The majority of Antarctic ice shelves are bounded by grounded ice rises.
These ice rises exhibit local flow fields that partially oppose the
flow of the surrounding ice shelves. Formation of ice rises is
accompanied by a characteristic upward-arching internal stratigraphy
(“Raymond arches”), whose geometry can be analysed to infer information
about past ice-sheet changes in areas where other archives such as rock
outcrops are missing. Here we present an improved modelling framework to
study ice-rise evolution using a satellite-velocity calibrated,
isothermal, and isotropic 3-D full-Stokes model including grounding-line
dynamics at the required mesh resolution (<500 m). This overcomes
limitations of previous studies where ice-rise modelling has been
restricted to 2-D and excluded the coupling between the ice shelf and ice
rise. We apply the model to the Ekström Ice Shelf, Antarctica, containing
two ice rises. Our simulations investigate the effect of surface mass
balance and ocean perturbations onto ice-rise divide position and
interpret possible resulting unique Raymond arch geometries. Our results
show that changes in the surface mass balance result in immediate and
sustained divide migration (>2.0 m yr-1) of up to 3.5 km. In contrast,
instantaneous ice-shelf disintegration causes a short-lived and delayed
(by 60–100 years) response of smaller magnitude (<0.75 m yr-1). The model
tracks migration of a triple junction and synchronous ice-divide
migration in both ice rises with similar magnitude but differing rates.
The model is suitable for glacial/interglacial simulations on the
catchment scale, providing the next step forward to unravel the
ice-dynamic history stored in ice rises all around Antarctica.
Introduction
Ice rises are parabolically shaped surface expressions along the margin of the
Antarctic ice sheet, and they form where the otherwise floating ice locally
regrounds. They are characterised by their local ice-flow centre – henceforth
referred to as ice-rise divide – that is independent of the main ice sheet and
the surrounding ice shelves, resulting in divergence of the main ice flow around
the obstacle. These obstacles act as a decelerating force that restricts ice
flow which is commonly referred to as “ice-shelf buttressing”. More than 700 ice
rises are distributed along the Antarctic
perimeter (Fig. a), providing additional buttressing to the ice
upstream. Ice rises archive their flow history in their characteristic internal
stratigraphy
e.g.,
making them potentially suitable sites for ice-core drilling such as for the
International Partnerships in Ice Core Sciences (IPICS) 2K and 40K Array
.
(a) Location map of ice rises along the margin of the Antarctic ice
sheet . The base map combines ice velocity of the ice
sheet and the bathymetry of the adjacent ocean regions
. The blue rectangle shows zoomed-in area of (b). (b) RADARSAT
image of the study area with locations mentioned in the main
text. Dashed lines (A-A', B-B') show the extent of the cross section shown in Fig. . Please note that (b) is rotated by 180∘ with respect to (a).
Due to very low deviatoric stresses near the bed of the divide region and the
power law rheology of ice, the effective viscosity (e.g. the stiffness of ice)
is significantly higher towards the ice sheet bottom under the divide than in
the surrounding areas. The stiff ice impedes downward flow under the divide in
comparison to the flank regions such that ice of
the same age is found at shallower depths in the ice column under the divide
compared to the flank regions . This results in
the formation of upward arches in the isochronal ice stratigraphy commonly
referred to as “Raymond arches”.
Previous studies have made much progress in interpreting the Raymond arches as
ice-dynamic archives. The arch amplitude and the tilt of multiple Raymond arches
– commonly referred to as Raymond stack – have been used to infer migration of
ice divides e.g., onset of local divide flow
e.g., and
ice-rise residence time
e.g..
Common to all these studies is the comparison of the observed isochrones derived
from radar data, with the predicted age fields from models with varying input
scenarios. The full stress solution of the Stokes equations is necessary because
both longitudinal and bridging stress gradients are important near ice divides
e.g..
Fast migration of ice divides results in abandoned Raymond stacks in the flanks, and a new Raymond stack starts to develop at the new divide position
Fig. d–f.
Slow migration of ice divides results in tilted Raymond stacks
.
For a new Raymond stack to form, a sudden displacement of 1–2 ice thickness is
required , but thus far a clear threshold between
both endmember scenarios remains elusive. Even though divide migration has been
an active focus in ice-rise research, most studies have focussed on the
ramifications of divide migration
rather than what causes the divide to migrate. Additionally, modelling thus far
has been restricted to 2-D and has not included interactions between ice rises and
the surrounding ice shelves
.
This means that the underlying processes resulting in different Raymond arch
geometries from radar observations are still incomplete and poorly constrained.
Another poorly understood phenomenon for ice-rise evolution is triple junctions
. Triple junctions are points where three
ice-divide ridges meet. They often coincide with the summits of ice domes. It
has been proposed that changes in triple junction position (e.g. merging of
two divide ridges) might explain observed relic arches in ice-divide flanks, which cannot be explained
with ice-divide migration in a 2-D setting .
Upper panel (a–c) shows schematic (a) steady state, (b) divide migration
induced by asymmetric surface mass balance forcing, and (c) divide migration induced by ocean
perturbation forcing. Buttressing in (c) is asymmetric. Solid red arrows
indicate approximate ice-flow path from the ice-rise divide to ice shelf. Grey
dashed lines in (b) and (c) display the steady-state geometry and divide position of (a). GL is the grounding line. Lower panel (d–f) shows schematic of expected internal stratigraphy for steady state (d), fast migration (e), and slow migration (f). (e) and (f) are not necessarily the result of forcing in (b) and (c),
respectively.
Here, we use the full-Stokes (FS) ice-sheet model Elmer/Ice in 3-D
and extend the model domain in contrast to
previous studies to include grounding-line dynamics and ice-shelf flow to study
potential causes for ice-rise divide migration. We apply the model to the
Ekström Ice Shelf catchment bounded by two large (15th and 16th largest in
Antarctica, ) ice rises. The model is calibrated
by tuning basal friction and ice viscosity so that modelled surface velocities
match today's flow field. Perturbations to the surface mass balance (SMB) and ice-shelf thickness are
applied in forward simulations to investigate the coupled transient response of
the two ice rises. The experiments are tailored to help improve our
understanding of what processes cause ice-divide migration. Specifically, we
address the following questions. Is the amplitude of divide migration controlled by the
SMB, ice-shelf buttressing, and/or is the divide position determined by the
subglacial topography? Do ice rises in close proximity of each other show a
similar response? Can we differentiate between the different trigger mechanisms?
Furthermore, we investigate if the triple junction at one of the ice rises in
the catchment – the Halvfarryggen Ice Rise – also migrates in synchronicity with the
main divide ridge.
Study area: Ekström Ice Shelf catchment
The Ekström Ice Shelf is located in the Atlantic sector of the East Antarctic ice
sheet and is a medium-sized ice shelf. The embayment is characterised by two
prominent ice rises: the Söråsen Ice Rise in the west and the Halvfarryggen Ice Rise, an
ice promontory sandwiched between the Ekström Ice Shelf and Jelbart Ice Shelf in the
east (Fig. b). While the Söråsen Ice Rise is made up of a single
ridge divide, Halvfarryggen consists of three main ridges that meet to form a
triple junction close to the summit . Both
ice rises belong to the larger ice rises in Antarctica with areas of about 5700
and 5500 km2 for Halvfarryggen and Söråsen, respectively
. The Söråsen Ice Rise is additionally buttressed in
the west by the Quar Ice Shelf (Fig. b). SMB across both divide
ridges is strongly asymmetric with the western downwind side of the divide
receiving much less accumulation (<0.6 m yr-1 ice equivalent) than the eastern
side (up to 2.9 m yr-1 ice equivalent)
. The catchment is a
suitable test site to study the dynamics of ice rises because the boundary input
datasets such as subglacial topography, ice thickness, and ice surface elevation
are well constrained (with many flight lines across the region due to the
vicinity of the Neumayer III station). In addition, the confined geometry of
both ice rises suggests that significant lateral buttressing is provided by
the Ekström and Jelbart ice shelves. This means that the buttressing the Ekström and
Jelbart ice shelves provide also restricts ice flow across the grounding line
from both ice rises, making the divide position potentially sensitive to ocean
forcing.
MethodsModel descriptionGoverning equations
Ice flow is dominated by viscous forces (e.g. low Reynolds number) permitting
the dropping of the inertia and acceleration terms in the momentum equations.
Using these simplifications, a complete description of ice flow is the
full-Stokes (FS) flow model. The Stokes equations for linear momentum are
divσ=-ρig,
where σ=τ-pI is
the Cauchy stress tensor, τ is the deviatoric stress
tensor, p=-tr(σ)/3 is the isotropic pressure,
I the identity matrix, ρi the ice
density, and g is the gravitational vector. Ice flow is assumed
to be incompressible, which simplifies mass conservation to
divu=0,
with u being the ice velocity vector. Here we model ice as an
isotropic material. Its rheology is given by Glen's flow law, which relates
deviatoric stress τ with the strain rate
ϵ˙:
τ=2ηϵ˙,
where the effective viscosity η can be expressed as
η=12EBϵ˙e(1-n)n.
In this equation E is the enhancement factor, B is a
viscosity parameter computed through an Arrhenius law, n is Glen's
flow law parameter (n=3), and the effective strain rate is defined as
ϵ˙e2=tr(ϵ˙2)/2. Although there is evidence that the
Glen flow parameter is >3, particularly near ice rises
, we stick to the current modelling standard of
n=3, because we focus on divide migration rather than Raymond arch amplitudes
here. The same holds for the enhancement factor, which is often used to account
for anisotropic effects but is set to 1 (isotropic conditions) in all
simulations. In future applications of this model, these assumptions should be
revisited.
Standard physical and numerical parameters used for the simulations.
A kinematic boundary condition determines the evolution of the upper and lower
surfaces zj:
∂zj∂t+ux∂zj∂x+uy∂zj∂y=uz+a˙j,
where a˙j is the accumulation/ablation term and j=(b,s), with s being the upper
surface and b being the lower surface (base) of the ice sheet.
The ice-shelf basal mass balance (a˙b) parameterisation follows
:
a˙b=Hα(ρfG+(1-ρf)A),
where H is the ice thickness; G and A are
tuning parameters to constrain melt rates at the grounding line and away from
the grounding line, respectively; and
α is a local tuning parameter (Table ). The parameter ρf(x,y)
decreases exponentially with distance from the grounding line, varying from
ρf(x,y)=1 at the grounding line to ρf(x,y)=0
some distance (∼40 km) away from it. This ensures that the highest melt rates are
always specified at the grounding line and decrease exponentially with distance
away from it. The tuning parameters are chosen such that basal melt rates
roughly agree with melt rates derived from satellite observations and mass
conservation . Melt rates under the shelf are generally
low, ranging from -0.2 to 1.1 m yr-1 near the grounding line
. No basal melting is applied to the grounded ice
sheet.
Where the ice is in contact with the bedrock a linear Weertman-type basal
sliding law is employed:
τb=C|ub|m-1ub,
where τb is the basal traction, m is the basal
friction component – set to 1 in all simulations, and C is the basal
friction coefficient inferred by solving an inverse problem (see Sect. , Model initialisation). Underneath the floating part (ice
shelves) of the domain basal traction is zero
(τb=0), but hydrostatic sea pressure is prescribed.
For the ice-shelf-front boundary, the true vertical distribution of the
hydrostatic water pressure is applied and the calving front is held fixed
throughout the simulations. We assume the ice is isothermal with a constant
temperature of -10∘C. We specify depth-independent horizontal ice velocities
taken from the MEaSUREs dataset at all lateral boundaries
other than the ice-shelf front. To solve the presented system of equations with
the appropriate boundary conditions, we use the open-source finite element code
Elmer/Ice .
Initial geometry and input data
To initialise the model geometry, subglacial topography was taken from the
BEDMAP2 dataset . Our surface elevation is a
merged product of CryoSat-2 and, where available, higher-resolution TanDEM-X
digital elevation models (Veit Helm, personal communication, 2018). SMB
(a˙s, Eq. ) in our simulations was taken from
regional climate model simulations with RACMO2.3 .
This model reproduces the commonly observed asymmetric SMB across ice rises
. Velocity observations to match modelled ice
velocities in the inversion procedure were taken from the MEaSUREs dataset
.
Model initialisation
We perform the model initialisation in two steps using two different mechanical
models: the shallow shelf approximation
(SSA; ) and the FS model. The SSA ice-sheet
model is initialised by solving an optimisation problem to simultaneously infer
the basal traction coefficient C and the viscosity parameter
B. This type of snapshot initialisation is well known and widely
employed in ice-sheet modelling and aims to match modelled velocities with
observed velocities
e.g..
In a more formal way, we seek a minimum of the objective function
J=Jm+Jp,
where Jm is the misfit between observed and modelled velocities and Jp is a
Tikhonov penalty function described by
Jp=λCJCreg+λBJBreg,
where λC and λB are the Tikhonov parameters and
JCreg and JBreg represent the smoothness
constraints of basal drag and viscosity
e.g.. The
smoothness constraints penalise the square of the first derivatives. An L-curve
analysis was performed to pick the Tikhonov parameters and avoid
overfitting or overregularisation . The selected
values are λC=107 and λB=106
(Fig. ). Velocity misfits obtained with these parameters are of similar
magnitude to previous studies e.g..
However, when the inferred basal traction coefficient and viscosity fields from
the SSA inversion were used in forward simulations, unrealistically high surface
lowering rates (200 m/century) in the vicinity of the divide were observed. To
circumvent this problem, the output fields from the SSA inversion were used as
input for a second inversion using the FS model, but only adjusting the basal
friction coefficient, while keeping the viscosity of the first inversion fixed.
The Tikhonov parameter from the SSA inversion was used for the FS inversion as
well. For both inversions a horizontal grid resolution of ∼1 km was used.
Following initialisation, as is commonly done in ice-sheet modelling, we
performed a short 10-year relaxation simulation to smooth out data
inconsistencies such as differences introduced by differing acquisition dates of
ice surface elevation and surface velocity. This simulation length for the
relaxation lies in the range of what other studies have done previously
e.g..
L-curve analysis to select Tikhonov parameters λB and
λC: (a) 2-D cross section for variable
λB and λC fixed at
107 Pa-2 m6 a-4. (b) Reverse case with constant
λB at 106 m4 a-2 and varying
λC. Jm (m4 a-2) and
JCreg (Pa2 m-2 a2) are colour-coded. JBreg is unitless. (c) shows
mismatch
between modelled and observed velocities after
FS inversion for the modelling domain (Fig. b, blue outline).
Experimental design
The forward simulations focus on two types of perturbation simulations: (1) perturbations to the SMB and (2) ocean perturbations resulting in changing
ice-shelf thickness and hence ice-shelf buttressing. For the SMB, we use
modelled values from RACMO2.3 . The spatial pattern of
the SMB is such that low accumulation rates are applied on the western
(downwind) side (<0.5 m yr-1) of the ice rises and high accumulation rates (>1 m yr-1) are applied on the eastern (upwind) sides. This asymmetric SMB pattern is
consistent with observations but does not
capture the correct magnitudes. Therefore, we adjust the SMB using the computed
model drift following the relaxation simulation. This means that for the reference simulation the SMB forcing consists of the RACMO2.3 field plus the computed spatial thickening/thinning
rate (model drift) at the end of the relaxation simulation. This approach ensures that the
model drift is eliminated and the divides stay at their initial positions. Since
without this model drift correction there is a change in divide position, we treat the unadjusted SMB as a simulation with a perturbed SMB. In the second type of experiments, we perturb the reference run by thinning ice shelves through an increase in ocean-induced melting. The extreme scenario,
where all ice shelves are removed, is simulated by cutting the numerical mesh
∼1 km downstream of the present grounding-line position. This ensures that
the same frontal boundary conditions still apply to this geometry. To test more
ice-shelf buttressing reduction scenarios, we performed additional model
simulations with intermediate shelf thinning scenarios, where shelf thickness
was reduced by 10 % and 50 % at the start of the perturbation simulation. For
both simulations the shelf geometry is kept constant for the remainder of the
simulation by applying a synthetic steady-state ocean forcing. To permit a more
direct comparison between ocean forcing and SMB forcing, an additional SMB
perturbation simulation is performed, where the SMB is unadjusted and the
initial grounding-line flux perturbation from the shelf removal simulation on
either side of Halvfarryggen is added to the SMB term (Table , Run 6). This is done such that the spatial pattern of the SMB remains unchanged, but the magnitude is different
by about a factor of 2 in comparison to the unadjusted SMB. In all perturbation
simulations the grounding line is permitted to freely evolve. All perturbation
simulations are run forward for 1000 years, which is about the characteristic
time T (T= ice thickness/accumulation) for the Halvfarryggen and Söråsen ice rises
. After one characteristic time, the formation
of synclines in the internal stratigraphy can usually be observed
. Most of the simulations are performed with two
different mesh resolutions (Fig. ). The first isotropic mesh
uses a horizontal resolution of ∼2 km throughout the domain (henceforth regular
mesh), whereas for the second mesh (henceforth refined mesh), we initially use
the same footprint as for the regular mesh but use the meshing software MMG
(http://www.mmgtools.org/, last access: 2 October 2019) to refine the mesh along the grounding line
and all ice-divide ridges to a resolution of ∼500 m. Mesh resolution then
decreases with distance away from the regions of interest to the lowest
resolution of ∼10 km (Fig. ). A second refined mesh, where we refine down to
∼350 m at the divides and grounding line, was used for two simulations. All
meshes are held fixed over time and no dynamic remeshing is performed. A summary
of all perturbation experiments is provided in Table .
List of all perturbation experiments including forcings and mesh resolution.
Run numberOcean forcingSMB forcingMesh (max. resolution)1Function of distance to GL (Eq. )AdjustedRegular mesh (2 km)2Function of distance to GL (Eq. )AdjustedRefined mesh (0.5 km)3Function of distance to GL (Eq. )UnadjustedRegular mesh (2 km)4Function of distance to GL (Eq. )UnadjustedRefined mesh (0.5 km)5Function of distance to GL (Eq. )UnadjustedRefined mesh (0.35 km)6Function of distance to GL (Eq. )Unadjusted + flux perturbation Run 6Refined mesh (0.5 km)7No shelf + function of distance to GL (Eq. )AdjustedRegular mesh (2 km)8No shelf + function of distance to GL (Eq. )AdjustedRefined mesh (0.5 km)9No shelf + function of distance to GL (Eq. )AdjustedRefined mesh (0.35 km)1090 % shelf thickness + adjusted ocean forcingAdjustedRefined mesh (0.5 km)1150 % shelf thickness + adjusted ocean forcingAdjustedRefined mesh (0.5 km)
The 3-D plot of the model domain showing the two main mesh resolutions
employed in this study. (a) shows uniform mesh of 2 km resolution and (b) shows
refined mesh in areas of interest such as the grounding line and the ice-rise
divides. In these areas the mesh resolution is ∼500 m and away from these regions the mesh resolution increases to 10 km.
ResultsEffect of mechanical model in initialisation on transient
simulations
We observed an order-of-magnitude difference in the basal drag coefficients
found by solving an inverse problem, depending on whether we use the SSA or the
FS approximation as a forward model (Fig. ). This means that
sliding is more restricted in the FS case in comparison to the SSA case. Most
notably, in comparison to the SSA inversion, the slippery regions in the FS
inversion do not extend as far upstream. For example, between the ice divide of
Halvfarryggen and the grounding line on either side, there are areas of stickier
bedrock conditions that are missing in the drag field of the SSA inversion
(Fig. ). Spuriously high ice surface lowering rates (200 m/century) near the divide were simulated when basal drag and viscosity
fields were used from the inversion using the SSA. Thinning rates were much
smaller (≤50 m/century) when using the FS forward model.
Inferred basal traction fields C using (a) the SSA model and (b) the FS
model.
Reference simulation (runs 1–2)
The reference simulation serves the purpose of ensuring that the applied
synthetic SMB does indeed result in a steady-state geometry in which divide
positions do not migrate and surface topography changes are minimal. It then
also functions as a baseline against which the perturbation simulations can be
compared. The thinning/thickening rates in this type of simulation for both
meshes are highest at the start of the simulation but never exceed 0.05 m yr-1
near the divide region. These low thinning/thickening rates result in a steady-state geometry of the model domain that is also characterised by stable
positions of the ice-rise divides. For both ice rises total divide-migration
amplitudes are <60 m throughout the forward simulation of 1000 years. Divide positions are
computed at every time step along two swath profiles (∼8 and ∼23 km
for Halvfarryggen and Söråsen, respectively; e.g. Fig. ). The
shorter swath profile for Halvfarryggen was chosen to permit a simple flux
balance analysis. The initial start point of the divide is the location of
highest surface elevation. From this point, the divide is tracked along the
swath profile by following the minimum direction of the aspect gradient until
the end of the swath. Computed mean divide-migration amplitudes are then
averaged along the swath profiles (e.g. Fig. ).
In simulations 3–5 (Table ), the SMB perturbation results in
immediate divide migration for both meshes (Fig. a). For
Halvfarryggen, we focus our analysis on the main (eastern) divide ridge
(Fig. b). Owing to the more positive SMB on the eastern side of
both divides, the divide migrates towards this region (Fig. a).
Almost all of the divide migration occurs over the first 200 years of the
simulation before a new steady-state position is reached
(Fig. a). During the first 200 years, the entire divide migrates
at an average rate of 16–20 m yr-1 to the east for the Halvfarryggen Ice Rise. This
range shows there is a clear mesh dependence on the magnitude of divide
migration over this timeframe (3.2–3.8 km, Fig. a), whereas this
difference is less pronounced in the steady-state divide positions at the end of
the simulation (2.5–2.8 km). For the two refined meshes (runs 4 and 5;
Table ), mesh dependence is still present, even if reduced, and first-order convergence between the simulations is absent. This indicates that a very fine mesh resolution is required to capture divide migration, but, in the light of the high computational costs to run all simulations at such a resolution, we restrict ourselves to a maximum resolution of 500 m. While the
averaged absolute magnitude along the swath profile is offset by about 300 m
between the refined grid simulations (runs 3 and 4; Table ), the
temporal pattern of divide migration is identical (Fig. a). This
is in contrast to the regular mesh, where divide migration reaches its maximum
(most eastern position) after ∼200 years and remains almost stable for the
remaining simulation period. In comparison, the refined mesh simulations reach
their maximum divide migration at a similar time (∼200 years) but start to
slowly migrate back towards its initial position until a steady state is reached
after 700 years (Fig. a). The SMB grounding-line flux
perturbation simulation (Run 6, Table ) has almost an identical steady state to the unadjusted SMB simulation (Fig. a), despite
the SMB flux difference between east and west of the divide being twice as high.
While this does not affect the steady-state divide position, it does result in
faster migration with a larger maximum amplitude of divide migration (∼3.9 km vs. ∼3.4 km).
Söråsen Ice Rise
The Söråsen Ice Rise shows a very similar response – both in absolute magnitude of
divide migration as well as temporal evolution of divide migration – to the
applied SMB perturbation (Fig. b) The most pronounced
differences of ∼ 4 m yr-1 are in the divide-migration rates (11–15 m yr-1) in
the first 200 years of the simulation, when most of the divide migration takes
place. The mesh dependence on divide-migration amplitudes is also present for
Söråsen, even though it is slightly reduced in comparison to Halvfarryggen
(2.9–3.3 km, Fig. b). Unlike Halvfarryggen, simulations with the
refined mesh do not show any backward migration of the Söråsen divide
(Fig. c) but reach a new steady-state position after
∼300–400 years with little migration beyond this point in all simulations
(Fig. b).
Ice-rise divide migration for (a) the Halvfarryggen Ice Rise and (b) the Söråsen
Ice Rise induced by surface mass balance perturbation for different mesh
resolutions. Positive numbers indicate migration to the east. Panels (c) and (d) show balance
fluxes for the eastern and western side of the divide to the grounding line for the Halvfarryggen Ice Rise. Grey shaded areas in (a) and (d) highlight the time lag
between the balance flux east being smaller than the corresponding flux west and the
start of backward migration. “Swath profile” indicates that computed divide
migration is averaged over cross sections shown in Fig. . For the grounding
line (GL) flux, this means that the flux is summed along the current
grounding-line position over the swath length. The surface mass balance (SMB)
flux is area averaged from the current divide position to the respective
grounding-line position in the east and west of the divide over the swath
length. Note different y-axis scales in (c) and (d).
Ice-rise divide positions using the refined mesh for the Halvfarryggen Ice Rise induced by (a) surface mass balance perturbation and (b) shelf removal perturbation. Panels (c) and (d)
show the same simulations for the Söråsen Ice Rise. Divide positions are plotted at
the start of the simulation, at the end of the simulation, and at the time of
maximum divide-migration amplitude for each simulation.
Ocean perturbation causes the adjacent ice shelves to thin and reduce their
ability to buttress the ice upstream. If buttressing of ice shelves on either
side of the ice rise is asymmetric, this should lead to an asymmetric increase
in grounding-line flux, which in turn should result in an asymmetric thinning
perturbation. This asymmetric thinning perturbation should then push the
divide in the direction of the smaller thinning perturbation
(Fig. c). In our model simulations, we test this hypothesis
using the real-world example of the Ekström Ice Shelf embayment.
Halvfarryggen Ice Rise
All ocean perturbation experiments result in an instantaneous divide migration.
For the extreme case of ice-shelf removal (runs 7–9, Table ) the maximum
mean ice-divide migration along the swath is -0.4 and -0.7 km with local maximum amplitudes of -1.7 and -2.3 km, occurring after 700 and 1000 years for the regular and refined mesh, respectively (Fig. a). Negative
numbers indicate westward migration (Fig. b, d). For the intermediate thinning
scenarios (runs 10–11, Table ), there is no divide migration for the 10 %
thinning scenario and -0.2 km divide migration for the 50 % thinning scenario. So
even though half of the shelf's thickness is removed, the magnitude of the
divide migration is reduced by 77 %, indicating that the response to ice-shelf
thinning is non-linear and requires a strong perturbation for a significant
response. Most of the divide migration takes place in the centuries following
the perturbation. A stable divide position is reached after ∼690 years for the
regular mesh simulation and after ∼480 years for the refined mesh simulation in
the shelf removal scenario (Fig. a). In the 50 % shelf thinning scenario, a
stable divide position is reached after ∼600 years. As was the case for the SMB,
the shelf removal simulations exhibit mesh dependence with mesh convergence present for the two refined meshes (Fig. a).
To shed light on why divide-migration amplitudes differ for the different shelf
thinning scenario, grounding-line fluxes for the eastern and western side were
computed for Halvfarryggen (Figs. c, d and c). In all simulations, an
initial sharp increase in grounding-line flux is computed, which then quickly
decays and is even lower for the eastern side of the divide than in the
reference simulation (Fig. c, d). The initial flux perturbation is largest for
the shelf removal scenario and is non-linear, where halving of the ice-shelf
thickness results in a flux reduction of 60 %.
Ice-rise divide migration for (a) the Halvfarryggen Ice Rise and (b) the Söråsen
Ice Rise induced by ocean perturbation (shelf removal) for different
mesh resolutions. Negative numbers indicate migration to the west. Panels (c) and (d)
show grounding-line (GL) fluxes for the eastern and western side of the
divide to the grounding line for the Halvfarryggen Ice Rise. “Swath profile”
indicates that computed divide migration is averaged over cross sections
shown in Fig. . For the GL flux, this means that the
flux is summed along the current grounding-line position over the swath
length. Note different y-axis scales in (c) and (d).
(a–c) Ice-rise divide migration for the Halvfarryggen Ice Rise induced by
different ocean perturbations (shelf removal, 50 % ice-shelf thickness,
and 90 % ice-shelf thickness) for the refined mesh. (a) shows
divide position at the end of the simulation period. Grey dashed lines
show the approximate area used for flux calculations in (c). (b) shows mean
divide migration for the different perturbations. Negative numbers
indicate migration to the west. (c) displays ice flux perturbation
across the grounding line for the eastern and western side of the divide
for the first two decades. Perturbation fluxes smaller than 0 indicate
that ice flux across the grounding line is reduced in comparison to the
reference simulation. (b, c) “Swath profile” indicates that computed
divide migration is averaged over cross sections as shown in (a). For the
ΔGL flux, this means that the flux is summed along the current
grounding-line position over the swath length.
Söråsen Ice Rise
For the extreme case of ice-shelf removal (runs 7–9, Table ) the
maximum ice-divide migration is -0.5 and -0.7 km with local maximum rates of -1.0 and -1.5 km for the regular and refined
mesh, respectively (Fig. b). Almost all of the divide migration
happens in the first half of the simulation period before a new steady-state
position is reached after 300 years for the regular mesh and after ∼400 years
for the refined mesh (Fig. b). This behaviour closely follows
Halvfarryggen both in terms of migration amplitudes and direction
(Fig. b). This is not the case for the
intermediate thinning scenarios (runs 10–11, Table ), where no
divide migration (<60 m) occurs in any of them.
(a, b) Ice-rise divide migration for the Söråsen Ice Rise induced by different
ocean perturbations (shelf removal, 50 % ice-shelf thickness, and 90 %
ice-shelf thickness) for the refined mesh. (a) shows divide
positions at the end of the simulation period. (b) shows mean divide
migration for the different perturbations. Negative numbers indicate
migration to the west.
Triple junction migration for the Halvfarryggen Ice Rise
In both experiments (runs 4 and 8), the triple junction immediately migrates in
response to the perturbations. The triple junction evolution closely follows the
temporal evolution of the main divide. The maximum migration amplitude is 3.3
and -1.2 km for the SMB and shelf removal simulations, respectively. Most of the
triple junction migration takes place over the first ∼400 years, before a new
steady-state position is reached (Fig. ). The temporal evolution of the
triple junction migration also shows a tendency to migrate back to its initial
position in the latter part of the SMB simulation (Fig. c). Because the SMB
is most positive to the east of the main divide arm, the eastward migration of
the divide leads to an increased angle between these two ridges of
3.5∘, translating into a widening of 5 %. In addition to the widening,
the minor ridge appears less distinct at the end of the simulation period
(Fig. b). This feature is absent in the shelf removal simulation. In
contrast to the SMB simulation, the westward migration of the divide leads to a
narrowing of the angle between the two ridges, albeit only by 1.1∘, corresponding
to a narrowing of 1.4 %. The main component of the migration is east–west, but
in both simulations the triple junction also migrates south.
Triple junction migration for the Halvfarryggen Ice Rise induced by (a–c) surface
mass balance perturbation for the refined mesh and (d–f) ocean perturbation
(shelf removal) for the refined mesh. Upper and middle panels (a, b, d, e) were regridded onto a 500 m Cartesian mesh and
show the triple junction position at the start and end of the simulation period,
respectively. Lower panel (c, f) shows mean divide migration for the respective
perturbation simulations. Negative numbers indicate migration to the west. Note
different y-axis scales in (c) and (f).
DiscussionEffect of mechanical model in initialisation on transient
simulations
Transient simulations using basal drag coefficient and ice viscosity fields from
the SSA inversion result in unrealistically large ice mass loss rates in the
decades following model initialisation. Knowing that the Ekström catchment is
likely close to a steady state , we attribute these differences in the transient simulations to the difference in force balance
approximation used in the inversion and transient simulation. This shows that
for the presented ice-rise modelling a FS inversion is necessary for plausible
transient simulations.
Surface-mass-balance-induced divide migration
The computed mean divide-migration rates of 2.5–3.5 m yr-1 for both ice rises are
higher than what has previously been inferred from the geometric analysis of
Raymond stacks (0.5 m yr-1, Siple Dome; ). This means a
realistic SMB perturbation, which could have occurred at the Last Glacial
Maximum (LGM), leads to fast divide migration – even more so because these
migration rates are even higher (11–20 m yr-1) if only the first 200 years are
considered. Such a perturbation would likely lead to an abandoning of the
Raymond stack and the formation of a new Raymond stack at the new steady-state
divide position (Fig. e), as the divide migrates >3 ice thicknesses away from
its initial position. Because the Raymond stack at Halvfarryggen and Söråsen are
well developed and it takes ∼10T (T= characteristic time, T=900 years,
) to form such a stack, we conclude that both ice rises have not
experienced a perturbation of such magnitude over the last ∼9000 years,
indicating stable ice-flow conditions in this embayment for at least the
Holocene time period.
For both ice rises, divide migration shows a clear dependence on mesh
resolution. As the regular mesh simulations are most likely under-resolved, we
will mainly focus here on the refined mesh simulations. In the latter half of
the simulation period (Fig. a), the refined mesh simulations
show a subtle backward migration trend. We attribute this backward migration
pattern to be a direct result of an imbalance in balance fluxes for the eastern
and western side of the divide (Fig. c, d). The more positive SMB
on the eastern side of the divide results in an initial increase in balance
flux, which quickly decays over time because the grounding-line flux compensates
for the increased ice thickness by discharging more ice into the shelf. After
∼80 years the balance flux on the eastern side is lower than the
corresponding flux on the western side. However, the balance flux continues to
decrease because of the continuing increase in grounding-line flux up to
∼250 years, before it recovers slightly. The negative balance flux in the
east leads to the computed subtle back migration trend from ∼185 years to
∼700 years. However, the timing of when the balance flux in the east starts
to be lower than the balance flux in the west is 120 years prior to this
(Fig. a, d). This means that there is a time lag of 120 years
before the divide reacts to changes in the balance fluxes. The time lag may be a
little smaller since the balance flux in the east is also lower than the balance
flux in the west for the regular mesh simulation, but it does not result in
backward migration of the divide. This indicates that there must be a certain
magnitude in the imbalance of the balance fluxes on either side of the divide
before the divide responds to this. The difference in the balance flux analysis
between the regular and refined mesh (Fig. c, d) also highlights
the importance for fine mesh resolution to resolve these processes,
corroborating that mesh resolutions finer than 500 m are required. A flux analysis
could not be performed for Söråsen because the selected model domain does not
completely cover the ice rise.
Ocean-perturbation-induced divide migration
Since divide-migration amplitude is ≤1 ice thickness away from the initial
divide position and the mean migration rates are <0.75 m yr-1, we interpret this
as shelf thickness perturbations resulting in slow divide migration. Especially
in the context of complete shelf removal as a rather extreme perturbation,
the intermediate thinning scenarios might provide a more realistic experiment
for the recent past of the Ekström catchment. The simulated small migration
amplitudes for the intermediate shelf thinning scenarios (<300 m) indicate
that, due to the wedged-in geometrical setting of the Ekström Ice Shelf, a large
portion of the shelf thickness needs to be removed before any flux increase
across the grounding line becomes apparent and leads to migration of the divide.
This means that shelf thickness perturbations in our experiment would most
likely result in a left-tilted Raymond stack rather than lead to the
abandoning of the initial Raymond stack (Fig. f). The low
migration amplitudes also show that the employed mesh resolution (∼500 m)
may be insufficient for the intermediate scenarios, but owing to computational
restrictions this is the highest resolution possible.
As all mean migration amplitudes are <1 km, we will restrict our discussion to the
refined mesh simulations (runs 8–9, Table ). Based on our asymmetric buttressing
hypothesis, a simple interpretation of our results would be that the Jelbart Ice
Shelf for Halvfarryggen and the Ekström Ice Shelf for Söråsen provide more
buttressing than their respective counterparts in the west, as the divides of
both ice rises migrate to the east. However, when using Schoof's flux formula
together with the computed initial fluxes to estimate
buttressing (Θ) for Halvfarryggen, the derived values for Θ are
similar for both shelves (Table ). Despite the similar stress
reduction through thinning or removal of the ice shelf, the increase in absolute
flux across the grounding line differs. This asymmetry is not induced by
asymmetric buttressing but is caused by the difference in initial flux across
the grounding line, which is almost an order of magnitude higher in the east
than the counterpart in the west. If now the stress is reduced by the same
percentage, the flux imbalance between east and west will widen
(Table ),
resulting in the divide migrating to the west. We infer from our model
simulations that while buttressing induces divide migration, it is by no means
necessary to have asymmetric buttressing for the divide to migrate. The more
important determining factor as to how far the divide is going to migrate is the
absolute flux imbalance between the two sides of the divide. If we use the flux
imbalance from the three different shelf thinning/removal simulations (runs
8 and 10–11, Table ), the relationship is almost linear between flux imbalance and the
resulting divide migration. If this linear fit equation and the modelled flux
imbalances are used to predict divide migration for the same three simulations,
migration amplitudes of -718, -277, and -16 m are predicted. This compares
reasonably well with the computed divide-migration amplitudes after 100 years of
-734, -235, and -43 m, respectively. This does not mean that this
relationship must be linear but underlines the fact that flux imbalance is much
more important than the buttressing provided by the ice shelf for divide
migration.
Flux calculations, derived buttressing factors, and stress reduction
calculations for both sides of the divide for the Halvfarryggen Ice Rise
from runs 8 and 10–11 (Table ). GL flux reduction in relation to the shelf thickness perturbation simulations (column 2) is computed by dividing
ΔGL flux (column 3) by GL flux (column 2), and the buttressing factor and
stress reduction are calculated from (Eq. 29).
SimulationGL reference fluxGL fluxΔGL fluxGL fluxDerivedStressat Year 0at Year 0at Year 0reductionbuttressingreduction(×108 m3 yr-1)(×108 m3 yr-1)(×108 m3 yr-1)(–)factor Θ (–)(%)No shelf east9.55323.6514.100.5920.2179.0No shelf west1.8745.2273.3530.6420.1881.050 % shelf east9.55315.215.6570.3720.3664.050 % shelf west1.8742.9571.0830.3660.3664.090 % shelf east9.55310.961.4070.1280.6040.090 % shelf west1.8742.3150.4410.1910.5248.0Comparison of SMB- and ocean-induced divide migration
Despite the fact that the SMB perturbation represents the more physically
realistic perturbation rather than the most extreme shelf thickness perturbation (shelf
removal), the SMB perturbation results in fast divide migration and the shelf
thickness perturbation leads to slow divide migration. This leads us to a
different interpretation for the resulting geometry of the Raymond stack, with
SMB perturbations leading likely to a Raymond stack abandoning and ocean
perturbations resulting in a left-tilted Raymond stack
(Fig. e, f).
The response of the divide position to ocean perturbations is primarily
controlled by the subglacial topography with lateral buttressing only being a
controlling factor of secondary order. The modelled short-lived response of the
increased grounding-line flux to all ocean perturbations is typical of drainage
basins located on prograde sloping bedrock (Fig. ), where the instantaneous removal of
all buttressing leads to a sudden but short-lived response. Similar results
have been obtained from the modelling of ice-shelf collapse in the Antarctic
Peninsula region .
As both ice rises in the model domain, and to the authors' knowledge most other
ice rises around Antarctica as well, are located on subglacial topography
plateaus, the potential for grounding-line retreat is limited. Because ice flux
across the grounding line is primarily a function of ice thickness at the
grounding line , the initial retreat of the grounding
line on prograde slopes often leads to thinner ice at the grounding line and in
turn leads to a reduction of ice flux across the grounding line that can even be
lower than before the perturbation. Similarly, the response of the divide
position to SMB perturbations also seems to be primarily controlled by the
subglacial topography with the magnitude of the flux imbalance between east and
west of the divide being a controlling factor of secondary order. Evidence for
this is provided by the unadjusted SMB perturbation (Run 5,
Table ) and the SMB grounding-line flux perturbation (Run 6,
Table ) simulations, which both converge to the same new
steady-state divide position, even though the forcing is different by a factor of 2.
As SMB perturbations directly affect surface topography, this type of
perturbation leads to quicker response times of ice-divide migration in
comparison with shelf thickness perturbation with e-folding times of 95–170
and 30–50 years for shelf thickness and SMB simulations, respectively.
This means that not only is the magnitude lower, but also the timing of ice-rise
divide migration is delayed in the case of shelf thickness perturbations.
Moreover, even though the magnitude of the initial perturbation is lower for the
SMB simulations, divide migration is larger by a factor of ∼3.4 and ∼3.9 for
Halvfarryggen and Söråsen, respectively. The divide-migration rate and amplitude
for SMB perturbations is most likely heavily dependent on the spatial pattern of
the perturbation, with SMB perturbations near the divide likely leading to
faster and larger divide migration than SMB perturbations that have their
maximum farther away from the divide .
Cross sections of grounded subglacial topography from the BEDMAP2 dataset for the (a) Halvfarryggen and (b) Söråsen ice rises, underlining the prograde sloping subglacial topography of both ice rises in the vicinity of the current grounding line.
In spite of the lower divide-migration rates computed from shelf thickness
perturbations, the magnitude of the migration is still large enough to affect
the geometry of Raymond stacks. The computed magnitudes are of similar amplitude to divide-migration rates inferred for Siple Dome .
Moreover, the Ekström Ice Shelf and Jelbart Ice Shelf belong to the smaller ice
shelves around Antarctica, making it likely that ice rises with larger ice flux
across the grounding line in combination with larger buttressing provided by the
surrounding ice shelves may well have the potential to experience larger divide-migration rates.
The rate of triple junction migration appears to be closely linked to the
migration rate of the main divide arm and thus seems to be similarly susceptible
to migration as divide ridges. There appears to be a widening/splitting trend
of the triple junction in the SMB simulations and a narrowing/merging trend for
the shelf thickness simulations. However, the migration amplitudes are
insufficient to evaluate if a merging or splitting of a divide triple junction
might explain observed radar features such as relic arches in the divide flank .
Migration of the triple junction also bears importance for selecting potential ice-core drilling
sites on these types of ice domes, and a tilted Raymond stack may indicate a
displacement of the triple junction as well.
Model limitations
By calibrating our ice-sheet model on the Ekström Ice Shelf catchment, we aim to
introduce commonly employed initialisation techniques in large-scale ice-sheet
modelling to ice-rise modelling. The advantage of the calibration is that
buttressing is simulated in a realistic fashion. Without the calibration, large
thinning/thickening rates would result in unrealistic model results. However,
the calibration matches observed horizontal velocities with modelled horizontal
velocities without any constraints on vertical ice velocities. This leads to the
situation that any errors in the horizontal velocities propagate into the
vertical velocity through mass conservation. As horizontal velocities in the
divide region are close to zero, small errors in horizontal velocities have a
large effect on vertical velocities, and therefore we were unable to solve for the age field (Raymond
stacks). In addition, due to computational constraints, only 10 equally spaced
vertical layers could be employed. For an ice thickness of ∼900 m at
the Halvfarryggen Ice Rise, this corresponds to a vertical resolution of ∼90 m. While this vertical resolution is sufficient for our ice-rise divide
migration purposes, a much higher vertical resolution (∼30–40 layers) would
be necessary to model Raymond arches at the required detail
. Similarly, despite refining the mesh locally down
to 350 m, this may still not suffice for some of the shelf removal simulations, where finer meshes than presented here (<350 m) may result in larger divide-migration amplitudes.
In spite of the advanced ice-sheet model employed, compromises in the complexity
of the experimental setup had to be made to make these simulations
computationally feasible. These simplifications or approximations were done with
the goal of focussing on ice-rise divide migration at the expense of accurately
simulating Raymond arch formation. In the following, we will list these
simplifications and regard each of them as future avenues to improve on the
simulations presented here. As suggested by many previous ice-rise divide
studies, the commonly used exponent of the ice rheology law (n=3) is not able to
reproduce the Raymond arch amplitudes from observations, but often a higher
exponent (n≈4.5) is chosen that better matches the arch amplitudes from
observations
.
Moreover, showed that the commonly employed approximation
of ice being an isotropic material in large-scale ice-sheet models is not valid
at ice divides, where a preferential orientation of the ice crystals leads to
enhanced ice deformation. Changes in ice deformation can also be caused by
changes in ice temperature, where warmer ice leads to enhanced ice deformation
and cold ice reduces ice deformation. While the effect of temperature and
enhanced/decreased ice deformation could introduce differences in divide
position, to the author's knowledge no one has comprehensively shown this.
However, previous studies have found that thermomechanically coupled models
exhibit warmer ice at the base under the divide ,
which could potentially indicate that divide migration may occur faster. As our
model is not thermomechanically coupled, these effects are ignored in the
simulations. Even though ice temperature and anisotropy have been identified as
important parameters to be able to reproduce the internal structure of ice
divides, it is still uncertain as to how much they affect divide migration.
We performed our simulations with only one type of sliding law (linear
Weertman), without testing alternative implementations. Even though other
modelling studies have shown that this type of sliding law generally results in
smaller grounding-line retreat than other sliding laws (e.g. pressure-limited
sliding law; e.g.), it remains difficult to
assess the importance of the sliding law for divide migration. On the one hand,
reduced basal drag may lead to enhanced grounding-line retreat
, but on the other hand
ice near the divide region might be frozen to the bed and sliding can be
neglected in these areas. Therefore, we believe that the choice of the sliding
law is likely to have a limited impact on our results. Given that many previous
ice-divide modelling studies assume no basal sliding at all
e.g., the largest impact of this simplification
will be on the grounding-line position in the ocean perturbation simulations.
But even there, owing to the prograde sloping subglacial topography, the effect
of different sliding laws should not be a major concern for the computed divide-migration rates.
Conclusions
We used a calibrated 3-D ice-sheet model including grounding-line dynamics and
shelf flow for the Ekström catchment to investigate the coupled transient
response of ice-rise divides and triple junctions to perturbations in the SMB
and ice-shelf thickness. Our perturbation simulations for the Ekström catchment
reveal that SMB perturbations result in fast divide migration (up to 3.5 m yr-1),
while shelf thickness perturbations only trigger slow divide migration (< 0.75 m yr-1). The amplitude of divide migration is predominately controlled by the
subglacial topography and SMB, with ice-shelf buttressing being of secondary
importance.
We find in our simulations that asymmetric buttressing is not a required
condition for ice-rise divide migration, but rather how much the divide
will migrate is determined by the flux imbalance between either side of the
divide. Both ice rises show a closely coupled response to the perturbations with
divide migration being similar in timing and magnitude. Based on our
simulations, the geometry of the Raymond stack could provide clues about the
forcing mechanism behind the divide migration, with an abandoned Raymond stack
being more likely linked to SMB perturbations. For tilted Raymond stacks the
interpretation of the internal structure remains difficult with either a smaller
SMB perturbation than prescribed here or shelf thickness perturbations equally
likely. It is important to further unravel (e.g. through synthetic experiments)
the different trigger mechanisms for different types of Raymond
arch geometries in order to fully unlock the potential of ice rises as
ice-dynamic archives, potential ice-core drilling site, and to better constrain
palaeo-ice-sheet models.
We find that a high mesh resolution (<500 m) is required in the vicinity of the
dome and the grounding-line to capture ice-rise divide migration at the desired detail, as mean maximum
migration amplitude is <4 km in our perturbation experiments. To avoid
unrealistic ice mass loss in transient simulations around the divide region,
where longitudinal and bridging stresses are important, the same force balance
approximation (e.g. FS for ice divides) should be used in the initialisation and
forward simulation of the ice-sheet model.
Finally, migration of the triple junction closely follows the migration pattern
of the main ridge, which may prove useful in the future selection of ice-core
drilling sites. However, more targeted simulations are required to determine
whether a merging or splitting of the triple junction can explain relic Raymond
stacks in the flanks of ice rises. The model setup is suitable for
glacial/interglacial simulations on the catchment scale, providing the next step
forward to unravel the ice-dynamic history stored in ice rises all around
Antarctica.
Code and data availability
The Elmer/Ice code is publicly available through GitHub (https://github.com/ElmerCSC/elmerfem, last access: 10 September 2019). All simulations were performed with version 8.3 (rev. 74a4936). Elmer/Ice sample scripts and input files for the runs 3–5 and 7–9 are available at 10.5281/zenodo.3469937 (Schannwell 2019, last access: 2 October 2019).
Author contributions
CS and RD conceived the study with input from OE, TAE, and
CM. Simulations were run by CS with assistance from FGC. The manuscript was
written by CS and RD, and all authors contributed to editing and revision.
Competing interests
Olaf Eisen is a co-editor in chief of TC.
Acknowledgements
We thank Veit Helm for providing us with the TanDEM-X digital elevation models. Clemens Schannwell
was supported by the Deutsche Forschungsgemeinschaft (DFG) grant EH329/11-1 (to TAE) in the framework of
the priority programme “Antarctic Research with comparative investigations in
Arctic ice areas”. Reinhard Drews was funded in the same project under MA 3347/10-1. Reinhard Drews is supported by the DFG Emmy Noether grant DR 822/3-1. The authors gratefully acknowledge
the compute and data resources provided by the Leibniz Supercomputing Centre
(https://www.lrz.de/, last access: 2 October 2019).
We thank the editor Carlos Martin and two anonymous reviewers for comments which improved the manuscript.
Financial support
This research has been supported by the Deutsche Forschungsgemeinschaft (grant nos. MA 3347/10-1 and EH329/11-1).This open-access publication was funded by the University of Tübingen.
Review statement
This paper was edited by Carlos Martin and reviewed by two anonymous referees.
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