TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-11-319-2017Marine ice sheet model performance depends on basal sliding physics and sub-shelf meltingGladstoneRupert Michaelhttps://orcid.org/0000-0002-1582-3857WarnerRoland CharlesGalton-FenziBenjamin KeithGagliardiniOlivierhttps://orcid.org/0000-0001-9162-3518ZwingerThomashttps://orcid.org/0000-0003-3360-4401GreveRalfhttps://orcid.org/0000-0002-1341-4777VAW, Eidgenössische Technische Hochschule Zürich, ETHZ, Zürich, SwitzerlandAntarctic Climate and Ecosystems Cooperative Research Centre, University of Tasmania, Hobart, AustraliaArctic Centre, University of Lapland, Rovaniemi, FinlandAustralian Antarctic Division, Kingston, Tasmania, AustraliaUniv. Grenoble Alpes, CNRS, IRD, IGE, 38000 Grenoble, FranceCSC – IT Center for Science Ltd., Espoo, FinlandInstitute of Low Temperature Science, Hokkaido University, Sapporo, JapanRupert Gladstone (rupertgladstone1972@gmail.com)31January201711131932919June20166July201611November20163January2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/11/319/2017/tc-11-319-2017.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/11/319/2017/tc-11-319-2017.pdf
Computer models are necessary for understanding and predicting marine ice sheet behaviour.
However, there is uncertainty over implementation of physical processes at the ice base,
both for grounded and floating glacial ice. Here we implement several sliding relations
in a marine ice sheet flow-line model accounting for all stress components
and demonstrate that model resolution
requirements are strongly dependent on both the choice of basal sliding relation and the
spatial distribution of ice shelf basal melting.
Sliding relations that reduce the magnitude of the step change in basal drag from grounded
ice to floating ice (where basal drag is set to zero) show reduced dependence on
resolution compared to a commonly used relation, in which basal drag is purely a power law
function of basal ice velocity. Sliding relations in which basal drag goes smoothly to zero
as the grounding line is approached from inland (due to a physically motivated incorporation
of effective pressure at the bed) provide further reduction in resolution dependence.
A similar issue is found with the imposition of basal melt under the floating part of the ice shelf:
melt parameterisations that reduce the abruptness of
change in basal melting from grounded ice (where basal
melt is set to zero) to floating ice provide improved convergence with resolution compared to
parameterisations in which high melt occurs adjacent to the grounding line.
Thus physical processes, such as sub-glacial outflow (which could cause high melt near the
grounding line), impact on capability to simulate marine ice sheets.
If there exists an abrupt change across the grounding line in either basal drag or basal
melting,
then high resolution will be required to solve the problem.
However, the plausible combination of a physical dependency of basal drag
on effective pressure, and the possibility of low ice shelf basal melt rates next to the
grounding line, may mean that some marine ice sheet systems can be reliably simulated
at a coarser resolution than currently thought necessary.
Introduction
Ice sheet models (ISMs) are increasingly being used in process studies, sensitivity studies and projections
of marine ice sheet (MIS) future behaviour , and model intercomparison projects
(MIPs) to investigate the ice sheet response to ocean forced basal melting
of ice shelves are currently in their design phase .
Past ISM studies have shown inconsistent grounding line behaviour at typical
resolutions .
Inconsistencies were very large (grounding line discrepancies of
≈ 100 km) for grid resolutions of ≈ 10 km
and typically still not converged for grid resolutions of ≈ 1 km.
Studies in which simulations are carried out
at multiple different mesh resolutions usually demonstrate convergent behaviour,
but very fine resolution is often needed to approach a converged
solution .
Practical solutions have been suggested, such as
parameterising the flux of ice across the grounding line as a function of ice
thickness , parameterising the grounding
line position at sub-grid resolution , or implementing adaptive mesh refinement to
provide very high resolution at and near the grounding line .
These solutions all have limitations, and the computational cost of running a sufficiently high-resolution ISM
to robustly represent grounding line motion remains high, even with adaptive refinement.
However, model-based MIS studies (e.g. ) typically use a simple basal traction prescription,
or “sliding relation” , which neglects
the impact of effective pressure at the bed (or equivalently “height above buoyancy”) on basal shear stress.
The inclusion of pressure dependency (reviewed by ) provides a physical motivation
for smoothing out what is otherwise a large step change in basal drag across the grounding line.
This was proposed over 30 years ago (e.g. ) and may affect the resolution
requirements for successful grounding line modelling.
Recent treatments of basal traction that also vanish smoothly at the grounding line
include and .
Furthermore, the implications of imposing basal melting (note that in the current
study “basal melting” refers
always to melting under the ice shelf and not the grounded part of the MIS) on resolution
requirements have not been explicitly investigated.
In the current study, we assess the impact of choosing between different sliding relations, and
between different approaches to parameterising basal melting,
on model resolution requirements in a Stokes flow ice dynamic model.
Methods
We use the ice dynamic model Elmer/Ice .
The Stokes equations for a viscous fluid with non-linear rheology are solved using the finite element
method over a two-dimensional
flow-line domain (one vertical and one horizontal dimension) in which lateral drag is
parameterised according to channel width, W,
and a contact problem is solved to determine the evolving grounding line position .
The rheology follows Glen's law
with viscosity calculated using a temperature-dependent Arrhenius law .
Temperature is held constant at -10 ∘C for all simulations in the current study.
We implement alternative sliding relations (Sect. ) and
a basal melt parameterisation (Sect. ) in Elmer/Ice.
Basal sliding
The form of the sliding laws used in the current study is motivated by early
laboratory sliding experiments and Antarctic simulations ,
which suggested modifying the original Weertman sliding relation by
incorporating a power law dependence of the drag on effective pressure at the bed as follows:
τbp=-Cubmz*q,
where
τb is basal shear stress,
ub is basal ice velocity,
z* is the height above buoyancy (related to effective pressure at the bed, N,
by N=ρigz*),
m, p and q are constant exponents,
and C is a constant sliding coefficient.
Besides the laboratory studies of ice sliding already mentioned, the introduction of
basal effective pressure into sliding in the 1980s, particularly in the context of a
marine ice sheet and the identification (via z*) with ocean pressure, was further
motivated by a characteristic feature of West Antarctica's fast-flowing ice streams:
increasingly rapid flow towards the grounding line (and generally decreasing z*)
despite a steadily decreasing surface slope and hence gravitational driving stress.
Various parameterisations were developed from the information about velocities,
surface slopes and ice thicknesses available then (see e.g. ,
and references therein).
In the current study we set m=1/3 and p=q=1 for all simulations.
These values for p and q are chosen for simplicity and deviate from the
original values tuned for large-scale ice sheet simulations .
We impose z*≥0 when calculating τb.
Ideally z* would be calculated using basal water pressure from a sub-glacial hydrology model.
In the current study, we simply use hydrostatic balance based on sea level,
z*=H,if b>=0H+bρoρiif b<0,
where
H is local ice thickness,
b is the bedrock elevation relative to sea level (positive upwards),
ρo is the density of ocean water, and
ρi is the density of ice.
This is equivalent to assuming a sub-glacial hydrology system fully connected to the ocean.
The four sliding relations used in the current study are given by
τb=-C1ub13,τb=-C2ub13z*,τb=-C3ub13z*H,τb=-C4ub13(z*+zo),
where zo is a thickness offset and Cn are sliding coefficients.
The first two sliding relations (given by Eqs. and ,
and henceforth referred to as
SR1 and SR2 respectively) are specific cases of Eq. ()
and derive from previously published sliding laws.
SR1 is widely used in model intercomparison studies, such as the
Marine Ice Sheet Model Intercomparison Project (MISMIP, ), and features an abrupt change
in basal shear stress from grounded to floating ice.
It is commonly referred to as “Weertman sliding” after .
SR2 implements a smooth transition of basal drag to zero as the grounding line is approached
from landwards.
The form of SR2 is based on modifying SR1 for our study.
It is motivated by parameterisations
of sliding relations for fast-flowing ice streams in West Antarctica,
where a linear relation between τ and z* was observed to hold
towards the grounding line, although their parameter choice was p=q=m=1.
It is worth noting that a number of other sliding relations have been published
in which the transition of basal
shear stress across the grounding line is less abrupt that in SR1.
For example implemented a fixed size transition zone.
Theoretical work for the case of sliding with cavitation has also been used
to develop sliding relations in
models , though it is not clear whether the assumptions
made are applicable in all real-world cases of glacier sliding.
The current study does not aim to promote use of any particular sliding relation, but rather
to explore a specific aspect of the sliding implementation, namely the abruptness
with which basal shear stress goes to zero as the grounding line is approached.
Sliding relations SR3 and SR4 implement further modifications to SR2 in order to explore this aspect of sliding.
SR3 (Eq. ) uses thickness scaling to give a law which
captures the smooth fade to zero of basal drag approaching the grounding line of SR2,
but which equates to the familiar SR1 for ice grounded above sea level, providing ice sheet
profiles more directly comparable to SR1.
SR3 can be regarded as restricting the assumption that basal water pressures is
directly tied to ocean pressures as one moves inland from the grounding line.
The aim of SR4 (Eq. ) is to provide a step-change
in basal drag from grounded to floating ice, but one with significantly smaller
magnitude than
would occur with a Weertman-type (SR1) sliding relation.
The sliding relations and their coefficient values are summarised in Table .
The coefficient
values were chosen to give approximately similar grounding line positions after the initial
spin-up and advance experiments.
Sliding relations and constants used in the current study.
We implement a parameterisation for ice shelf basal melt rate, mb,
similar to that used in the Marine Ice Sheet Ocean Model Intercomparison Project
phase 1 (MISOMIP1), and described in the MISOMIP1 experimental setup .
This parameterisation in its original form increases with depth
due to the pressure-driven depression of the seawater freezing point along
the ice shelf base and hence would
generally give a maximum in mb adjacent to the grounding line.
However, a
parameterisation that only considers the pressure-enhanced thermal driving does
not account for the sub-ice cavity geometry that may limit oceanic heat transport
right to the grounding line.
Nor does it account for the impact of a sub-glacial outflow that may trigger or strengthen
a buoyant meltwater plume at the ice–ocean interface.
Ice shelf melt rates close to (within 20–30 km of) the grounding lines of major
Antarctic outlet glaciers are typically much higher than ice shelf average values,
sometimes by an order of magnitude .
However, did not investigate the spatial patterns of
melt rates within the regions close to the grounding line.
A plume model study suggests that, in the presence of sub-glacial
outflow at the grounding line, significant melting can occur adjacent to the
grounding line and that the maximum in melting is likely to occur close to,
but not adjacent to, the grounding line.
A 3-D ocean modelling study is in agreement with this result and further
indicates that a reduction in strength of sub-glacial outflow can reduce the strength
of melting adjacent to the grounding line.
Simulations using a plume model with no sub-glacial outflow
show a melt rate that peaks tens of kilometres from the grounding line
and decays to zero at the grounding line.
Here we implement an optional melt-scaling parameter, Sw, used to reduce mb
smoothly to zero as the grounding line is approached from the ice shelf.
By carrying out simulations both with and without
this melt scaling we effectively implement two opposite end members for the melt distribution:
a smooth transition to zero melting at the grounding line and
maximum melting adjacent to the grounding line.
The melt rate mb is calculated in m a-1 ice equivalent and is parameterised as a
function of depth by
mb=SwSicwγTLΩΔT,
where
ΔT is the “thermal driving”,
L is the latent heat of fusion of ice,
cw is the heat capacity of seawater,
γT=10-4 is a heat transfer coefficient,
Ω is a dimensionless tuning parameter,
and Sw and Si are scaling factors.
The thermal driving is the far field to local temperature difference,
ΔT=Tf-To,
where Tf is the local freezing point of seawater,
and To is the far-field ocean temperature.
Tf is approximated here in degrees Celsius using Tf=-1.85+7.61×10-4zi,
where zi is the depth of
the ice base relative to sea level (positive upwards).
We set To=2.0∘C for our simulations.
The scaling factor Sw is implemented as a function of water column thickness,
Hw (given by Hw=zi-b), to reflect the influence of cavity geometry,
Sw=tanheHwHw0,
where Hw0 is a reference water column thickness.
S approaches 1 in deeper water (Hw>Hw0).
We present simulations both with and without the water column thickness scaling.
Where it is used we set Hw0=100 m.
Where it is not used we set Sw=1.
Iceberg calving is not represented in the current study, and the ice shelf front position remains fixed.
This results in a vanishingly thin ice shelf in some simulations and can cause numerical instabilities
and model failures.
Si is an ice-shelf depth-scaling parameter introduced to avoid the occurrence of a vanishingly thin
ice shelf.
Si is given by
Si=maxtanhezi0-zizs,0,
where zi0 is a reference ice base height relative to sea level (positive upwards)
and zs is a (directionless) scaling depth.
In practice the use of Si gives zero melting for zi>zi0.
The Si scaling is used in all simulations with values zi0=-40m
and zs=100m.
In the simulations presented here melt is applied to all mesh nodes in the floating part
of the ice sheet.
Simulations were also carried out in which melting was also applied to the last grounded
node, and this was found not to cause a large difference: the results and interpretation
presented here hold for both cases.
The experiments are described in Sect.
and Table .
Model resolutions used in the current study.
ResolutionNumber of elementsElement size inin the horizontalthe horizontalR02507.2 kmR15003.6 kmR210001.8 kmExperiment design
The experimental set-up involves an 1800 km domain with linear down-sloping bedrock,
b, given in kilometres relative to sea level by
b=0.2-0.9x1800,
where x is horizontal distance in kilometres from the ice divide.
This gives a bedrock elevation varying between z=200m and
z=-700m, where z is the vertical coordinate measured
relative to sea level.
Net surface accumulation, a, is given in m a-1 by
a=x1800ρoρi.
The upstream boundary represents the ice divide, and a Dirichlet condition is used here to set
the horizontal component of velocity to zero, ensuring flow symmetry.
An external hydrostatic pressure distribution imposed by the ocean (below sea level)
is prescribed at the spatially fixed downstream calving front.
This external pressure is also applied to the base of the ice shelf.
The mesh is composed of quadrilateral elements with 11 equally spaced layers in the vertical direction.
Each experiment has been run three times with different resolutions in the
horizontal (Table ).
The resolutions chosen are indicative of resolutions that could be achieved by large-scale
Stokes simulations of ice sheets with the current generation of models.
Thus they are coarser than is commonly considered to be required for self-consistent simulations
involving grounding line movement .
This is intentional so that the current study may assess the potential for different sliding laws or
basal melt parameterisations to achieve resolution-independent behaviour at coarser resolutions than is required
with sliding relations similar to SR1.
Summary of experiments.
The experiment name is given in bold for experiments whose results are analysed in the current study.
Experiments not given in bold provide spin-up/initialisation for those analysed.
The basal melt forcing is described in Sect.
and the experimental design in Sect. .
W is parameterised channel width,
Sw is the water column thickness scaling of basal melt (Eq. ), and
Ω is a basal melt tuning parameter.
ExperimentDescriptionInitial conditionWSw used?ΩSPINInitial spin-upUniform slab (H=300 m)1000 km––ADVAAdvance due to buttressing increaseSPIN final state150 km––RBRetreat due to buttressing reductionADVA final state1000 km––RHWRetreat due to high melt (with water column scaling)ADVA final state150 kmYes0.045RHWBRetreat due to high melt (with water column scaling)ADVA final state1000 kmYes0.045and buttressing reductionRLWRetreat due to low melt (with water column scaling)ADVA final state150 kmYes0.009ALWAdvance due to lowering of melt (with water columnRHWB final state150 kmYes0.009scaling)RHBRetreat due to high melt (no water column scaling)ADVA final state1000 kmNo0.045and buttressing reductionRLRetreat due to low melt (no water column scaling)ADVA final state150 kmNo0.009ALAdvance due to lowering of melt (no water columnRHB final state150 kmNo0.009scaling)
The experiments are summarised in Table .
Spin-up is performed in two stages:
the first stage (“SPIN”, Table ) is from a uniform
thickness (300 m) slab of ice for 40 ka with parameterised channel width
1000 km (very low buttressing).
The second stage (“ADVA”, Table ) constitutes a further 40 ka
with parameterised channel width of 150 km
(significant buttressing).
This two-stage spin-up is carried out separately for each resolution and sliding law.
The purpose of including buttressing is to provide a mechanism for basal melting under the shelf to
impact on grounded ice.
This impact is through ice shelf thickness change: a thicker ice shelf provides more
buttressing.
Note that basal melting is zero during both stages of the spin-up.
Retreat simulations are then carried out (experiment names beginning with “R” in
Table ), which form the main focus for this study.
The cause of retreat is a change in forcing.
In the “retreat due to buttressing reduction” experiment (RB), the forcing change is a reduction in
the lateral drag back to a
parameterised channel width of 1000 km.
In the “retreat due to high melt with water column scaling” experiment (RHW)
basal melting (Sect. ) is imposed under the ice shelf.
Both forcing changes are applied together in the
“retreat due to high melt with water column scaling and buttressing reduction” experiment (RHWB).
For the melt induced retreat simulations we set Ω to 0.045 or 0.009 (Table ),
resulting in typical melt rates between 1 and 10 m a-1.
A variation on RHW is the “retreat with low melt with water column scaling” experiment (RLW).
We also carry out re-advance experiments
(ALW and AL in Table )
to test whether simulations reach the same steady-state
in advance as in retreat under identical forcing.
The full set of experiments and corresponding parameters are given in
Table .
Individual simulations are referred to in the results section by their “simulation code”,
made up of the experiment name (Table ),
the sliding relation used (Table ) and the resolution
(Table ).
For example, SPIN_SR1_R1 is the initial spin-up with Weertman
sliding and an element size of
3.6km.
Results
Our main criterion for assessing the results is resolution dependency.
The model used here, Elmer/Ice, has been demonstrated
in previous studies (e.g. )
to give convergent behaviour with resolution: its output approaches a self-consistent solution as resolution is made increasingly fine.
In the current study we do not attempt to demonstrate convergence in all cases (indeed convergence
is certainly not achieved in all cases), but instead
consider the dependence on resolution across the three resolutions used (Table ),
under the premise that weaker
dependence on resolution is an indicator of being closer to the converged solution.
The causes of strong resolution dependency in the current study will be discussed in
Sect. .
Specifically, we consider experiments in which the grounding line position
differs between simulations of different resolution by distances of
approximately the same magnitude as the size of a single
element not to have significant dependency on resolution.
Conversely, we consider experiments with grounding line differences of several element sizes or greater
to have significant dependence on resolution.
For example, differences in grounding line position of the order of 100 km between
simulations at different resolutions
are considered to indicate significant
resolution dependency, whereas differences of the order of 1 km are not.
Similarly, when we say “near to the grounding line” we are also talking in terms of
element size. For example “high melt near the grounding line” can be interpreted as
“high melt within a very small number of elements of the grounding line”.
We focus mainly on the evolution of grounding line position.
The spin-up simulations (SPIN and ADV, Table ) do not
vary significantly with resolution, and so our analysis focuses on retreat
and re-advance simulations.
The fact that the spin-up simulations show very little dependency on resolution
is not an indicator that the retreat simulations should show equally low dependency
on resolution.
Previous studies have shown that ice sheet models often demonstrate much
higher resolution dependency in retreat than advance, or vice
versa .
This discrepancy between retreat and advance experiments implies multiple possible
steady states for a given forcing .
Ice geometry and velocity magnitude (m a-1) at steady state from the
ADVA experiment.
Resolution R1 is shown, but these profiles do not vary significantly with resolution.
These profiles provide the starting point for the retreat simulations.
The final state of the melt-induced experiment RHW is overlain in grey outline.
Bedrock is shaded in grey.
Vertical exaggeration is 150 times.
However, the ice geometry in the spin-up simulations does
vary significantly with choice of sliding relation (Fig. ).
The steady-state ADVA_SR1 profiles have their steepest surface slope close to the
grounding line
due to the step change in basal drag from grounded to floating ice
(Figs. and ).
The steady-state SR2 and SR4 profiles (Fig. )
have their greatest surface slope further inland
where the overburden pressure becomes important.
The steady-state SR3 profile is similar to SR1 towards the ice sheet interior
but is thinner due to having lower drag close to the grounding line.
It is very similar to SR2 near the grounding line.
Stress tensor component σxz shown at the end of the ADVA experiment
for SR1 (top) and SR3 (bottom) for the seaward 1000 km of the domain.
At the base of the ice this approximates the basal shear stress, given the low slope of the bed.
Note the different colour scales.
Distance from the ice divide is shown along the bottom in kilometres.
Height relative to sea level is shown at the right end of the plots in metres.
Bedrock is shaded in grey.
Vertical exaggeration is 200 times.
Figure also shows the shear stress component of the Cauchy
stress tensor, σxz, for SR1 and SR3.
Given the low gradient of the bed, σxz at the bed is an approximation to basal drag.
Shear stress peaks at the grounding line for SR1 due to the step change in basal drag.
The inclusion of dependence on effective pressure at the bed via z*
in SR3 leads to a gradual change and a much
larger transition zone.
The impact of choice of sliding relation on the way the modelled ice sheet
responds to changing resolution is
shown for retreat simulations in Fig. .
The sliding relations featuring a step change in basal drag across the grounding line
(SR1, SR4a and SR4b, shown in teal, black and blue respectively – the top
three families of curves in the upper plot)
do not exhibit consistent behaviour with resolution, with the
RB_SR1_R0 simulation in particular showing no retreat of the grounding line after the
buttressing reduction.
Note that of these three simulations the magnitude of this basal drag
step change is smallest for SR4b and largest for SR1.
The results in Fig. are consistent with a smaller step change in basal drag
being indicative of better convergence with resolution, similar to a previous result
when using a “shelfy-stream” ice sheet model .
However, even SR4b still shows significant resolution dependency,
indicating that much finer resolution than is considered here would be
required for a reliable simulation.
The sliding relations in which basal drag goes smoothly to zero as the grounding line is approached
(SR2 and SR3, shown in green and red respectively)
show the most consistent behaviour with resolution in experiments RB and RHW.
Evolution of grounding line position relative to the inland boundary
during retreat simulations with different
sliding relations.
Sliding relations are described in Sect. and Table .
Experiments are described in Sect. and Table .
Resolutions (Table )
are coarse (R0, dashed line), medium (R1, dotted line) and fine (R2, solid line).
The vertical ordering of the families of curves matches that of the legend tables.
The purpose of the RHW experiment (Fig. , lower panel),
as distinct from the RB experiment, is to test dependence
on resolution in the presence of basal melting.
Because SR1 and SR4a showed strong resolution dependence already in RB they have been omitted
from the RHW experiments.
In general the consistency across different resolutions appears weaker in the case of the
melt-induced retreat simulations (RHW) than the reduced buttressing simulations
(RB).
The ice sheet profiles at the end of the RHW simulations are outlined in grey in
Fig. for sliding relations SR2, SR3 and SR4b.
The SR3 retreat simulations are unique in exhibiting an overshoot:
after a strong initial grounding line retreat a small advance is seen.
In the RHW_SR3_R2 simulation damped oscillations can be seen.
The reason for this behaviour is not clear, but the lower-resolution simulations fail to
exhibit this behaviour, indicating at least some resolution dependency in this experiment.
The grounding line positions in the RHW_SR3 simulations also do not show a monotonic
progression with increasing resolution.
Figure shows more clearly the resolution dependency in the final
grounding line positions for the RB and RHW experiments, after 40 ka.
Again, R1, R4a and R4b show much stronger resolution dependence than
R2 and R3.
Final grounding line position (after 40 ka)
against resolution (Table )
for the retreat due to buttressing reduction
(RB, left panel) and retreat due to basal melting
(RHW, right panel) with the different sliding relations
(Table ).
The y axis range is identical in both panels.
The vertical ordering of the families of curves matches that of the legend tables.
We now look more closely at the impact of basal melting
on resolution dependency.
We compare retreat and re-advance simulations.
These experiments involve the lower melt and greater buttressing scenarios,
applied to different starting configurations (see Table ).
We also investigate
the impact of the water column scaling, in which
zero melt is approached close to the grounding line.
Sliding relation SR2 is used for these experiments as it has shown much weaker resolution dependency
than SR1 and SR4 and has not shown difficult-to-interpret behaviour such as the
damped oscillations in RHW_SR3_R2.
Figure shows both retreat (RL and RLW, red lines) and advance
(AL and ALW, black lines)
simulations with water column scaling either on (RLW and ALW) or off (RL and AL).
The advance simulations have identical inputs to the corresponding retreat simulations
in all respects except for initial conditions.
Note that while the majority of these simulations were run for 20 ka, the AL simulations
were run for 40 ka because 20 ka was not long enough to approach a steady state.
Evolution of grounding line position during the sub-shelf melting simulations
with effective pressure dependency in the basal sliding relation.
Right-hand panels show results with the water column scaling factor Sw, active.
Left-hand panels show the abrupt melting transition.
The upper panels show the detail of the early stages of the retreat simulations,
whereas the lower panels show the full simulations.
The sliding relation (SR2) is described in Sect. and Table .
Experiments are described in Sect. and Table .
Resolutions (Table )
are coarse (R0, dashed line), medium (R1, dotted line) and fine (R2, solid line).
In the presence of water column scaling
the advance and retreat simulations approach the same grounding line position at
all resolutions, showing no significant resolution dependency
(right-hand plots of Fig. ).
This is consistent with the premise that a unique solution exists,
which might be expected behaviour on a linear down-sloping bed
, although this has not been
proven in the presence of buttressing and basal melting
that depend on ice shelf geometry.
However, where there is a large step change in basal melt across the
grounding line (AL and RL, left-hand plots of Fig. ),
the advance and retreat grounding lines do not approach the same final
position. Behaviour is strongly resolution dependent, especially in the
re-advance experiment. It is unclear whether retreat and re-advance
simulations would eventually converge
to the same solution, and
finer-resolution simulations would be required to determine this with confidence.
Dependency on resolution appears to be stronger in the case of advance
experiments than retreat experiments.
This is in sharp contrast to the SPIN and ADVA experiments, which are a kind of
advance experiment (in that the grounding line position is advancing through
the simulation toward its final position), in which no significant
resolution dependency was observed.
This suggests that it is specifically the melting
which causes resolution dependence and that it causes greater resolution dependence in
advance than in retreat.
The step changes in grounding line position during the early stages of
retreat (Fig. upper panels) are typically indicative
of a single element retreat for RLW_SR2 but are typically multiple element
retreat steps in RL_SR2.
Discussion
As in previous studies with different ice dynamic models (e.g.
), a step change in basal drag across
the grounding line causes strongly resolution-dependent behaviour in the
current study using the Elmer/Ice finite element Stokes flow model. A large
step change causes stronger resolution dependency than a smaller step change.
A comparable resolution dependency on basal melt is shown in the current
study: a step change in basal melt across the grounding line causes
significant resolution-dependent behaviour, worse for larger step changes.
Cases demonstrating strong resolution dependence at the resolutions presented
here are of low interest to the current project, which aims to identify
situations where such resolutions are viable. Much weaker resolution
dependence is found in the current study in the case where both basal drag
and basal melt approach zero as the grounding line is approached from
landward and seaward respectively.
A change in value of sliding coefficient for a given sliding relation can also impact on
resolution requirements .
But since SR1 will typically give a global maximum basal shear stress at the
grounding line, and SR2 will typically give a global minimum basal shear stress at the
grounding line, it is expected (and this is the result of
comparing to the current study) that choice of sliding relation has
much greater impact on resolution requirements than the magnitude of the sliding coefficient.
The results of the melting experiments have important implications for
application of model studies to real marine ice sheet systems. We have shown
that, even when the ice sliding relation permits resolution-independent
simulations at the widely achievable resolutions used in the current study,
this situation can be negated by the abruptness of spatial onset of ice shelf
basal melting. In RHW_SR2, RLW_SR2 and ALW_SR2 experiments, where the
onset of basal melting was gradual due to the scaling factor Sw
(Eq. ), acceptable behaviour was observed over the sequence of
resolutions we explored. However, even in low melt rate scenarios, the
absence of this gradual transition gave rise to much more significant
resolution dependence and a failure of retreat and readvance simulations to
arrive at a unique grounding line location. Clearly more studies are required
to explore the influence of abrupt spatial onset of melting. As discussed
earlier (Sect. ) high melt rates are observed within tens
of kilometres of the grounding lines of major Antarctic outlet glaciers, with the
likelihood that such melt rates occur immediately adjacent to the grounding
line in the presence of strong sub-glacial outflows. Accordingly, marine ice
sheet systems with low surface slopes near the grounding line (indicating low
basal drag approaching the grounding line) and with low basal melting near
the grounding line (such as might be the case in the absence of strong
sub-glacial outflow) would likely be more easily achievable targets for
modelling studies at the resolutions explored in the present study. For model
studies of less tractable systems, very high resolution would be needed near
the grounding line. While sub-grid parameterisations for grounding line
position or cross-grounding line ice flux have been developed (e.g.
), there is clearly a new
challenge to handle the influence of onset of basal melting on the near
grounding line dynamics. Furthermore, parameterisations that work well in the
absence of ice shelf basal melting will need to be tested in the presence of
melting and may need to be modified.
The basal melt parameterisations presented here, in particular the choice of
whether or not to implement water column scaling, are intended to provide
opposing end members in terms of melt distribution near the grounding line.
The current study has demonstrated the impact this choice has on required
model resolution but does not advocate a particular melt parameterisation.
Similarly, the sliding laws SR1 and SR2 are opposing end members in terms of
basal shear stress near the grounding line. The choice of sliding relation
has been shown to impact on resolution requirements, but a specific sliding
relation is not advocated here. The choice of both melt parameterisation and
sliding relation should be governed by the physical processes, not by
numerical convenience. Our aim has been to demonstrate that different
physical systems can have different resolution sensitivities.
The sliding relations presented here in which dependence on effective
pressure at the bed is incorporated would have a stronger physical
justification if used in conjunction with a computer model for sub-glacial
hydrology, to replace the assumption that the hydrologic system is everywhere
in contact with the ocean with a physically justifiable effective pressure
distribution. It might be expected that, for the case of efficient
channelised sub-glacial drainage , a strong
hydrologic connection to the ocean may exist. However, in such a case, there
may be very high local variations in basal water pressure, resulting in
sticky spots (relatively low basal water pressure and hence high basal shear
stress) in between active channels. To simulate such a system a very high
model resolution would need to be used to represent basal processes, and
potentially also for the grounding line, if these sticky spots are present
close to the grounding line. For the case of less efficient “distributed”
drainage , a lower resolution would suffice for the
hydrology system, and perhaps also for the grounding line, since there would
likely be uniformly high basal water pressures (i.e. low effective pressure)
near the grounding line. Studies of grounding line behaviour in a coupled
hydrology–ice sheet model would be of great benefit to further understand
this issue.
The results from the current study appear to be in conflict with the findings
of , who found that imposing a fixed length transition
zone near the grounding line (similar to that proposed
by ), where the basal drag is scaled linearly to zero as
the grounding line is approached (from landward), did not significantly
reduce the resolution requirements. There are, however, a number of
significant differences between the current study and ,
such as the use of a direct physical motivation to impose the drag reduction
in the current study, rather than imposition of linearity. We speculate that
the key factor is that the imposed linear transition zone of
is typically of the same order of magnitude as the
element size, meaning that the step change in basal drag across the grounding
line, while moderately reduced, is not reduced by an order of magnitude or
more, as in the current study for SR2 and SR3. The effect of incorporating
dependence on basal effective pressure on the basal stress gradient
approaching the grounding line is evident for the current study in
Fig. . The transition zone is several hundred kilometres for
SR3. A future study with further simulations will be needed to fill the gap
in experiment design between the two studies to confirm whether this
difference in transition zone size is the actual explanation for the
differences in resolution dependence between the two studies.
Conclusions
We have demonstrated that resolution requirements for marine ice sheet simulations
with an evolving grounding line
are highly sensitive to the physical implementation of both basal sliding and ice shelf
basal melting.
In particular a large step change in either basal drag or basal melting across the grounding line
can cause strong dependence of model behaviour on resolution.
Any marine ice sheet modelling studies whose outcomes involve a moving grounding line should
demonstrate convergent behaviour with resolution over the region of parameter space relevant
to their experimental setup, bearing in mind that basal drag and basal melt can both
cause resolution dependence
and that resolution dependence may differ for an advancing and a retreating grounding line.
A significant implication of the current study is that conducting transient Stokes flow simulations
of whole marine ice
sheets, such as century-scale simulations of the West Antarctic
Ice Sheet for example, is a potentially tractable problem
where evidence supports both basal drag and basal melting decreasing smoothly to zero as
the grounding line is approached from respectively grounded and floating regions.
Conversely, if there is a sharp onset of basal melting immediately beyond the grounding line,
high resolution might be required regardless of the character of the basal sliding relation.
Olivier Gagliardini is a member of the editorial board of the journal. All other authors declare that they have no conflict of interest.
Acknowledgements
The authors wish to thank Stephen Cornford and Bill Budd for useful discussions about the simulations.
The authors wish to acknowledge CSC – IT Centre for Science, Finland, for computational
resources.
This research utilised the NCI National Facility in Canberra, Australia,
which is supported by the Australian Commonwealth Government.
Rupert Gladstone was funded from the European Union Seventh Framework Programme (FP7/2007-2013)
under grant agreement number 299035.
This research was supported in part by Academy of Finland grant number 286587.
Ralf Greve was supported by a UTAS (University of Tasmania) Visiting Fellowship
(September–December 2014), and by a JSPS (Japan Society for the Promotion of Science)
Grant-in-Aid for Scientific Research A (no. 25241005).
This work was supported in part by the Australian Government's Cooperative Research Centres
Programme through the Antarctic Climate and Ecosystems Cooperative Research Centre (ACE
CRC).
Edited by: F. Pattyn
Reviewed by: V. C. Tsai and J. Bassis
ReferencesAsay-Davis, X. S., Cornford, S. L., Durand, G., Galton-Fenzi, B. K.,
Gladstone, R. M., Gudmundsson, G. H., Hattermann, T., Holland, D. M.,
Holland, D., Holland, P. R., Martin, D. F., Mathiot, P., Pattyn, F., and
Seroussi, H.: Experimental design for three interrelated marine ice sheet and
ocean model intercomparison projects: MISMIP v. 3 (MISMIP +), ISOMIP v. 2
(ISOMIP +) and MISOMIP v. 1 (MISOMIP1), Geosci. Model Dev., 9, 2471–2497,
10.5194/gmd-9-2471-2016, 2016.
Budd, W., Keage, P. L., and Blundy, N. A.: Empirical studies of ice sliding,
J. Glaciol., 23, 157–170, 1979.
Budd, W., Jenssen, D., and Smith, I.: A 3-dimensional time-dependent model
of
the West Antarctic Ice-Sheet, Ann. Glaciol., 5, 29–36, 1984.
Cornford, S. L., Martin, D. F., Graves, D. T., Ranken, D. F., Le Brocq,
A. M.,
Gladstone, R. M., Payne, A. J., Ng, E., and Lipscomb, W. H.: Adaptive mesh,
finite volume modeling of marine ice sheets, J. Comput.
Phys., 232, 529–549, 2013.Durand, G., Gagliardini, O., de Fleurian, B., Zwinger, T., and Le Meur, E.:
Marine ice sheet dynamics: Hysteresis and neutral equilibrium, J.
Geophys. Res.-Earth, 114, F03009, 10.1029/2008JF001170,
2009.Favier, L., Gagliardini, O., Durand, G., and Zwinger, T.: A three-dimensional full Stokes model of the grounding line dynamics: effect of a pinning point beneath the ice shelf, The Cryosphere, 6, 101–112, 10.5194/tc-6-101-2012, 2012.Favier, L., Durand, G., Cornford, S. L., Gudmundsson, G. H., Gagliardini, O.,
Gillet-Chaulet, F., Zwinger, T., Payne, A. J., and Le Brocq, A. M.: Retreat
of Pine Island Glacier controlled by marine ice-sheet instability, Nature
Climate Change, 4, 117–121, 10.1038/NCLIMATE2094, 2014.Feldmann, J., Albrecht, T., Khroulev, C., Pattyn, F., and Levermann, A.:
Resolution-dependent performance of grounding line motion in a shallow model
compared to a full-Stokes model according to the MISMIP3d
intercomparison, J. Glaciol., 60, 353–360, 10.3189/2014JoG13J093,
2014.
Fowler, A. C.: Weertman, Lliboutry and the development of sliding theory,
J. Glaciol., 56, 965–972, 2010.Gagliardini, O., Cohen, D., Raback, P., and Zwinger, T.: Finite-element
modeling of subglacial cavities and related friction law, J.
Geophys. Res.-Earth, 112, F02027, 10.1029/2006JF000576,
2007.Gagliardini, O., Durand, G., Zwinger, T., Hindmarsh, R. C. A., and Le Meur,
E.:
Coupling of ice-shelf melting and buttressing is a key process in ice-sheets
dynamics, Geophys. Res. Lett., 37, F04014, 10.1029/2010GL043334,
2010.Gagliardini, O., Zwinger, T., Gillet-Chaulet, F., Durand, G., Favier, L., de
Fleurian, B., Greve, R., Malinen, M., Martín, C., Råback, P.,
Ruokolainen, J., Sacchettini, M., Schäfer, M., Seddik, H., and Thies, J.:
Capabilities and performance of Elmer/Ice, a new-generation ice sheet model,
Geosci. Model Dev., 6, 1299–1318, 10.5194/gmd-6-1299-2013, 2013.Gagliardini, O., Brondex, J., Gillet-Chaulet, F., Tavard, L., Peyaud, V., and
Durand, G.: Brief communication: Impact of mesh resolution for MISMIP and
MISMIP3d experiments using Elmer/Ice, The Cryosphere, 10, 307–312,
10.5194/tc-10-307-2016, 2016.
Galton-Fenzi, B.: Modelling Ice-Shelf/Ocean Interaction, PhD thesis,
University of Tasmania, 2009.Gladstone, R. M., Lee, V., Vieli, A., and Payne, A.: Grounding Line Migration
in
an Adaptive Mesh Ice Sheet Model, J. Geophys. Res.-Earth, 115, F04014, 10.1029/2009JF001615,
2010a.Gladstone, R. M., Payne, A. J., and Cornford, S. L.: Parameterising the
grounding line in flow-line ice sheet models, The Cryosphere, 4, 605–619,
10.5194/tc-4-605-2010, 2010b.Gladstone, R. M., Payne, A. J., and Cornford, S. L.: Resolution requirements
for
grounding-line modelling: sensitivity to basal drag and ice-shelf
buttressing, Ann. Glaciol., 53, 97–105,
10.3189/2012AoG60A148, 2012.
Glen, J. W.: Experiments on the deformation of ice, J. Glaciol., 2,
111–114, 1952.Gong, Y., Cornford, S. L., and Payne, A. J.: Modelling the response of the
Lambert Glacier–Amery Ice Shelf system, East Antarctica, to uncertain
climate forcing over the 21st and 22nd centuries, The Cryosphere, 8,
1057–1068, 10.5194/tc-8-1057-2014, 2014.Hewitt, I. J., Schoof, C., and Werder, M. A.: Flotation and free surface flow
in a model for subglacial drainage. Part 2. Channel flow, J. Fluid
Mech., 702, 157–187, 10.1017/jfm.2012.166,
2012.Jenkins, A.: Convection-Driven Melting near the Grounding Lines of Ice
Shelves
and Tidewater Glaciers, J. Phys. Oceanogr., 41, 2279–2294,
10.1175/JPO-D-11-03.1,
2011.Joughin, I., Smith, B., and Holland, D.: Sensitivity of 21st century sea
level
to ocean-induced thinning of Pine Island Glacier, Antarctica, Geophys.
Res. Lett., 37, L20502, 10.1029/2010GL044819, 2010.Leguy, G. R., Asay-Davis, X. S., and Lipscomb, W. H.: Parameterization of basal friction near grounding lines in a one-dimensional ice sheet model, The Cryosphere, 8, 1239–1259, 10.5194/tc-8-1239-2014, 2014.
Mclnnes, B. and Budd, W.: A Cross-Sectional Model for Antarctica, Ann.
Glaciol., 5, 95–99, 1984.Parizek, B. R. and Walker, R. T.: Implications of initial conditions and
ice-ocean coupling for grounding-line evolution, Earth Planet.
Sc. Lett., 300, 351–358, 10.1016/j.epsl.2010.10.016,
2010.
Paterson, W.: The physics of glaciers, Pergamon, Oxford, 3rd Edn., 1994.Pattyn, F., Huyghe, A., De Brabander, S., and De Smedt, B.: Role of
transition
zones in marine ice sheet dynamics, J. Geophys. Res.-Earth, 111, F02004, 10.1029/2005JF000394,
2006.Pattyn, F., Schoof, C., Perichon, L., Hindmarsh, R. C. A., Bueler, E., de
Fleurian, B., Durand, G., Gagliardini, O., Gladstone, R., Goldberg, D.,
Gudmundsson, G. H., Huybrechts, P., Lee, V., Nick, F. M., Payne, A. J.,
Pollard, D., Rybak, O., Saito, F., and Vieli, A.: Results of the Marine Ice
Sheet Model Intercomparison Project, MISMIP, The Cryosphere, 6, 573–588,
10.5194/tc-6-573-2012, 2012.Pollard, D. and DeConto, R. M.: Modelling West Antarctic ice sheet growth
and
collapse through the past five million years, Nature, 458, 329–332,
10.1038/nature07809, 2009.Rignot, E. and Jacobs, S. S.: Rapid Bottom Melting Widespread near Antarctic
Ice Sheet Grounding Lines, Science, 296, 2020–2023,
10.1126/science.1070942, 2002.Schoof, C.: The effect of cavitation on glacier sliding, P.
Roy. Soc. A-Math. Phys., 461,
609–627, 10.1098/rspa.2004.1350, 2005.Schoof, C.: Ice sheet grounding line dynamics: Steady states, stability, and
hysteresis, J. Geophys. Res.-Earth, 112, F03S28,
10.1029/2006JF000664, 2007.Schoof, C., Hewitt, I. J., and Werder, M. A.: Flotation and free surface flow
in a model for subglacial drainage. Part 1. Distributed drainage, J.
Fluid Mech., 702, 126–156, 10.1017/jfm.2012.165,
2012.Seroussi, H., Morlighem, M., Larour, E., Rignot, E., and Khazendar, A.:
Hydrostatic grounding line parameterization in ice sheet models, The
Cryosphere, 8, 2075–2087, 10.5194/tc-8-2075-2014, 2014.Tsai, V. C., Stewart, A. L., and Thompson, A. F.: Marine ice-sheet profiles
and
stability under Coulomb basal conditions, J. Glaciol., 61,
205–215, 10.3189/2015JoG14J221,
2015.Vieli, A. and Payne, A.: Assessing the ability of numerical ice sheet models
to simulate grounding line migration, J. Geophys. Res.-Earth, 110, F01003, 10.1029/2004JF000202,
2005.
Weertman, J.: On the sliding of glaciers, J. Glaciol., 3, 33–38, 1957.Werder, M. A., Hewitt, I. J., Schoof, C. G., and Flowers, G. E.: Modeling
channelized and distributed subglacial drainage in two dimensions, J.
Geophys. Res.-Earth, 118, 2140–2158,
10.1002/jgrf.20146,
2013.