Introduction
Marine-terminating glaciers export ice from the interior of the Greenland Ice
Sheet (GrIS) through deep valleys terminating in fjords .
Mass loss from the GrIS has increased significantly during the last two
decades, contributing increasingly to sea-level rise . The
observed increase in mass loss has broadly been associated with large-scale
atmospheric and oceanic warming
.
About half of the current mass loss from the GrIS is due to dynamic ice
discharge , which is impacted by several processes partly
linked to air and ocean temperatures. A warmer atmosphere enhances surface
runoff, which may cause crevasses to penetrate deeper through
hydrofracturing, which in turn can promote iceberg calving
. A warmer ocean
strengthens submarine melt below ice shelves and floating tongues
, which can potentially
destabilize the glacier via longitudinal dynamic coupling and upstream
propagation of thinning . Increased air and
fjord temperatures can additionally weaken sea ice and ice mélange in
fjords, affecting calving through altering the stress balance at the glacier
front . Most of these processes are still
poorly understood, as well as heavily spatially and temporally undersampled
e.g.,.
Despite widespread acceleration and retreat around the GrIS, individual
glaciers correlate poorly with regional trends
. For example, four glaciers alone have
accounted for 50 % of the total dynamic mass loss since 2000; Jakobshavn
Isbræ in West Greenland is the largest contributor .
Even if exposed to the same climate, individual glaciers can respond
differently, because inland mass loss can be regulated by individual glacier
geometry . It is well known that grounding line stability
and ice discharge is highly dependent on trough geometry, with retrograde
glacier beds potentially causing unstable, irreversible retreat
e.g.,. The impact of
glacier width, however, is less studied. Lateral buttressing
and topographic bottlenecks
have been
suggested to stabilize grounding lines on reverse bedrock slopes. Despite
these studies showing the importance of geometry, limited knowledge is
available of the interplay between bedrock geometry, channel-width variations
and external controls on glacier retreat. A poor understanding of the
heterogeneous response of individual glaciers inhibits robust projections of
sea-level rise due to mass loss from ice sheets. So far, there has been a
strong emphasis on the role of ice–ocean interactions as a key control on the
retreat of marine-terminating glaciers, disregarding the influence of trough
geometry
e.g.,.
Also, studies that focus on the control of geometry so far only model
synthetic glaciers e.g.,, prohibiting
validation and justification of model parameters. In this paper, we therefore
use a real-world glacier geometry to study the geometric controls on glacier
retreat.
Several attempts to model Jakobshavn Isbræ have been made to understand
the dynamics behind the observed acceleration and retreat
. These studies
focus on the time period after 1985 and partly into the future. However,
given the current exceptional rapid changes, our understanding and model
capacity should span long (centennial) timescales if we are to predict
changes into the future. Jakobshavn Isbræ has a history of stepwise and
nonlinear retreat. We aim to understand this history, by comparing our model
results with observations starting with the Little Ice Age maximum (LIA; ca.
1850) and into the present.
Since the deglaciation of Disko Bugt
between 10 500 and 10 000 years before
present , Jakobshavn Isbræ has
experienced alternating periods of fast and slow retreat with the formation
of large moraine systems e.g., at Isfjeldsbanken, Fig.
;. Most observations exist after
the LIA (Fig. ), when the glacier reached a
temporal maximum extent followed by a retreat. From 2001 until May 2003 it
accelerated significantly after the disintegration of its 15 km long
floating tongue . Today,
it is the fastest flowing glacier in Greenland , with a
maximum velocity of 18 kmyr-1 measured in summer
2012; and ice discharge rates of about
27–50 km3yr-1
. With a contribution of
4 % to global sea-level rise in the 20th century (IPCC, 2001), Jakobshavn
Isbræ is the largest contributor in Greenland . It is
also one of the most vulnerable glaciers in Greenland, with recent thinning
potentially propagating as far inland as one third of the distance across the
entire ice sheet . Combining these centennial
observations with dynamic ice flow modeling is crucial for putting the recent
dramatic changes into a long-term perspective, as well as for interpreting
records of the past and projections for the future.
Glacier front positions of Jakobshavn Isbræ from
(1850–1985) and CCI products derived from ERS, Sentinel-1
and Landsat data by ENVO (1990–2016). The background map is a Landsat-8
image from 16 August 2016 (from the U.S. Geological Survey). Location names
that occur in the text are marked. The inset shows the location of Jakobshavn
Isbræ in Greenland.
The aim of this study is to investigate the external, glaciological and
geometric controls on Jakobshavn Isbræ in response to a linear forcing on
a centennial timescale. We use a simple numerical ice flow model
e.g., with a fully dynamic treatment of the
calving front to assess the relative impact of fjord geometry and climate
forcing on the retreat of Jakobshavn Isbræ from the LIA maximum to
the present day. Geometric controls are isolated by (a) using a linear forcing
to avoid complex responses and (b) artificially straightening the trough
width and depth. The model experiments are run over several centuries to
account for internal glacier adjustment. The application of the model on a
real glacier enables a comparison of model results with long-term observed
velocities and front positions, but also ensures the use of realistic values
for the width–depth ratio and the model parameters.
Section documents the numerical ice flow model, followed by
an outline of the specific model setup used for the simulations in
Sect. . Section describes the results
of the experiments with varying climate forcing and fjord geometry. The
importance of trough width versus depth and forcing is discussed in
Sect. , followed the limitations of the model and the
implications of our results for understanding the past.
Modeling approach
We use a dynamic depth- and width-integrated numerical ice
flow model constructed for marine-terminating glaciers
. Despite many assumptions
required, this model is well suited to study the long-term (centennial)
retreat pattern of an outlet glacier with high basal motion (such as
Jakobshavn Isbræ). It is based on mass continuity and a balance between
the driving stress, longitudinal stress gradient, and basal and lateral drag.
The model benefits from a robust treatment of the grounding line
consistent with and a fully dynamic
marine boundary . It is also more efficient than complex
models , which enables multiple model runs
covering several centuries. The physical calving law applied in the model has
been successfully tested on several outlet glaciers where there are
observational data available . The calving law also has the
advantage of allowing for a dynamic and free migration of the glacier
terminus, given changes in climate forcing. The climate forcing is
implemented as a slow linear change in surface mass balance (SMB), crevasse water
depth, submarine melt and buttressing by sea ice – model parameters that
represent the impact of changes in temperature. In this section, the physical
approach, parameterizations and the implementation of climate forcing are
described.
Numerical ice flow model
The numerical ice flow model as described in calculates the
time-varying ice thickness H from the along-flow ice flux and mass balance,
using a depth- and width-integrated continuity equation:
∂H∂t=-1W∂(HUW)∂x+B˙.
U is the width- and depth-averaged velocity, t the time and x the
along-flow component. The width W is assumed to be symmetric around the
central flow line. The mass balance B˙ includes the surface mass
balance and submarine melt below the floating tongue (described in
Sect. ).
The ice flux is controlled by a balance of lateral and basal resistance,
along-flow longitudinal stress gradient and driving stress. Lateral
resistance is parameterized using a width-integrated horizontal shear stress
and we use a Weertman-type basal sliding law based on
effective pressure . The longitudinal stress gradient is
dependent on the effective viscosity ν, which is nonlinearly dependent
on the longitudinal strain rate ϵ˙xx and the rate factor A
. The stress balance is calculated as
2∂∂xHν∂U∂x-AsH-ρsρiDU1/m-2HW5UEAW1/n=ρigH∂s∂x,
where s is the surface elevation; g is the gravitational acceleration; D
is the depth of the glacier below sea level; and ρi and ρs are the
densities of ice and ocean water, respectively. n and m are the exponents
for Glen's flow law and sliding relations, respectively. The lateral
enhancement factor E, controlling the lateral resistance, and the basal
sliding parameter As are model parameters that are adjusted to roughly
match the observed ice flow and thickness for the present fjord geometry.
Both parameters are constant along the flow line and in time. The dependency
of the basal resistance on effective pressure is accounted for through the
term H-ρsρiD.
The grounding line position is calculated with a flotation criterion based on
hydrostatic balance . Its treatment relies on a moving
grid: at each time step the grid adjusts freely to the new glacier length,
continuously keeping a node at the calving front
. This allows for a precise simulation of
the glacier front and grounding line position using high grid resolution. The
grid size is Δx = 302 m initially and reduces to Δx = 292 m
at the present-day position due to the use of a stretched grid. At the marine
terminus, a dynamic crevasse-depth calving criterion is used as described in
Sect. .
List of variables, physical parameters and constants used in the
model. The forcing parameters with their initial (LIA) values are given in
the lower part. Parameter values used for the glacier retreat experiments are
listed in Table .
Symbol
Definition
Value
Unit
H
glacier thickness
m
t
time
yr
W
glacier width
m
x
along-glacier coordinate
m
U
velocity
myr-1
B
mass balance
myr-1
ν
viscosity
Payr
D
depth below sea level
m
s
surface elevation
m
db
depth of basal crevasses
m
ds
depth of surface crevasses
m
Rxx
tensile deviatoric stress
Pa
ϵ˙xx
longitudinal strain rate
myr-2
QL
lateral ice flux
myr-1
a
surface mass balance (SMB)
myr-1
s0
transition height for SMB
1600
m
g
gravitational acceleration
9.8
myr-1
ρi
ice density
900
kgm-3
ρs
ocean water density
1028
kgm-3
ρw
fresh water density
1000
kgm-3
m
sliding exponent
3
n
Glen's flow law exponent
3
A
rate factor taken from
A(-20 ∘C) –
yr-1Pa-3
A(-5 ∘C)
As
basal resistance parameter
120
Pam-2/ms-1/m
E
lateral enhancement
10
dx
grid size
250–300
m
dt
time step
0.005
yr
Perturbation parameters with their initial LIA values
m
submarine melt rate
175
myr-1
dw
crevasse water depth
160
m
Gl
lower SMB gradient
0.0011
myr-1
Gu
upper SMB gradient
-0.002
myr-1
a0
maximal SMB
0.64
mw.e.yr-1
fi
sea ice buttressing factor
1
Calving law
The fully dynamic crevasse-depth criterion calculates
calving where the sum of surface and basal crevasse depth (ds and
db, respectively) penetrates the whole glacier thickness
. The depth of the surface crevasses is given by
ds=Rxxρig+ρwρidw, with Rxx=2ϵ˙xxA1/n.
The depth of the surface crevasses is calculated from the tensile deviatoric
stress Rxx and the pressure from melt water filling up the crevasses
(Eq. 3) . Note that the water depth in crevasses
dw is not a physical quantity, but a forcing parameter within the
calving model that links calving rates to climate. ρw is the
density of freshwater. The tensile deviatoric stress is the difference
between tensile stresses that pull a fracture open and the ice overburden
pressure. It is calculated via Glen's flow law from the longitudinal
stretching rate ϵ˙xx, which is responsible for the opening
of crevasses by
ϵ˙xx=∂U∂x=fiAρig4H-ρsρiD2Hn,
which depends on a sea ice factor fi, accounting for reduced buttressing
due to weakening of ice mélange. The depth of basal crevasses is calculated
from tensile deviatoric stresses and the height above buoyancy
:
db=ρiρs-ρiRxxρig-(H-ρsρiD).
Water in crevasses and sea ice buttressing are both model parameters that
impact the glacier response by changing the calving rate. Because the
parameters are linked to different processes, they are kept separate in the
model to enable a distinct forcing.
Atmosphere and ocean forcing
The model SMB, a, is derived from observed monthly mean SMB data at
Jakobshavn Isbræ . The SMB data are based on a combination
of meteorological station records, ice cores, regional climate model output
and a positive degree-day model. Its implementation in our model consists of
a piecewise linear function of surface elevation separated by a transition
height s0: in the steep lower part of the glacier, the SMB increases with
elevation; and, in the flat upper part of the glacier, where the precipitation
is low, the SMB decreases with elevation (Eq. ).
a(x)=a0-dadx⋅s0+dadx⋅s(x), with dadx=Glat s(x)≤s0Guat s(x)>s0.
SMB profiles along Jakobshavn Isbræ's main flow line at the LIA
(1840–1850 average) and present day (2002–2012 average) from observations
by and the linear fit used in the model. Thin dotted lines
show position of the equilibrium line altitude (ELA) for the present-day and
LIA fit.
Figure shows the observed and estimated linear SMB profiles for
the LIA (1840–1850 average) and for the present day. The corresponding values
for the vertical gradients Gl and Gu as well as the SMB a0 at
the height s0 are given in Tables and
.
Submarine melt is implemented in the model as a vertical melt rate that
decreases the glacier thickness seaward of the grounding line and is assumed
to be spatially uniform. The induced artificial step decrease in ice
thickness at the grounding line is smoothed out in the model by a
sufficiently small time step. The submarine melt rates are
one order of magnitude smaller than the grounding line flux.
Sensitivity analyses with along-flow variations in submarine melt
show similar results, as long as the constant submarine
melt rate is comparable to the along-flow averaged submarine melt rate.
Lateral ice flow
The model domain covers the full drainage basin towards the ice divide at
about 520 km upstream of the present-day position. For the lowermost 77 km,
we restrict the model width to the pronounced narrow channel seen in bed
topography data to realistically account for lateral and basal stresses.
Lateral ice flow into this narrow channel from the surrounding ice sheet and
tributary glaciers is implemented as an additional SMB similar to previous
studies , giving a realistic mass flux
into the lower channel. This lateral influx QL,0 is initially
calculated as the sum of the northern and southern lateral fluxes. These are
given by the observed ice velocity UL,0 and thickness HL,0
at each grid point along the lateral
boundary of the narrow main channel, divided by the width of the main trough
WJI (Eq. ). The strength of the initial influx
is indicated by the arrows in Fig. and locally accounts for
about 100 times the SMB, with a maximum of 120 myr-1.
We assume that the relative contribution of the lateral flux to the overall
flux is constant in time; therefore, we scale it with the change in the
overall flux with time (Eq. ).
QL,0(x)=UL,0(x)⋅HL,0(x)WJI(x)QL,t(x)=QL,0(x)⋅QJI,t(x)QJI,0(x)
QJI,0 and QJI,t are thereby the initial overall flux through the
main trunk and the flux after time t, respectively. Note that the constant
relative contribution by side fluxes is a rough approximation. A thinning of
the main trunk could initiate a speed-up in the tributary glaciers due to
increased surface slope.
Model setup
Despite the general focus of this study on the
external versus geometric controls on glacier retreat, we apply the model to
Jakobshavn Isbræ – a well-studied glacier on west Greenland. The
intention is to use a realistic along-flow glacier geometry to compare
modeled ice thickness, length and velocity with observations.
Observations of ice velocities, calving front positions, ice thickness and
ice discharge are used to
tune model parameters. In the following, we distinguish between constant
parameters (basal sliding, rate factor and lateral enhancement factor) and
climate-related perturbation parameters (SMB, submarine melt rate, crevasse
water depth and sea ice buttressing). For the model experiments, the
perturbation parameters are changed linearly from their LIA values to
simulate increasing temperatures. Importantly, the calving front and
grounding line evolve freely during retreat. Only combinations of forcing
parameters that simulate a total retreat rate matching the observed retreat
of about 43 km from the LIA to 2015 are considered. In the following, the
choice of tuning parameters and the perturbations are described, together
with relevant observations.
Model glacier geometry
Jakobshavn Isbræ extends 520 km inland
towards the ice divide and can be distinguished from the surrounding ice
sheet by its high velocities along the deep trough. The geometry of the model
glacier consists of a narrow (in average about 5.4 km wide) and deep
(1.3 km at the deepest) trough; further upstream, it widens gradually with a
relatively flat and shallow bottom. The fjord width in today's ice-free
area is obtained from satellite images (Fig. ).
The channel width in the fast-flowing part (77 km upstream of the 2015
position) is defined as the trough width at the present-day sea level from
topography data by . Further upstream, where the
catchment widens gradually, the width is defined following .
For the one-dimensional glacier depth in the deep trough and fjord, we use
the along-flow bed topography profile as it is presented in
. The fjord bathymetry is obtained from Operation
IceBridge gravity data and the subglacial trough profile from
high-sensitivity radar data by . For the bed in the wider
catchment area, 150 m resolution data by are averaged
over the glacier width.
Constant parameters
Most observations only exist for the present day. Parameters that are constant in
time (basal resistance, lateral enhancement factor and rate factor) are tuned
with observations to obtain a steady-state glacier corresponding to the
observed present-day glacier geometry. After tuning the constant parameters,
the climate-related perturbation parameters are reduced to colder
temperatures to achieve an initial steady state corresponding to the observed
LIA front position. For the LIA steady state, the only constraints are given
by the LIA front position and the height of the LIA trimline
found at the GPS station KAGA Fig. ;
by .
Basal sliding – as implemented in the model – influences ice flow and hence
the surface slope and thickness. The basal sliding parameter
As=120Pam-2/3s-1/m is chosen to achieve an observed
present-day thickness of 3065 m at the ice divide ; the
present-day thickness in the interior is also valid for the LIA
initialization as the ice sheet is assumed to be in a steady state above
2000 m of elevation within this time period . We keep the
basal sliding parameter constant in time, because the impact of increased
melt on basal sliding on interannual timescales is still unclear
. Also, the model takes into account the
dependency of basal sliding on the effective pressure, which is calculated
explicitly. The actual degree of basal resistance at the bed of Jakobshavn
Isbræ is highly debated, with some studies explaining high surface
velocities as reflecting a slippery bed ,
whereas other studies ascribe the high velocities to weakened shear margins
e.g.,, or an interplay of both processes
.
The surface profile and ice velocity are determined by the lateral resistance
and the rate factor. A uniform lateral enhancement factor is applied along
the entire glacier, controlling the strength of the transmission of lateral
drag to the sides. A value of E=10 gives a simulated present-day surface
with a best fit to observations . The rate factor for Glen's
flow law is to a first approximation a function of ice temperature. Here it
is set to values corresponding to temperatures of -20 ∘C at the
ice divide, linearly increasing to -5 ∘C at the terminus
. This gives a good fit of simulated present-day glacier
surface and ice velocities to observations . The
rate factor is kept constant in time.
Forcing experiments and perturbation parameters
The climate-related perturbation parameters are tuned for the LIA
steady state to simulate the observed glacier length and velocities, or the
ice discharge. Starting from the initial LIA glacier configuration, a retreat
is triggered by simultaneous linear changes in SMB, crevasse water depth,
submarine melt rate and sea ice buttressing. The parameter perturbations are
combined in order to obtain a total retreat of 43 km from 1850 to 2015,
corresponding to the observed retreat. Nine different parameter combinations
that satisfy the observational data, and cover a wide range of perturbations,
are presented here. Table shows the parameter values
reached by the year 2015 in the nine different model runs.
SMB is the only purely physical and well-known variable both for LIA and
today . The piecewise linear function presented in Sect.
is a good approximation to the observed profiles (Fig. ) and is therefore used here. All model experiments use the same
gradual changes of the SMB gradients and maximal SMB from the LIA values to
present-day values (Table ).
Submarine melt is influenced by ocean temperatures. In Disko Bugt, ocean
temperatures have increased from about 1.5 ∘C in 1980 to
3 ∘C in 2010 , including a 1 ∘C warming in 1997 .
estimates a doubling of melt rates underneath the floating tongue of
Jakobshavn Isbræ (depending on initial conditions and the way in which
melting is applied), when considering a 1 ∘C warming and
steepening of the glacier front. Submarine melt rates may be additionally
enhanced by increased subglacial ice discharge
, although this may be a local
effect and negligible when width averaged . Observations of
submarine melt rates beneath Jakobshavn Isbræ's floating tongue suggest
an annual melt rate of 228 ± 49 myr-1 between 1984 and 1985
and 2.98 md-1 (1087 myr-1)
averaged over the melt seasons in 2002 and 2003 . Since
the submarine melt rate is otherwise poorly constrained, especially further back
in time, we employ a large range of linear forcing, from no increase to a
2-fold increase in the LIA value of 175 to
340 myr-1 in 2015. Note that the model neglects submarine melt
at the vertical calving front.
The crevasse-water depth has not been measured and is applied as a
nonphysical model parameter regulating discharge fluxes. It is likely to be
exaggerated in the model, accounting for the lack of submarine melt at the
vertical glacier front. For the LIA steady state, the crevasse water depth is
set to 160 m, which produces a calving rate of 34 km3yr-1 in
1985 after the applied linear forcing. This is the same order of magnitude as
the observed calving rate of 26.5 km3yr-1 in 1985
, as well as the more recent values between 24 and
50 km3yr-1 . The
increase in crevasse water depth with time is unknown, but may be related to
runoff, which has increased by 63 % since the LIA . To
account for such a large range, we increase the crevasse water depth from its
LIA value to values between 185 and 395 m in 2015. The
crevasse water depth is tuned depending on the combination of sea ice
buttressing and submarine melt rate to reach the observed retreat (Table ).
Ice mélange in the fjord can apply a buttressing stress to the calving
front of about 30–60 kPa, or one-tenth of the driving stress
. With increasing air and ocean temperatures, ice mélange
can weaken or break up, thereby influencing iceberg calving
e.g.,. However, the correlation between ice
mélange and iceberg calving is poorly known. Breakup of ice mélange is
thought to impact frontal migration on a daily to seasonal timescale, leaving
annual fluxes unaffected
. We conduct
experiments with unchanged buttressing by sea ice (fs=1; also used for
the LIA steady state), as well as decreased buttressing by a factor of 2 and
3 compared to the LIA value in 2015.
Nine combinations of the perturbation parameters used in this study.
Values shown here are those reached in 2015 after a linear perturbation from
their LIA value shown in Table . Values for the SMB are
perturbed to the same 2015 values for all model runs: Gl=0.0019yr-1, Gu=-0.00013yr-1, a0=0.64mw.e.yr-1. Run 5 (in bold) is presented in more detail
in the paper.
Run ID
fs
m
dw
(myr-1)
(m)
Initial steady-state values in year 1850
0
1
175
160
Linear forcing: values reached in 2015
1
1
180
395
2
1
260
370
3
1
340
340
4
2
180
295
5
2
260
275
6
2
340
255
7
3
180
225
8
3
260
210
9
3
340
185
Step forcing: values applied in 1850
10
2
260
250
The observed retreat position in 2015 is reached with all the parameter
combinations presented in Table . The 2015 values for
each parameter depend on the values for the other parameters. This means,
for example, that in the case of reduced sea ice buttressing and a small crevasse water
depth, a low submarine melt rate is needed. Similarly, if sea ice buttressing
is high and submarine melt is low, the crevasse water depth must be large.
In addition to experiments with linearly increased parameters, we also
conduct one experiment with a step increase in the four parameters starting
from the LIA maximum. The step increases in sea ice buttressing, submarine
melt rate, crevasse water depth and SMB applied starting at 1850 are
comparable to those reached in the model year 2015 in run 5, with slightly
different values to reach the right front position in 2015. All experiments
shown in Table are run until 2100 in order to test the
temporal and spatial response to the underlying geometry.
Despite a relatively high number of frontal observations since the LIA (Fig. ), only the observed calving front positions in
1850 and 2015 are used to tune the model parameters; in between these two
time slices, the forcing parameters increase linearly and the glacier length
evolves freely. We present the time evolution of the simulated front
positions together with observations. To obtain one-dimensional observed
front positions, we assume the trough to be approximately east–west oriented.
We calculate the mean latitudinal coordinate of each observed calving front
(Fig. ) with the corresponding longitudinal
position at that latitude. The positions of the resulting one-dimensional
front positions lie approximately in the center of the trough. The
uncertainty of the front positions is calculated as the maximal spread of
each front in cross-trough direction.
Geometric experiments
In addition to the effect of climate forcing, we
investigate the effect of fjord geometry and the relative importance of bed
topography versus channel width. The experiments are designed with a smoothed
width and depth in the deep and narrow trough. Four different geometry
combinations are constructed and shown in Fig. .
Different model geometries used to investigate the impact of
topography on ice dynamics. (a) Original geometry,
(b) straight width, (c) straight bed, and
(d) straight width and bed. Arrows indicate the tributary ice flux,
with their length representative for the influx volume.
Original geometry: observed width and depth of the trough as described in
Sect. .
Straight width: the width until 80 km inland of today's front is set to a
constant value of 5.4 km. Only at the LIA front position, a wide section is
kept in order to reach a steady state with the same parameters. The depth is
kept as in a.
Straight bed: the bed of the deep trough to 120 km inland of today's front
is smoothed to get an almost straight bed, linearly rising inland. The width
is kept as in a.
Straight width and bed: both the width and the bed are straight.
The runs with simplified geometry start from a steady state at the LIA front
position with the same parameters and forcing as for the original model setup
(Table ). Due to the changed topographies, the glacier
surfaces and velocities differ from the original geometry and the LIA front
position is slightly changed.
Results
In this section, we present the steady-state glacier at
the LIA maximum extent and the glacier retreat simulated with run 5 (Table ) as an example. In addition, the response to different
forcing parameter combinations, more simplified geometries and a step forcing
are presented.
Jakobshavn Isbræ at the LIA maximum
The initial steady-state glacier as shown in Figs. a and
a is reached with the parameters in Table . The glacier has an uneven surface that reflects the
trough geometry, which is common for fast-flowing ice streams
. At the position of KAGA, the surface elevation
reaches about 400 m compared to the 300 m of the LIA-trimline height
Fig. a;; however, the side
margins are expected to be lower than the centerline and the model glacier
has a – probably overestimated – surface bump at this position. The LIA
glacier terminates with a 9 km long floating tongue, where it has a velocity
of 5 kmyr-1 and a grounding line flux of 35 kmyr-1.
The modeled width-averaged basal shear stress for the LIA is about 128 kPa
at 40 km inland of the present-day front position and the driving stress is
290 kPa at the same location, when applying a 3 km moving average to smooth
the surface bumps. In comparison, other modeling studies obtain lower basal
resistance and data assimilation methods
imply basal stresses at the bed of the deep trough of about 65 kPa at 50 km
upstream of the calving front, equivalent to only 20 % of the driving
stress . However, these estimates are from the present day and
it is unknown how much the relative contribution of the stresses has changed
over the time period. During the speed-up, the basal shear stress might have
been reduced in the lowermost 7 km, and not changed further upstream
. Note also that the stresses provided by the model are
width averaged.
Nonlinear glacier response to linear forcing
Modeled retreat of Jakobshavn Isbræ in response to a gradual
change of the forcing parameters (run 5 in Table ).
Yearly profiles are shown for (a) the along-flow glacier profile and
the elevation of the KAGA LIA trimline in green,
(b) the front positions in a top view and (c) the
along-glacier annual velocities including the yearly grounding line (GL) flux
(gray circles from dark to light with time). Observed yearly velocities are
plotted at the calving front from 1985 to 2003 and at
seven different points upstream from the glacier front from 2009 to 2013
.
Figure a and b show that the modeled front position
retreats nonlinearly in response to the linear external forcing (shown here
is run 5 in Table ). It retreats 21 km during the first
163 years, after which a 16 km long floating tongue forms. During the
break-off of the tongue in 2013 to 2014, the front retreats a further 23 km.
Throughout the retreat, the glacier terminus alternates between a floating
tongue and a grounded front. The front velocities (Fig. c) only increase by 3 kmyr-1 during the
first 163 years and more than double from 8 to
19 kmyr-1 when the floating tongue breaks off. This acceleration
is overestimated, as the simulated tongue breaks off faster than observed.
However, velocity observations by shown in
Fig. c are smaller than that simulated in the early
1990s but are in between the simulated velocities before and after the
break-off. The model simulations show that the acceleration continues until
the retreat of the front slows down. The grounding line flux, calculated as
the grounding line velocity times the grounding line gate area, increases
from 35 to 65 km3yr-1 from the LIA until
2015 compared to observed values of about 32–50 km3yr-1 between
2005 and 2012 . Beyond 2015 it
increases to 100 km3yr-1 and finally stagnates at a flux of
77 km3yr-1.
The various parameter combinations presented in Table
– and many more that are in between those presented here – reproduce the
observed total retreat since the LIA. Figure
shows the retreat of the glacier front and grounding line with time for the
nine parameter combinations applied. The simulated temporal retreat pattern
of the glacier front is similar for all experiments and shows the strong
nonlinearity of the frontal retreat – despite the linear forcing (Fig. a). The response to the different forcing
experiments differs mainly in the timing of the phases of rapid retreat,
especially the final retreat just after 2050. All model runs show a very
abrupt retreat of at least 23 km within a few years, which corresponds to
the observed retreat of 19 km after year 2000. The simulated frontal
positions differ from those observed, which is expected due to the strong
simplification of the forcing. The aim here is to study the geometric
controls on rapid retreat, rather than tuning the model until the simulated
retreat fits the observations. The reasons for the deviation of the
simulations from the observations are discussed in Sect. .
The grounding line retreats in a more stepwise manner (Fig. b) compared to the glacier front. Before 2015,
it stabilizes at distances of 32, 25 and 20 km from the 2015 frontal
position for all experiments. It retreats more gradually beyond 2015 with
short still-stands at 8, 12 and 18 km upstream of the present-day
position. The forcing parameter combination thereby determines the timing of
the grounding line displacement.
Simulated position of (a) the front and (b) the
grounding line (GL) for nine different gradual forcing combinations presented in
Table . The colors for the different model runs are
random. Black dots show the observed front positions at the centerline with a
spread (gray shading) corresponding to the across-fjord variation of each
front position (Fig. ).
Control of fjord geometry on front and grounding line retreat
The residence time of the
grounding line is analyzed for the different geometries introduced in Fig. . Residence time is thereby quantified by the amount of
time that the grounding line rests within a distance of 1 km. Figure a shows the original
geometry with the most pronounced pinning points at distances of 32,
25, -10 and -13 km from the 2015 position. Only the length of
grounding line still-stand thereby varies among the nine different model runs
(Table ), whereas the pinning point locations coincide
(also seen in Fig. b). Artificially
straightening the width removes the pinning points at 25 km and those beyond
the 2015 position (Fig. b).
Instead, the glacier rests at the present-day position. The geometry with the
straightened bed causes a similar response to the linear forcing as with the
original geometry, only with a wider spread of pinning points (Fig. c). Straightening the bed
and the width removes all pinning points (Fig. d) and leads to a linear
retreat. Note that all geometries have an initial pinning point at the LIA
position to allow a steady state at the LIA position. Generally, a reduction
in the complexity of the fjord geometry, for example, straightening the bed and/or
width, reduces the number of pinning points.
Residence time of the grounding line (GL) for the different
geometries presented in Sect. : (a) the
original geometry, (b) straightened width, (c) straightened
bed, and (d) straightened width and bed. The bars represent the time
that the grounding line rests within 1 km, and the colors correspond to the
model runs in Table . Only residence times of more than
2 years are included.
Delayed abrupt glacier response
In addition to the linear increase in climate forcing, the response to a step
forcing (Table ) is presented in Fig. . With the step forcing, the glacier front
remains at a distance of 22 km for 60 years, before it retreats rapidly to
its new pinning point. This unprovoked rapid retreat – after centuries of
constant forcing – demonstrates the long response time of the glacier
. The long response
time is caused by a slow adjustment of the glacier volume to external
changes. The corresponding accumulated volume loss, also shown in Fig. , adjusts steadily to the initial changes in
forcing, despite the constant grounding line position. During the rapid
frontal retreat, the volume decreases by 300 Gt and continues even after the
grounding line reaches a still-stand. This emphasizes that a constant
grounding line position does not imply a steady state. Similarly, an observed
rapid retreat of a marine-terminating glacier might be the delayed response
to historic temperature changes.
Discussion
For the example of Jakobshavn Isbræ, our results
show the importance of lateral and basal topography and their implications
for the evolution of glacier retreat in fjords. This knowledge can be used
for a better understanding of the recent observed retreat history; however,
it is hard to isolate the relative impact of changes in ocean forcing, SMB
and internal factors including the fjord geometry. Here, we discuss the
impact of fjord geometry on glacier front retreat and compare the simulated
glacier response to the recorded long-term glacier retreat history. In
addition, we explore the implications of our results for the future response
of Jakobshavn Isbræ to changes in climate.
We argue that fjord geometry, and in particular fjord width, to a large
degree dictates the retreat history of marine-terminating glaciers.
Nevertheless, changes to the external forcing of the glacier are important,
because their magnitude controls the onset and overall rate of the retreat
(Fig. ).
Geometric control on glacier stability
Our simulations show that once a glacier
retreat is triggered, through changes at the marine boundary, or at the
glacier surface, a nonlinear response unfolds due to variations in the fjord
geometry with a complexity given by the bed topography and the trough width.
For a retrograde bed, where water depth increases as the glacier retreats,
the ice discharge increases, leading to further unstable glacial retreat in
the case of constant lateral stresses .
Previous studies show that changes in the width of a glaciated fjord impact
the lateral resistance as well as the ice flow, thereby stabilizing the
glacier where narrow sections occur
.
These findings are corroborated by our model results. However, most of these
earlier studies use synthetic glaciers that do not allow for a validation of
the model against observations. Further, the shorter time periods considered
neglect the long-term adjustment of the glaciers. Figure shows
that the timescale of glacier adjustment can be several
decades. However, in reality temperature changes
are likely smaller and less abrupt than we have imposed. Nevertheless, our
study demonstrates that the observed recent retreat could have been triggered
and sustained by an earlier warm event. This finding is consistent with
, who studied Antarctic ice stream retreat on millennial
timescales. Depending on the local geometry of the underlying bed, individual
glaciers exhibit different response times and spatial extensions of dynamic
thinning .
The geometry experiments in Fig. assess the relative role
of glacier width versus glacier length on Jakobshavn Isbræ. The width
seems to be the leading factor for grounding line still-stand, as
artificially straightening the fjord removes the pinning points that cause
a slowdown of the grounding line retreat. Flattening the bed topography is
less efficient in linearizing the grounding line retreat compared to
straightening the fjord. It has to be considered that the glacier trough is
an order of magnitude wider than it is deep, with larger variations in the
width compared to the bed, increasing the importance of the glacier width.
Relative role of forcing parameters
Only certain parameter combinations simulate the observed retreat pattern of
Jakobshavn Isbræ since the LIA (Table ). If the
submarine melt rate is increased, the crevasse water depth has to be reduced
and/or the sea ice buttressing increased. Similarly, if the sea ice
buttressing is reduced, the crevasse water depth and submarine melt rate have
to be smaller (Table ). Importantly, none of the forcing
parameters can trigger the retreat alone, unless the change to the parameter
is unreasonably large relative to its LIA value. Changed individually, the
submarine melt rate would have to reach 650 myr-1 in 2015 – an
increase of 370 % from the LIA, the crevasse water depth has to increase to
400 m (250 % larger than the LIA value), and the sea ice buttressing
factor has to be more than quadrupled (value 4.2 relative to LIA factor of 1)
in 2015 to force a strong enough retreat. Absolute values for the parameters
have to be taken with caution, as they do not necessarily correspond to
physical variables. For example, to reach the observed grounding line flux,
the value for the crevasse water depth is likely too high in our study. This
is because it is a parameter for calving that has to balance the neglected
submarine melt along the calving front in the model. The change in parameters
required to trigger the retreat is also dependent on the initial parameter
choices and what forcing is needed to unpin the grounding line from the
initial pinning point. As shown by , non-unique parameter
combinations can exist for the same front positions. This implies that
real-world observations are vital to reduce uncertainty in transient model
simulations.
Note that the SMB contribution to the frontal retreat is insignificant, even
if the lower SMB gradient Gl is doubled and the SMB curve is lowered by
50 %. Taken together this gives a SMB at the terminus of
-6 compared to -1.1 mw.e.yr-1 during
the LIA (cf. Fig. ). In our model of Jakobshavn Isbræ,
variations in air temperatures contribute mainly through runoff and the
filling of crevasses with water, rather than directly through surface
ablation. For the specific geometry of Jakobshavn Isbræ, the influx of
ice at the lateral boundaries is a factor of 100 larger than the local SMB and
could be important for the sensitivity of the glacier to changes in climate
forcing. However, the lateral influx is an order of magnitude smaller than
the flux through the main trough and a sensitivity study shows that the
lateral flux has a minor impact on the retreat pattern (not shown here). If
all other parameters are kept fixed, the lateral influx has to decrease by
nearly 70 % from its LIA value in order to simulate the observed retreat.
Model limitations and comparison to observations
In order to isolate the effect of geometry on glacier retreat, a relatively
simple – but physically based – model is forced with a linearly changing
external forcing. Notwithstanding a number of assumptions, the model is well
suited, as it is computationally inexpensive and allows for a large set of
ensemble simulations starting from the LIA in 1850. Studying long time
periods is vital in order to capture internal glacier adjustments to changes
in external forcing beyond the last few decades. Unfortunately, few
observations exist to validate the model for such a long time period, which
supports our chosen idealized model setup. The model parameters are
calibrated with the few observations that exist, and the modeled retreat of
Jakobshavn Isbræ is compared to the observed retreat.
Both modeled and observed calving front positions show a highly nonlinear
retreat and the rapid disintegration of a several-kilometer-long floating
tongue (Fig. ). The model results show a
robust dependency of this nonlinear retreat on the trough geometry,
especially the trough width. However, the modeled glacier front retreats more
slowly (deviating up to 13 km from the observations) and exaggerates the
break-off of the floating tongue. For the dynamic interpretation of the
nonlinear retreat, a perfect agreement is not essential, especially given
the one-dimensionality of the model and the uncertainties in the
width-averaged observed front positions.
For the interpretation of the model results, the assumptions made in the
model have to be considered. The most obvious assumption is the
one-dimensionality that does not account for across and vertical variation in
geometry. The residence time of the grounding line at pinning points may be
partly overestimated due to this width and depth integration. Local bedrock
highs that partly pin the floating tongue are not properly
represented in a width-averaged setting, and the width is regarded as
symmetric around the central flow line. In reality, one lateral margin might
narrow down and pin the grounding line while the other lateral margin widens
up, causing an asymmetric calving front retreat
(Fig. ). Here, we only focus on the large-scale
dynamics; lateral and vertical variations in the ice flow are seen as
second-order processes, considering the high basal motion and high velocities
in the deep and narrow channel at the lowermost 100 km of the model domain.
As the glacier retreats further upstream “into” the ice sheet, the lateral
ice flux becomes more significant and the whole drainage area should
explicitly be modeled, favoring the use of a three-dimensional model for
future projections.
Simulated front and grounding line (GL) positions with accumulated
volume loss for the step forcing (Table ).
The depth and width integration also applies to internal glacier properties;
ice temperatures are in reality high at the bottom , so
most deformation happens there, whereas the model assumes a vertically
constant shearing and a constant rate factor. Along the margins of a real
glacier, ice viscosity drops significantly in response to acceleration and
calving front migration , and marginal crevasses can form,
which are not considered here. However, lateral drag and weakened margins
mostly affect the timing and not the details of the retreat, as has been
tested in an idealized setting with the same model . Ice
viscosity is a response to dynamic changes rather than a cause, and it is
therefore not expected to change the retreat pattern significantly. However,
ice viscosity may slightly alter the timing and residence time of the
grounding line.
Several parameterizations of physical processes are used in the model, such
as submarine melt and buttressing by ice mélange. This complicates direct
model validation with observed values. However, these processes are still
crudely implemented, if at all represented in glacier models. For example,
many models prescribe the position of the calving front
e.g.,, or only focus on grounding line migration,
whereas our model uses a physical calving law. Also, few observations exist
of submarine melt, calving rates and basal sliding, especially over the long
time period studied here. The impact of plume dynamics on submarine melt
could be implemented in our model , or an along-flow
variation in submarine melt rate . However, the number of
observations on ocean temperatures is sparse and the model results are
similar when using along-flow variations in submarine melt, compared to a
constant value along the floating part (not shown here). Also interannual
variability of calving rates due to submarine melt, runoff and ice mélange
is neglected and not considered as important when looking at centennial
timescales. Although many of the model parameters are only indirectly linked
to observations, existing observations such as velocities, ice discharge and
thickness are used to tune the parameters and to reproduce the glacier
behavior as close as possible. Note that the change in forcing parameters
required to dislodge the grounding line from its stable LIA position might be
overestimated, due to large variations in bed topography and width. Also,
many parameter combinations can simulate the same stable position but lead
to different glacier retreat . Therefore, we include a
large range of parameter perturbations, leading to different residence times
for the grounding line, but with no reduction in the importance of the
geometry in defining locations of intermittent slowdown in the overall
grounding line retreat.
The choice of the model is dependent on the questions raised; if the
objective is to accurately predict or reconstruct the time evolution of
glacier retreat e.g.,, a more sophisticated
model has to be used. Note that also the observations contain uncertainties.
The front position can vary by several kilometers seasonally
e.g., and this position varies by several kilometers
across the trough (Fig. ). For the calculation of
the one-dimensional front position, we assume a west–east orientation of the
trough, which gives an offset at the most recent calving fronts; however, the
deviation is only a few kilometers and within the spread of the across variation of
the calving front. Most importantly, the bed topography – especially in the
densely ice-covered fjord and a sediment-rich subglacial bed
– is challenging to obtain. Due to the strong control
of the fjord geometry on the glacier retreat, small uncertainties in the
trough geometry can cause a very different retreat pattern. This highlights
the importance of detailed knowledge of the underlying bed topography
e.g.,.
Glacier front reconstructions based on trough width
Figure illustrates the
potential in using the model simulations in a geomorphological context.
Marine-terminating glaciers continuously erode their beds and deposit
sediments, forming submarine landforms such as moraines. The rate of sediment
deposition and resulting proglacial landforms are functions of climatic,
geological and glaciological variables, though these functions remain poorly
quantified due to sparse observational constraints. Proglacial transverse
ridges tend to form during gradual grounded calving front retreat, whereas
more pronounced grounding zone wedges are associated with episodic grounding
line retreat .
The abundance of ice mélange in front of Jakobshavn Isbræ renders
studies of submarine geomorphology difficult. Studies of this kind are
lacking in the fjord, though evidence of the style of deglacial ice sheet
retreat in Disko Bugt does exist . Our study raises generic
questions about the links between trough geometry and moraine positions. We
suggest that likely locations for moraine formation can be predicted from the
glacier width, which largely determines the position of grounding line
slowdown. The finding of the very robust influence of width on the retreat
patterns (Fig. ) means that
investigating the detailed fjord geometry allows for the location of expected
slowdowns or step changes . This is extremely
useful for reconstructions and interpreting paleo-records, for example, from
adjacent land records, moraines and proglacial lake sediments.
To this end, our study clearly highlights the potential of combining
long-term modeling studies with geomorphological and sedimentary evidence to
understand the nonlinear response of marine ice sheet margins. This needs to
be considered when inferring climate information based on glacier retreat
reconstructions.