. The crevasse
penetration depth may then be estimated from (Benn and
others, 2007b)
ds¼Rxx
igþw
i
dw:ð4Þ
In the initial form of the crevasse-depth model, the position
of the calving front is defined as the point where d
s
equals
the glacier freeboard above sea level, based on the obser-
vation that many subaerial calving events occur when the
depth of surface crevasses approaches the waterline,
followed by calving of the subaqueous toe (Motyka, 1997).
On the other hand, observations on Greenland outlet
glaciers show infrequent calving of larger pieces or tabular
icebergs interspersed with numerous smaller calving events
(personal communication from L. Stearns, 2009). Production
of these larger bergs likely involves full-thickness fracturing.
Therefore we introduce a modification to the Benn and
others (2007b) calving model, in which calving occurs when
surface crevasses reach the depth to which basal crevasses
penetrate upward into the ice. Significant upward propa-
gation of basal crevasses is possible only where the glacier is
at or near flotation and stretching rates are large (Van der
Veen, 1998a), so for grounded termini this implies that the
water-filled surface crevasse must penetrate nearly the full
ice thickness before a calving event occurs. On floating
glacier tongues, however, basal crevasses can penetrate
upward a significant fraction of the ice thickness, thus
facilitating the calving process.
To estimate the height of basal crevasses, the net normal
stress must be considered. This stress is the sum of the
lithostatic stress, the pressure of water filling the crevasse,
and the tensile stress associated with flow stretching. Adding
these three contributing stresses yields (Van der Veen,
1998a, equation (16))
nðzÞ¼igðHzÞþpgðHpzÞþRxx :ð5Þ
In this expression,
n
is the net normal stress, zis the height
above the glacier or ice-shelf base, H
p
is the piezometric
head, or the height above the base to which water in a
Nick and others: Calving model applied to marine outlet glaciers 783
borehole to the bed will rise, and
p
is the density of sea or
lake water (depending on the proglacial water body into
which the glacier calves). For a floating ice tongue, the
piezometric head corresponds to sea level. The assumption
is made here that where the terminus is grounded, a full and
easy connection between the subglacial drainage system
and the adjoining sea or lake exists. In that case, H
p
equals
the depth, D, of the glacier sole below sea level.
As in the case of surface crevasses, where basal crevasses
are closely spaced, stress concentrations at the crevasse tips
may be ignored and the penetration height may be estimated
from the requirement that the net longitudinal stress is zero
at that height (Weertman, 1980; Jezek, 1984). After some
rearranging, this gives
db¼i
pi
Rxx
igHab

,ð6Þ
where H
ab
represents the height above buoyancy, defined as
Hab ¼Hp
i
D:ð7Þ
For a floating ice tongue, H
ab
= 0, and the height of the
bottom crevasses is determined solely by the tensile stress,
R
xx
. On grounded glaciers, H
ab
> 0 and basal crevasses will
penetrate less far upward.
The modified calving model is now complete. Because
the depth of surface crevasses is estimated from the
longitudinal stress and water level in the crevasses (Equa-
tion (4)), the model allows links between calving rate and
changes in climate conditions to be explored. Water levels
may be expected to increase during the melt season, starting
from zero at the end of the winter, as surface meltwater
collects in the crevasses. Progressive melting allows surface
crevasses to penetrate deeper into the ice, thus providing a
mechanism for increased calving losses during summer.
For the present applications, we do not specify surface
melt and freezing rates, but rather prescribe directly water
level within crevasses as simple forcing scenarios. In
principle, though, water depths in crevasses could be
modelled explicitly.
3. ICE-FLOW MODEL
The calving model has been incorporated into a numerical
ice-flow model that calculates the flow and geometric
evolution, based on the model used by Nick and others
(2009). The ice-flow model is briefly outlined below. A list of
model parameters and their values is given in Table 1.
3.1. Continuity and force balance
Considering a flowband of width Wand thickness H,
conservation of mass is expressed by the depth-integrated
continuity equation (Van der Veen, 1999; Oerlemans, 2001),
@H
@t¼ 1
W
@q
@xþa,ð8Þ
where tis time, xis the distance along the central flowline
and ais the surface mass balance. Neglecting the effect of
sloping side-walls, the horizontal ice flux through a cross
section of the flowband is given by q=HWU, with Uthe
vertically averaged horizontal ice velocity.
Conservation of momentum requires
2@
@xH@U
@x

AsHp
i
D

U

1
m
2H
W
5U
AW

1
n
¼igH @h
@x,ð9Þ
where is the strain-rate dependent effective viscosity,
defined as
¼A1
n@U
@x
1n
n
:ð10Þ
Equation (9) states that the driving stress (right-hand side) is
balanced by resistive forces associated with gradients in
longitudinal stress (first term on the left-hand side), drag at
the glacier bed (second term) and lateral drag (third term).
The assumption is made that basal drag depends on sliding
velocity and effective basal pressure (Bindschadler 1983;
Van der Veen and Whillans, 1996; Vieli and Payne, 2005).
The sliding parameter, A
s
, and the friction parameter, ,may
be related to bed roughness and basal water, respectively.
The value m=3 is chosen for the nonlinear sliding relation.
Resistance from drag along the lateral margins is estimated
by integrating the force-balance equation over the width of
the flowband assuming that lateral drag supports the same
fraction of driving stress along a transect across the glacier
(section 5.5 of Van der Veen, 1999).
3.2. Boundary conditions
The up-glacier boundary, x= 0, corresponds to the ice divide
where the surface slope and horizontal velocity are set to
zero. At the calving front, the longitudinal stress is balanced
by the difference between hydrostatic pressure of the ice and
water, giving for the depth-averaged stress
Rxx ¼1
2igHp
i
D2
H

Bð11Þ
in which Dis the depth of the glacier base below sea level
and
B
is a back pressure from sea ice or sikkusak. Applying
Glen’s flow law and rearranging, the corresponding stretch-
ing rate at the terminus is
@U
@x¼Aig
4Hp
i
D2
HB
ig

n
:ð12Þ
The second boundary condition at the terminus is provided
by the calving criteria discussed in section 2, to account for
mass loss at the terminus. The crevasse-depth model allows
formation of an ice shelf or a floating tongue when ice
Table 1. Values of model parameters
Constant or
parameter
Value Notes
i
920 kg m
–3
Ice density
p
1028 kg m
–3
Sea density
w
1000 kg m
–3
Meltwater density
g9.8 m s
–2
Gravitational acceleration
m3 Bed friction exponent
n3 Exponent in Glen’s flow
A5.6 10
–17
Pa
–3
a
–1
Glen’s law coefficient
m1 Friction parameter
A
s0
100 Pam
–2/3
s
–1/3
Sliding parameter
x300–500 m Variable grid size
t0.001 years Time-step
Nick and others: Calving model applied to marine outlet glaciers784
thickness is less than the flotation thickness. The transition
between grounded ice and shelf is achieved through setting
basal resistance to zero; that is, the friction parameter, ,in
Equation (9) is set to zero when the ice thickness becomes
less than the flotation thickness.
In the crevasse-depth model, the local water depth or
front geometry influences the glacier flow and strain rate and
eventually calving rate. But, contrary to previous calving
models, there is no direct dependency of calving rate on the
water depth.
Model calculations are performed on a moving grid,
which allows the glacier front to be followed continuously
(Nick and Oerlemans, 2006). The initial horizontal grid
spacing is 300 m; this distance changes with every time-step
as a new grid is defined to fit the new glacier length. For
cases with a floating ice tongue, grounding-line motion has
been checked for spatial grid-size independency by model
experiments with refined grid resolutions. The problem of
grid-size dependency (Vieli and Payne, 2005) is here
overcome by the chosen high spatial resolution. Since basal
resistance depends on effective basal pressure (second term
in Equation (9)), it decreases as the ice thickness thins down
to the flotation thickness. Therefore there is a smooth
transition in basal resistance from the grounded to the
floating ice tongue. Equation (9) is solved using a standard
Newton iteration method. The fluxes and velocities are
computed on a staggered grid between the gridpoints where
thickness is calculated (more detail can be found in Vieli
and Payne, 2005).
4. MODEL EXPERIMENTS
Our objective here is to evaluate how different calving
criteria affect glacier dynamics. In particular, we seek to
explore how a floating ice tongue affects the stability of the
terminus in the presence of a basal overdeepening and
whether the flowline model can produce a seasonally
advancing and retreating terminus in such a geometric
setting. Three calving models are considered: (1) the
crevasse-depth model (CD) in which a calving event occurs
when the combined depth of surface and bottom crevasses
equals the ice thickness; (2) the waterline crevasse-depth
model (CDw), with calving occurring when a surface
crevasse extends down to the waterline (Benn and others,
2007b); and (3) the height-above-buoyancy or flotation
model (FL) in which the glacier thickness at the terminus
cannot be less than a given limit, H
c
. Following Vieli and
others (2001), the critical thickness is defined as a small
fraction, q, of the flotation thickness plus the flotation
thickness:
Hc¼p
i
ð1þqÞD:ð13Þ
At each time-step, the position of the terminus is shifted to
the location where the thickness equals this critical
thickness.
For the following model experiments and besides chan-
ging surface mass balance, the main forcing for terminus
migration is through the variation of water level within
crevasses, d
w
, or of back pressure at the calving front,
B
.
These forcings reflect processes such as surface melt and the
existence of sea ice or sikkusak and are both linked to
climate and, to some extent, oceanic conditions. Our choice
for the model forcing does not imply that these are the only
important drivers for outlet glacier change. Indeed, obser-
vations indicate that other processes, such as changes in
basal lubrication or oceanic melting beneath floating ice
tongues or at the grounding line, may play an important role
as control for calving-front dynamics (Motyka and others,
2003, 2009; Holland and others, 2008; Joughin and others,
2008c). While such processes should be included in
attempts to model realistic behaviour of actual outlet
glaciers, our study represents a first step towards this
objective. The limiting of forcing processes to variations in
water level and in back pressure still targets our main aim of
investigating the primary implications of calving criteria on
marine outlet glacier dynamics on the relevant timescales.
An idealized geometry (Fig. 1) is used consisting of a wide
accumulation area and a narrow outlet channel to the sea.
Except near the ice divide, the bed is below sea level with
two overdeepenings. The size of the model glacier is
purposely kept small (total catchment area of ~100 km
2
)to
minimize possible stabilizing effects from increased dis-
charge from the interior. For all three calving models, steady-
state profiles were produced using parameter values given in
Tables 1 and 2. The profile shown in Figure 1 corresponds to
an initial glacier length of 46 km and was used as the initial
geometry for modelling advance past the proglacial over-
deepening. A second equilibrium profile with a glacier
length of 69 km served as the initial profile to investigate
glacier retreat (Fig. 4).
4.1. Terminus advance
In a first set of experiments for the CD and CDw models,
glacier advance is forced by reducing the water level in
surface crevasses (d
w
in Equation (4); see Table 2). For both
the CD and CDw models, an ice shelf forms and the
grounding line advances through the basal depression to
reach a steady-state position just beyond the deepest point,
on the upsloping bed (Fig. 2). In this particular model
experiment, the grounding line does not advance all the way
to the next bed high; by decreasing the crevasse water level
further, advance to the next bed high occurs (not shown
here). The FL model is forced by reducing the critical
thickness (smaller value of qin Equation (14)), but it only
produces minor advance and does not advance beyond the
bedrock low, as was also found in earlier model studies
(Nick and others, 2007).
In a second set of experiments, glacier advance is
achieved by increasing the accumulation rate in the
catchment area (i.e. ain Equation (8) is doubled where
a> 0). The terminus advances across the basal trough to the
next basal high (Fig. 3). During the initial stage of advance
into deeper water, a small ice shelf forms for the CD and the
Table 2. Overview of model comparison experiments for the three
calving criteria
CD CDw FL
Initial glacier, L=46km d
w
=80m d
w
=10m q= 0.1
Advance scenario d
w
=60m d
w
=7m q= 0.0001
a>0: a=2aa>0: a=2aa>0: a=20a
Initial glacier, L=69km d
w
=80m d
w
=10m q= 0.1
Retreat scenario d
w
= 125 m d
w
=14m q= 0.3
a>0: a= 0.7aa>0: a= 0.7aa>0: a= 0.7a
Nick and others: Calving model applied to marine outlet glaciers 785
CDw model (Fig. 4). Again, in the FL model, increasing
surface accumulation does not allow glacier advance into
deeper water, even if an extreme increase (by a factor of 20)
is applied.
4.2. Terminus retreat
To investigate glacier retreat, model runs start from a steady-
state glacier with a length of 69 km, similar to the most
advanced profile shown further below in Figure 4. Model
parameters used to obtain the initial profile are listed in
Table 2.
Increasing the water level in crevasses results in deeper
penetration and initially higher calving rates. For both the
CD and CDw models, the terminus retreats a few kilometres
behind the basal high before reaching a new equilibrium
position on the upward slope (Fig. 5). Similar retreat results
Fig. 1. Initial steady-state geometry: (a) glacier surface and basal elevation along the central flowline; (b) glacier width; and (c) annual
surface mass balance.
Fig. 2. The simulated advance forced by decreasing water level in crevasses or critical height. (a) Glacier length evolution in time for
different calving criteria, the CD, CDw and FL models (black, blue and red, respectively). The black and blue dashed curves show position of
the grounding line for CD and CDw models, respectively. (b) Bed elevation at glacier front. Arrow indicates direction of advance.
Nick and others: Calving model applied to marine outlet glaciers786
from a decrease in accumulation rate in the catchment basin
(Fig. 6). For the FL model, any increase in critical thickness
(Fig. 5) or decrease in accumulation rate (Fig. 6) moves the
terminus over the basal high into the basal overdeepening
where it retreats rapidly and is halted only when the
terminus reaches the shallower bed. This unstable behaviour
is the direct consequence of the terminus position being
linked to water depth in the FL model.
To further illustrate the difference between the CD and
FL calving criteria, Figure 7 shows retreat forced by a
decrease in surface accumulation with a bed geometry
characterized by a longer, deep basal depression. In the CD
model, the glacier reaches a new equilibrium characterized
by the presence of a small ice shelf (Fig. 8). Retreat is
arrested well before the grounding line reaches shallower
water. In contrast, the FL model predicts retreat all the way
Fig. 3. The simulated advance forced by an increase in the accumulation rate by factor 2. (a) Glacier length evolution in time for different
calving criteria, the CD, CDw and FL models (black, blue and red, respectively). The black and blue dashed curves show position of the
grounding line for CD and CDw models, respectively. The dashed red curve refers to the case in which an extreme increase in accumulation
rate (by factor 20) is applied. (b) Bed elevation at glacier front. Arrow indicates direction of advance.
Fig. 4. The simulated surface profiles along the central flowline for experiment using the CD model and increased accumulation rate
(corresponding to the black curve in Fig. 3). The time interval between the profiles is 50 years.
Nick and others: Calving model applied to marine outlet glaciers 787
through the basal depression. Because terminus position in
the FL model is directly linked to water depth, retreat does
not stop until the calving flux reduces as it reaches the
shallower bed.
4.3. Stability on a reversed bed
As shown in Figures 2 and 5, when using the CD or CDw
model, the glacier can reach a steady state on an upsloping
bed, which is not possible for the height-above-buoyancy
Fig. 5. The modelled retreat forced by increasing water level in crevasses or critical height. (a) Glacier length evolution in time for different
calving criteria, the CD, CDw and FL models (black, blue and red, respectively). (b) Bed elevation at glacier front. Arrow indicates direction
of retreat.
Fig. 6. The simulated retreat forced by a decrease in the accumulation rate by factor 0.7. (a) Glacier length evolution in time for different
calving criteria, the CD, CDw and FL models (black, blue and red, respectively). (b) Bed elevation at glacier front. Arrow indicates direction
of retreat.
Nick and others: Calving model applied to marine outlet glaciers788
model (Vieli and others 2001; Nick and Oerlemans, 2006).
This is because in both the CD and CDw models, the calving
flux is not directly related to water depth and terminus retreat
into deeper water does not necessarily increase the calving
flux. To investigate this issue further, the CD model is used
starting from a steady-state geometry with a glacier length of
65 km and terminating on the upward bed slope shown in
Figure 1a with a constant water level of 120m in surface
crevasses. Note that for this case no floating ice tongue
occurs and the terminus is grounded. A step increase in water
level by 5 m allows crevasses to penetrate deeper and a
sudden increase in calving flux occurs, causing the terminus
Fig. 7. The modelled retreat on a bed geometry characterized by a long deep depression. Glacier retreat is forced by applying a decrease in
the accumulation rate by factor 0.7. (a) The black and red curves indicate position of the glacier front for the CD and FL models, respectively.
The dashed curve shows position of the grounding line for the CD model. (b) Bed elevation at glacier front. Arrow indicates direction of
retreat.
Fig. 8. The simulated surface profiles along the central flowline for the experiment using the CD model and forcing a retreat by decreasing
the accumulation rate by factor 0.7. The time interval between the profiles is 50years.
Nick and others: Calving model applied to marine outlet glaciers 789
to retreat into deeper water (Fig. 9a and b). However, as the
terminus retreats further, the calving flux does not increase
accordingly and, in fact, decreases rapidly as the glacier front
thickens (Fig. 9c). The retreat slows down and approaches a
new equilibrium with the grounding line on the reversed bed.
Note that for this experiment no floating tongue formed
during retreat and the terminus position coincides with the
grounding line. Glacier retreat first results in a step increase
in speed (Fig. 9d) as resistive stresses are reduced, but the
speed-up is short-lived and the velocity starts slowly
decreasing again. This reduction in frontal velocity stabilizes
ice discharge while the terminus thickens (Fig. 9e), which in
turn slows terminus retreat. Figure 9f illustrates how the ice
flux suddenly increases after increasing water level in the
surface crevasses, but stabilizes shortly thereafter. In the new
steady state, the ice flux at the terminus is greater than before
the perturbation was imposed, to compensate for the loss of
part of the ablation zone.
The important finding is that the terminus and grounding
line stabilize on a reverse bed slope, even for the case
where no floating ice tongue or ice shelf exists that could
potentially impose a buttressing force due to lateral
resistance. This contradicts common views regarding
stability of marine-based glaciers. Also, the model results
show that the system is sensitive to the formulation of the
calving model.
The model results above and presented in Figures 2–8
show little difference in response between the calving model
based on full thickness crevasse penetration (CD) and the
model based on penetration to the waterline (CDw). When
externally forced, both models predict grounding-line ad-
vance or retreat, with retreat not necessarily being unstable
or controlled by the geometry of the bed. The height-above-
buoyancy calving model cannot produce terminus advance
over a deep basal depression, and retreat, once initiated, is
halted only where the bed becomes sufficiently shallow.
Consequently, the FL model cannot reproduce significant
seasonal terminus advance and retreat as typically observed
for Greenland outlet glaciers (Howat and others, 2008a,b;
Joughin and others, 2008c). In the following set of
experiments, which explores the dynamic response to a
seasonal forcing, we therefore only present results obtained
from the CD model.
4.4. Response to seasonal forcing
Two types of seasonal forcings are imposed, namely a
periodic change in water level in surface crevasses (repre-
senting simplified seasonal variations in surface melting and
runoff), and a periodic change in the magnitude of back
pressure at the glacier terminus (reflecting seasonal changes
in the concentration of sea ice and sikkusak in front of the
calving terminus).
The initial geometry for these experiments is the steady-
state glacier with a length of 69 km, as in the retreat
experiments shown in Figures 5 and 6. The model is then run
for 200 years with seasonal forcing to reach a stable seasonal
variation in terminus position. The results in Figures 10 and
11 show the last few years of this time integration.
Fig. 9. Glacier retreat on an upsloping bed in response to a step change in water level in surface crevasses at time t= 10 years.
Nick and others: Calving model applied to marine outlet glaciers790
We first consider the dynamic response to sinusoidal
variations in water level of 20m amplitude (Fig. 10c).
Increasing the water level increases calving losses, forcing
the terminus to retreat (Fig. 10a). Retreat continues until the
water level starts decreasing and the terminus advances.
During advance, calving goes almost to zero and the
Fig. 10. Seasonal glacier variation using the CD model. Modelled ice-front position (a) and ice velocity at different locations behind the ice
front (b) for a seasonal variation in water level in surface crevasses (c).
Fig. 11. Modelled ice-front position (a) and ice velocity at different locations behind the ice front (b) in response to a seasonal variation in
back pressure at the glacier front (c).
Nick and others: Calving model applied to marine outlet glaciers 791
terminus advances with the speed of the glacier. Increasing
the water level does not promptly result in a retreat; the
glacier continues advancing until crevasse depth is sufficient
to penetrate through the glacier thickness and then retreat is
initiated. Associated with frontal advance and retreat are
periodic changes in ice velocity (Fig. 10b), with velocity
increases occurring simultaneously with terminus retreat as
a result of reduced basal and lateral resistance and greater
frontal height. Speed variations are greatest at the terminus
and are rapidly muted upstream of the calving front
(Fig. 10b).
Confining sikkusak in front of the terminus may exert a
back pressure on the glacier front (Equation (12)). As the sea-
ice concentration reduces during the summer, this back
pressure may be expected to decrease. Again, a sinusoidal
seasonal variation is imposed with 20 kPa amplitude
(Fig. 11c), resulting in cyclic response of terminus position
(Fig. 11a) and glacier speed (Fig. 11b). An increase in back
pressure lowers the stretching rate at the terminus and results
in shallower crevasse depths and lower calving rates.
Conversely, reducing or eliminating the back pressure
causes stretching rates to increase with deeper penetration
of surface crevasses and including terminus retreat. The
change in back pressure adopted here (20 kPa) may be
unrealistically large, although the observation that confining
sea ice may halt the rotation of large icebergs (Amundson
and others, 2010) suggests an appreciable back pressure
from sikkusak in Greenlandic fjords. Adopting a significantly
smaller change in back pressure does not produce a clear
seasonal glacier response.
Comparison of the results shown in Figures 10 and 11
shows that seasonal variations in water level and in back
pressure produce more or less the same dynamic glacier
response. The glacier responds dynamically to short-term
fluctuations in climate, as has been observed for some outlet
glaciers in Greenland with ~15% seasonal change in
velocity (e.g. Luckman and Murray, 2005; Joughin and
others, 2008c). It is worth noting that in these experiments
no floating ice tongue forms during seasonal advance,
indicating that the presence of such a tongue is not a
necessary requirement for terminus advance.
5. DISCUSSION
The model experiments described above show that crevasse-
depth calving criteria allow a greater range of glacier
behaviour to be modelled than has hitherto been possible.
By allowing the formation of a floating ice tongue, the model
can simulate glacier advance into deep water, unlike water-
depth and height-above-buoyancy calving models which
necessarily apply only to grounded termini (Vieli and others,
2001; Nick and others, 2007, 2009). For the height-above-
buoyancy model (as well as for the water-depth model; those
results are not shown here), the model results suggest that
retreat into deeper water, once initiated, is sustained by
increased calving until the terminus reaches water depths
sufficiently small to reduce calving and arrest the terminus
(Figs 5–8). Subsequent terminus advance through deeper
water is not possible (Figs 2 and 3) unless sedimentation at
the grounding line is invoked to reduce the local water depth
at the terminus, lowering calving rate (Nick and others,
2007). This inherently unstable behaviour is eliminated by
implementation of crevasse-depth calving criteria. Lowering
the calving rate by decreasing the amount of water in surface
crevasses or increasing accumulation in the catchment area
allows an ice shelf to form and the grounding line to advance
into or across a basal deepening (Figs 2–4). Increasing the
calving rate or lowering accumulation forces the terminus to
retreat, but this retreat is halted well before the terminus
retracts to an upward slope (Figs 5 and 6).
Our model results indicate that the retreat and advance of
calving outlet glaciers is not simply dictated by the bed
topography. According to earlier results (e.g. Vieli and
others, 2001; Nick and others, 2007, 2009), terminus retreat
into deeper water is irreversible and will be halted only
where the bed becomes shallower. The main reason for this
difference is that the current model eliminates any direct
dependence of calving rate on the local water depth or ice
thickness, and allows the system to respond to a broader
range of climatic, topographic and glaciological controls.
Another feature of the new calving model is that it allows
calving rate to be linked to seasonal variations in air
temperature or surface melting. This, in turn, results in
seasonal advance and retreat of the glacier terminus, as has
similarly been observed on Greenland outlet glaciers. In the
present simulations, seasonal variations in the amount of
water in crevasses and in back pressure are prescribed
directly, rather than these parameters being linked to climate
parameters such as air temperature and temperature of the
water in the fjord. This allows for a straightforward evalu-
ation of model sensitivity. Nevertheless, for future studies,
more realistic parameterizations or explicit modelling is
appropriate. For example, there is a time lag between
minimum sea-ice concentration and maximum air tempera-
ture, and the amount of sea ice formed during the winter will
have an impact on concentrations during the following
summer (Gough and Houser, 2005). Our estimate for the
back pressure on the glacier terminus due to the presence of
confining sikkusak is based on observed seasonal changes in
stretching rate at the terminus of Helheimgletscher (Howat
and others, 2007, fig. 2b). A more quantitative assessment is
desirable when attempting to model observed behaviour
more closely. Similarly, surface ablation likely depends on
factors other than air temperature, such as previous snowfall,
cloudiness, incoming solar radiation, etc.
Further, oceanic melt at the calving face and beneath
floating ice tongues or ice shelves has purposely not been
considered in this study, but observations suggest that it may
play a crucial role in forcing marine outlet glacier change
(Motyka and others, 2003, 2009; Thomas and others, 2003;
Shepherd and others, 2004; Holland and others, 2008).
Within the present model, basal melt can easily be included
as a forcing process beneath the floating part and should
certainly be considered and explored for realistic numerical
models of marine outlet glaciers. Recent studies indicate
that an accurate knowledge of the magnitude and more
importantly the spatial pattern of basal melt is crucial for
understanding its effect on the grounding-line dynamics
(Walker and others, 2008; O. Gagliardini and others,
unpublished information), but our current understanding
of basal melt processes is limited and prognostic models are
in their infancy.
It must be repeated that the crevasse-depth calving criteria
are not intended to represent the exact physical processes
underpinning individual calving events. Calving occurs
through a wide range of mechanisms, and the propagation
of surface and basal crevasses in response to longitudinal
stretching should only be regarded as a large-scale, first-
Nick and others: Calving model applied to marine outlet glaciers792
order control on the position of the calving front (Benn and
others, 2007a). Similarly, our approach to modelling the
overdeepening of water-filled crevasses must be regarded as
a simplification of complex processes. However, by relating
the position of calving fronts to crevasse depths, our model
provides a way of linking calving losses to ice dynamics and
surface melting in a physically plausible but workable way,
unlike previously used calving criteria which rely on poorly
tested empirical relationships.
Data presented by Mottram and Benn (2009) show that
the function for calculating the depth of dry crevasses
(Equation (3)) yields reasonable results. However, to our
knowledge, no equivalent data are available for water-filled
crevasses, or the way in which crevasse depths vary
throughout the year. Obtaining accurate field observations
of crevasse depths is not an easy undertaking, so it will be
very difficult to obtain direct observations that could be used
to test our hypothesis that surface melt rates exert a direct
control on calving losses by deepening surface crevasses.
However, some indirect validation is provided by a rough
agreement of model behaviour with observations on marine
outlet glaciers, particularly their seasonal fluctuations in
terminus position.
Calving is, of course, a 3-D process, and calving events
require both lateral and vertical propagation of fractures.
Such effects are not taken into account in our present model,
which only considers fractures in two dimensions. A future
goal, therefore, is the development of a full, 3-D, time-
evolving calving model. On the basis of our modelling
studies to date, we believe that crevasse-depth calving
criteria provide the most promising means of representing
the calving process in such models (Otero and others, 2010).
The current model experiments apply to an idealized
geometry and employ a limited range of input variables, in
order to investigate how the new calving criterion affects
glacier dynamics and behaviour. The next step should be to
apply the model to actual geometries of Greenland outlet
glaciers (e.g. Helheimgletscher or Jakobshavn Isbræ) and
simulate observed temporal variations more closely. Such
validation will be crucial and should be possible with the
much increased spatial and temporal resolution of flow
velocity, front position and thickness observations that are
becoming available from remote-sensing and field-moni-
toring programmes. Modelling exercises such as that
reported in this paper can play an important role in guiding
data collection, by highlighting which variables could be
controlling observed behaviour. Taken together, model
simulations and observations may provide important insights
into the relative importance of different forcing processes
and feedback mechanisms, and allow us to make further
significant progress on the long-standing ‘calving problem’.
6. CONCLUDING REMARK
The main conclusion that can be drawn from the model
experiments described in this contribution is that the choice
of calving model in numerical prognostic ice-flow models
crucially determines behaviour and stability of the model
glacier. This means that better understanding of processes
controlling calving from grounded and floating termini is
needed. Observations to validate proposed models are
needed, in particular seasonal progression of crevasse
depths following the onset of surface melting, but also the
inclusion of oceanic processes such as basal melt.
ACKNOWLEDGEMENTS
This paper is published with the permission of the Geo-
logical Survey of Denmark and Greenland. The research was
funded mainly by the Danish Ministry of Climate and Energy
through the Programme for Monitoring the Greenland Ice
Sheet (PROMICE), by the UK Natural Environment Research
Council (NERC) New Investigators Grant NE/E001009/1,
and in part by US National Science Foundation grants ANT-
0424589 and ARC-0520427 (University of Kansas) and the
ice2sea project funded by the European Commission’s 7th
Framework Programme through grant No. 226375. F.M.N. is
grateful for comments by H.M. Nick and D. van As, who
helped to improve the model.
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MS received 12 February 2010 and accepted in revised form 15 July 2010
Nick and others: Calving model applied to marine outlet glaciers794
... A variety of different approaches have been used to model fracture crevasse, in ice. Aside from early heuristic "zero-stress" type models (Nye, 1957;Nick et al., 2010), these are primarily discrete element models (which do not pretend to represent ice as a continuum), linear elastic fracture mechanics models, which focus on one or a few discrete cracks, and continuum damage mechanics models, which treat calving as the result of the density of microfractures accumulating to generate a macroscopic 25 crevasse that penetrates through the ice thickness (Larour and Aubry, 2004;Benn et al., 2007;Borstad et al., 2012Borstad et al., , 2013Krug explores the dependence of steady state configurations on parameter variations, allowing us to describe how changes in forcing can precipitate calving in section 4.5. In section 5, we use these results to formulate calving laws that can be used in large scale models, focusing on the distinction between calving laws that require a knowledge of the history of the ice shelf from those that can be formulated purely in terms of current forcing parameters in the form of extensional stress, ice thickness, and hydrology. ...
... The first involves a prescribed water level h w below the ice surface as previously used by van der Veen (1998a), while the second involves a prescribed water volume in the top crack. The latter is motivated by Nick et al. (2010), who use the somewhat more difficult-to-justify assumption of a prescribed water column height above the bottom of the crevasse (see also Schoof et al., 2017) one would not generally expect the column height to be prescribed in nature, while water volume might be. As we will see in this paper, we obtain very different behaviour depending on whether water level or 140 water volume is prescribed. ...
... These could be implemented in a large-scale ice sheet model, where thickness and stress are dynamical variables, and a surface hydrology model could conceivably be developed to predict water level h w . In fact, structurally, these calving laws are analogous to others such as that in Nick et al. (2010) and Schoof et al. (2017), in which calving happens at a critical thickness H that depends on extensional stress and a hydrological parameter analogous to h w : equation (37) defines an implicit relationship between H and the remaining model parameters. ...
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Calving is one of the main controls on the dynamics of marine ice sheets. We solve a quasi-static linear elastic fracture dynamics problem, forced by a viscous pre-stress describing the stress state in the ice prior to the introduction of a crack, to determine conditions under which an ice shelf can calve for a variety of different surface hydrologies. Extending previous work, we develop a boundary-element-based method for solving the problem, which enables us to ensure that the faces of crevasses are not spuriously allowed to penetrate into each other in the model. We find that a fixed water table below 5 the ice surface can lead to two distinct styles of calving, one of which involves the abrupt unstable growth of a crack across a finite thickness of unbroken ice that is potentially history-dependent, while the other involves the continuous growth of the crack until the full ice thickness is cracked, which occurs at a critical combination of extensional stress, water level and ice thickness. We give a relatively simple analytical calving law for the latter case. For a fixed water volume injected into a surface crack, we find that complete crack propagation almost invariably happens at realistic extensional stresses if the initial crack 10 length exceeds a shallow threshold, but we also argue that this process is more likely to correspond to the formation of a localized, moulin-like slot that permits drainage, rather than a calving event. We also revisit the formation of basal cracks, and find that, in the model, they invariably propagate across the full ice shelf at stresses that are readily generated near an ice shelf front. This indicates that a more sophisticated coupling of the present model (which has been used in a very similar form by several previous authors) needs modification to incorporate the effect of torques generated by buoyantly-modulated shelf flexure in the far field.
... In a floating ice shelf, iceberg calving can occur when the combined depth of surface and basal crevasses at a location reaches the full ice thickness [60]. Therefore, we consider the propagation of a surface and a basal crevasse within close proximity of each other and near the calving front. ...
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There is a need for computational models capable of predicting meltwater-assisted crevasse growth in glacial ice. Mass loss from glaciers and ice sheets is the largest contributor to sea-level rise and iceberg calving due to hydrofracture is one of the most prominent yet less understood glacial mass loss processes. To overcome the limitations of empirical and analytical approaches, we here propose a new phase field-based computational framework to simulate crevasse growth in both grounded ice sheets and floating ice shelves. The model incorporates the three elements needed to mechanistically simulate hydrofracture of surface and basal crevasses: (i) a constitutive description incorporating the non-linear viscous rheology of ice, (ii) a phase field formulation capable of capturing cracking phenomena of arbitrary complexity, such as 3D crevasse interaction, and (iii) a poro-damage representation to account for the role of meltwater pressure on crevasse growth. A stress-based phase field model is adopted to reduce the length-scale sensitivity, as needed to tackle the large scales of iceberg calving, and to adequately predict crevasse growth in tensile stress regions of incompressible solids. The potential of the computational framework presented is demonstrated by addressing a number of 2D and 3D case studies, involving single and multiple crevasses, and considering both grounded and floating conditions. The model results show a good agreement with analytical approaches when particularised to the idealised scenarios where these are relevant. More importantly, we demonstrate how the model can be used to provide the first computational predictions of crevasse interactions in floating ice shelves and 3D ice sheets, shedding new light into these phenomena. Also, the creep-assisted nucleation and growth of crevasses is simulated in a realistic geometry, corresponding to the Helheim glacier. The computational framework presented opens new horizons in the modelling of iceberg calving and, due to its ability to incorporate incompressible behaviour, can be readily incorporated into numerical ice sheet models for projecting sea-level rise.
... In a floating ice shelf, iceberg calving can occur when the combined depth of surface and basal crevasses at a location reaches the full ice thickness [60]. Therefore, we consider the propagation of a surface and a basal crevasse within close proximity of each other and near the calving front. ...
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Plain Language Summary Geothermal heat flow (GHF) is important in controlling both the ice temperature and the production of meltwater at the base of the Antarctic Ice Sheet, which impacts how rapidly ice flows. However, GHF estimates are generally low resolution and highly uncertain. This uncertainty in GHF impacts the reliability of model simulations of the flow of ice and meltwater into the ocean, which in turn impacts estimates of the contribution of ice sheets to future sea level rise. We use an ice sheet model to investigate how spatial variations in GHF impact meltwater production in the Aurora Subglacial Basin (ASB), East Antarctica. GHF fields with spatial variations lead to consistently higher melt rates at the ice sheet base than a constant GHF field with the same mean value. We determine the minimum heat at the ice sheet base in the ASB required to cause the ice to melt, and highlight regions where small variations in GHF will have greater impacts on ice melting. These results show where measurements to constrain GHF should be prioritized to improve ice sheet model simulations.
... However, many existing calving front position parameterizations are without a convincing physical basis (for example, the calving laws based on height or thickness above floatation), while calving rate parameterizations like the von Mises stress formulation are desirable because they take the stress and strain rate fields into account. The crevasse depth model (Benn et al., 2007;Nick et al., 2010) should evolve in time as surface melt increases. We have not used it here for these reasons, but we acknowledge that it could alleviate some of the problems we find with the von Mises calving law, which could potentially justify accepting a decreased ability to match observed retreat rates. ...
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Humboldt Glacier, North Greenland, has retreated and accelerated through the 21st century, raising concerns that it could be a significant contributor to future sea-level rise. We use a data-constrained ensemble of three-dimensional higher-order ice sheet model simulations to estimate the likely range of sea-level rise from the continued retreat of Humboldt Glacier. We first solve for basal traction using observed ice thickness, bed topography, and ice surface velocity from the year 15 2007 in a partial differential equation constrained optimization. Next, we impose calving rates to match mean observed 2007-2017 retreat rates in a transient calibration of the exponent in the power-law basal friction relationship. We find that power law exponents in the range of 1/7-1/5-rather than the commonly used 1/3-1-are necessary to reproduce the observed speedup over this period. We then tune an iceberg calving parameterization based on the von Mises stress yield criterion in another transient calibration step from 2007-2017 to approximate both observed ice velocities and terminus 20 position in 2017. Finally, we use the range of basal friction relationship exponents and calving parameter values to generate the ensemble of model simulations from 2007-2100 under three climate forcing scenarios from CMIP5 (two RCP 8.5 forcings) and CMIP6 (one SSP5-8.5 forcing). Our simulations predict 5.5-9.2 mm of sea-level rise from Humboldt Glacier, significantly higher than a previous estimate (~3.5 mm) and equivalent to a substantial fraction of the 40-140 mm predicted by ISMIP6 from the whole Greenland Ice Sheet. Our larger future sea-level rise prediction results from the transient 25 calibration of our basal friction law to match the observed 2007-2017 speedup, which requires a semi-plastic bed rheology. In many simulations, our model predicts the growth of a sizable ice shelf in the middle of the 21st century. Thus, atmospheric warming could lead to more retreat than predicted here if increased surface melt promotes hydrofracture of the ice shelf. Our data-constrained simulations of Humboldt Glacier underscore the sensitivity of model predictions of Greenland outlet glacier response to warming to choices of basal shear stress and iceberg calving parameterizations. Further, 30 transient calibration of these parameterizations, which has not typically been performed, is necessary to reproduce observed behavior. Current estimates of future sea-level rise from the Greenland Ice Sheet could, therefore, contain significant biases.
Article
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Mass loss from ice shelves is a strong control on grounding-line dynamics. Here we investigate how calving and submarine melt parameterizations affect steady-state grounding-line positions and their stability. Our results indicate that different calving laws with the same melt parameterization result in more diverse steady-state ice-sheet configurations than different melt parameterizations with the same calving law. We show that the backstress at the grounding line depends on the integrated ice-shelf mass flux. Consequently, ice shelves are most sensitive to high melt rates in the vicinity of their grounding lines. For the same shelf-averaged melt rates, different melt parameterizations can lead to very different ice-shelf configurations and grounding-line positions. If the melt rate depends on the slope of the ice-shelf draft, then the positive feedback between increased melting and steepening of the slope can lead to singular melt rates at the ice-shelf front, producing an apparent lower limit of the shelf front thickness as the ice thickness vanishes over a small boundary layer. Our results illustrate that the evolution of marine ice sheets is highly dependent on ice-shelf mass loss mechanisms, and that existing parameterizations can lead to a wide range of modelled grounding-line behaviours.
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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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Laterally confined marine outlet glaciers exhibit a diverse range of behaviours. This study investigates time-evolving and steady configurations of such glaciers. Using simplified analytic models, it determines conditions for steady states, their stability and expressions for the rate of the calving-front migration for three widely used calving rules. It also investigates the effects of ice mélange when it is present. The results show that ice flux at the terminus is an implicit function of ice thickness that depends on the glacier geometric and dynamic parameters. As a consequence, stability of steady-state configurations is determined by a complex combination of these parameters, specifics of the calving rule and the details of mélange stress conditions. The derived expressions of the rate of terminus migration suggest a non-linear feedback between the migration rate and the calving-front position. A close agreement between the obtained analytic expressions and numerical simulations suggests that these expressions can be used to gain insights into the observed behaviour of the glaciers and also to use observations to improve understanding of calving conditions.
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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.
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Inaccurate representations of iceberg calving from ice shelves are a large source of uncertainty in mass-loss projections from the Antarctic ice sheet. Here, we address this limitation by implementing and testing a continuum damage-mechanics model in a continental scale ice-sheet model. The damage-mechanics formulation, based on a linear stability analysis and subsequent long-wavelength approximation of crevasses that evolve in a viscous medium, links damage evolution to climate forcing and the large-scale stresses within an ice shelf. We incorporate this model into the BISICLES ice-sheet model and test it by applying it to idealized (1) ice tongues, for which we present analytical solutions and (2) buttressed ice-shelf geometries. Our simulations show that the model reproduces the large disparity in lengths of ice shelves with geometries and melt rates broadly similar to those of four Antarctic ice shelves: Erebus Glacier Tongue (length ~ 13 km), the unembayed portion of Drygalski Ice Tongue (~ 65 km), the Amery Ice Shelf (~ 350 km) and the Ross Ice Shelf (~ 500 km). These results demonstrate that our simple continuum model holds promise for constraining realistic ice-shelf extents in large-scale ice-sheet models in a computationally tractable manner.
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An experimental study of the fracture mechanics and rheology of ice from the Ronne Ice Shelf, Antarctica, is currently being undertaken. The apparent critical stress-intensity factor (or apparent fracture toughness, K Q) for crack propagation has been measured using a three-point bend method for inducing crack growth perpendicular to the axis of cylindrical ice-core specimens. Tensile crack nucleation under applied uniaxial compressive stress has also been evaluated. Both methods have allowed a profile of ice elastic and fracture properties with depth through the ice shelf to be constructed. Preliminary results indicate that the measured elastic modulus increases with depth through the firn and upper meteoric ice before reaching a constant value in the deeper, dense meteoric and basal marine ice. The resistance to fracture, as measured by changes in apparent fracture toughness and crack-nucleation stress, increases with depth right through the firn and meteoric ice layers. A simple fracture mechanics model applied to the Ronne Ice Shelf indicates that crevasses form from small surface cracks, less than 40 cm deep, which quickly grow to depths of 40–60m and then remain stable.
Article
A simple model is developed based on the notion that on active ice streams the resistance to flow is partitioned between basal drag and lateral drag. The relative roles of these sources of resistance is determined by a friction parameter that effectively describes the strength of the bed under the ice stream. Reduction in the basal strength is caused by meltwater production, taken proportional to the product of basal drag and ice speed. The width of the ice stream is governed by the balance between entrainment or erosion of ice from the slow-moving inter-stream ridges and advection from the ridges into the ice stream. Entrainment of ridge ice is parameterized as a function of the shear stress at the lateral margins, in one case proportional to the lateral shear stress and in the second case scaled to ice-stream width. In the first formulation, the model rapidly becomes unstable but, using the second formulation, a steady state is reached with lateral drag providing all or most of the resistance to flow. The results point to the great importance of achieving an understanding of entrainment. With the second model and a wide range of parameter values, there is no cyclic behavior, with rapid flow being followed by a quiescent phase.
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An approximate calculation is made of the rate at which a bottom crevasse in a cold ice shelf or tabular iceberg can close shut by freezing of water and can creep open through the creep deformation of ice. In all but the thickest ice shelves and icebergs, those with a thickness greater than about 400 m, the freezing process is the more important mechanism if the ice is cold (
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The elastic stress intensity factor is a parameter used in fracture mechanics to describe stress conditions in the vicinity of the tip of a sharp crack. By superimposing solutions of stress intensity factors for different loading conditions, equations are derived which model crevasses in ice. Solutions are presented for the theoretical depth of isolated crevasses, free from or partially filled with water. Close agreement exists with a previous calculation by Weertman using a different technique. The effect of crevasse spacing is investigated and it is demonstrated that closer spacing always reduces crevasse depth, but over a wide range of spacing the predicted variation in depth is slight.
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An important phase in a project to utilise icebergs as a freshwater source is the separation of the ‘berg into smaller pieces to promote faster melting and easier transport. Any scheme involving cleavage should produce the greatest savings of energy. This paper introduces, in a simple way, the subject of fracture mechanics, which enables estimates to be made of the forces required for various cleaving methods. It is shown that the huge forces required to cleave an iceberg arise not from the strength of the ice itself, but from compressive stresses due to the ice weight. A technique involving wedging with water pressure is shown to be promising in so far as it should be possible to extend a crack at least to the water line in this way. However, formidable engineering problems remain to be investigated before claims could be made that a solution to the problem has been found.
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Fundaments of Glacier Dynamics presents an introduction to modelling the flow and dynamics of glaciers. The emphasis is more on developing and outlining procedures than on providing a complete overview of all aspects of glacier dynamics. Derivations leading to frequently-used equations are presented step-by-step to allow the reader to grasp the mathematical details and approximations involved and gain the understanding needed to apply similar concepts to different applications. The first four chapters discuss the background and theory needed for glacier modelling. The central part of the book discusses simple analytical solutions and time-evolving numerical models that are used to study general aspects of glacier dynamics and important feedback mechanisms. The final three chapters discuss applications specific to smaller mountain glaciers, the Greenland Ice Sheet, and the Antarctic Ice Sheet, respectively. This book will be suitable for graduate courses in geophysics and will also serve as a reference volume for scientists active in all aspects of glaciology and related research. Standard undergraduate mathematics and physics are sufficient background for studying the text.
Article
Nye has estimated that the depth L of crevasses is equal to Tjog, where T is the tensile stress causing extending flow, Q is the density of ice, and g is the gravitational acceleration. This expression for L is derived on the assumption that the crevasses are closely spaced and free of water. It is shown in this paper that the depth of an isolated crevasse is a factor sr/2 greater than the depth calculated by Nye for closely spaced crevasses. It is shown further that the presence of water in a crevasse can increase its depth. A crevasse filled with water up to at least 97.4% of its depth can penetrate to the bottom of a glacier. Water-filled cavities can exist directly beneath water filled crevasses. RÉSUMÉ. Une crevasse emplie d'eau peut-elle atteindre la surface basale d'un glacier ? Nye a estimé que la profondeur L des cre-vasses est égale à Tjog, où Test la contrainte d'élongation causant un écoulement d'extension, est la densité de la glace et g est l'accélération de la pesanteur. Cette expression pour L est dérivée en supposant que les crevasses sont proches et vides d'eau. On montre ici que la profondeur d'une crevasse isolée est plus grande d'un facteur de JT/2 que la profondeur calculée par Nye pour les crevasses proches l'une de l'autre. Il est montré de plus que la présence d'eau dans une crevasse peut augmenter sa profondeur. Une crevasse emplie d'eau d'au moins 97,4% de sa profondeur peut pénétrer jusqu'au fond d'un glacier. Des cavités pleines d'eau peuvent exister directement sur des crevasses emplies d'eau.
Article
Calving speed is the difference between glacier speed and the rate of terminus advance. The mean yearly calving speed calculated in this way for 12 glaciers in Alaska ranges from 220 to 3700 m/a. Yearly calving speeds estimated by using balance flux and thinning flux for three additional glaciers that recently underwent rapid retreat extend the range of calving speed to 12 500 m/a. A statistical analysis of calving speed and mean yearly values for water depth, cliff height, and glacier thickness at the terminus indicates that calving speed is fit best by a simple proportionality to average water depth at the terminus.-from Authors
Article
Bottom crevasses are fractures that extend upward into floating ice shelves. They form when seawater penetrates the base of the ice shelf and ruptures the ice up to the level at which englacial stresses equal the stress of the seawater. For a freely floating ice shelf, the penetrating level of closely spaced crevasses is estimated at about half the ice thickness h; for an isolated crevasse the level is about πh/4. However, an analysis of the heights and locations of bottom crevasses in the Ross Ice Shelf shows that none of the crevasses approach the predicted limits, perhaps because the existing theory does not include the back stress σb which is present in bounded ice shelves. By reformulating the theory to include a back stress term, σb can be evaluated experimentally from radar measurements of crevasse height and ice thickness. The magnitude of σb (2 bars in the grid northwest corner of the ice shelf) suggests the ice shelf is playing an important role in buttressing the inland ice sheet.