The magnitude of the Antarctic ice sheet's contribution to global sea-level rise is dominated by the potential of its marine sectors to become unstable and collapse as a response to ocean (and atmospheric) forcing. This paper presents Antarctic sea-level response to sudden atmospheric and oceanic forcings on multi-centennial timescales with the newly developed fast Elementary Thermomechanical Ice Sheet (f.ETISh) model. The f.ETISh model is a vertically integrated hybrid ice sheet–ice shelf model with vertically integrated thermomechanical coupling, making the model two-dimensional. Its marine boundary is represented by two different flux conditions, coherent with power-law basal sliding and Coulomb basal friction. The model has been compared to existing benchmarks.

Modelled Antarctic ice sheet response to forcing is dominated by sub-ice
shelf melt and the sensitivity is highly dependent on basal conditions at the
grounding line. Coulomb friction in the grounding-line transition zone leads
to significantly higher mass loss in both West and East Antarctica on
centennial timescales, leading to 1.5 m sea-level rise after 500 years for a
limited melt scenario of 10 m a

Removing the ice shelves altogether results in a disintegration of the West
Antarctic ice sheet and (partially) marine basins in East Antarctica. After
500 years, this leads to a 5 m and a 16 m sea-level rise for the power-law
basal sliding and Coulomb friction conditions at the grounding line,
respectively. The latter value agrees with simulations by

The chosen parametrizations make model results largely independent of spatial resolution so that f.ETISh can potentially be integrated in large-scale Earth system models.

Projecting future sea-level rise (SLR) requires ice sheet models capable of
exhibiting complex behaviour at the contact of the ice sheet with the
atmosphere, subglacial environment and the ocean. Some of these interactions
demonstrate non-linear behaviour due to feedbacks, leading to self-amplifying
ice mass change. For instance, surface mass balance interacts with ice sheets
through a powerful melt–elevation feedback, invoking non-linear response as
a function of equilibrium line altitude, such as a positive feedback on
ablation that can be expected as the ice sheet surface becomes lower

Another powerful feedback relates to the contact of ice sheets (especially
marine ice sheets with substantial parts of the bedrock lying below sea
level) with the ocean.

Other feedbacks relate ice sheet dynamics to basal sliding through
thermoviscous instabilities, which may lead to limit-cycle behaviour in ice
sheets

In this paper, I present a new ice sheet model that reduces the
three-dimensional nature of ice sheet flow to a two-dimensional problem,
while keeping the essential (or elementary) characteristics of ice sheet
thermomechanics and ice stream flow. Processes controlling grounding-line
motion are adapted in such a way that they can be represented at coarser
resolutions. This way, the model can more easily be integrated within
computationally demanding Earth system models. A new grounding-line algorithm
based on the zero effective pressure conditions reigning at the contact with
the ocean has been implemented

I start by giving a detailed overview of the model and its components. The
initialization procedure for the Antarctic ice sheet is then given and,
finally, the sensitivity of the Antarctic ice sheet to sudden atmospheric and
ocean warming is presented on centennial timescales. The appendices further
describe results of known benchmarks for grounded ice flow

The model consists of diagnostic equations for ice velocities and three
prognostic equations for the temporal evolution of ice thickness, ice
temperature and bedrock deformation beneath the ice. Prescribed boundary
fields are equilibrium bedrock topography, basal sliding coefficients,
geothermal heat flux and sea level. Present-day mean surface air
temperatures and precipitation are derived from data assimilation within
climate models. Ablation is determined from a positive degree-day (PDD) model. A
list of model symbols is provided in
Tables

Model symbols, units and nominal values.

Model symbols, units and nominal values (continued).

Model symbols, units and nominal values (continued)

General Cartesian geometry of the f.ETISh model.

For the coupled ice sheet–ice shelf system the surface elevation

The ice sheet–ice shelf model has several modes of operation, depending on
the boundary conditions that are applied. The most elementary flow regime of
the grounded ice sheet is according to the shallow-ice approximation

A second mode of operation is the hybrid mode, in which the flow regime of
the grounded ice sheet is governed by a combination of SIA, responsible for
ice-deformational flow, and SSA for basal sliding

The SIA by

The flow velocity in an ice shelf or an ice stream characterized by low drag
is derived from the Stokes equations

The ice shelf velocity field is needed for determining the effect of buttressing in the grounding-line flux conditions (see below) as well as for the thickness evolution of the ice shelf. For the purpose of buttressing, velocity gradients downstream from the grounding line are used to determine the longitudinal stretching rate, which is compared to the stretching rate of a freely floating ice shelf to determine a so-called buttressing factor.

Both SIA and SSA velocities are combined to obtain the velocity field of the
grounded ice sheet according to the hybrid model

Basal sliding is introduced as a Weertman sliding law, i.e.

Basal friction within the HySSA equations can also be calculated based on a
model for plastic till

The most comprehensive approach to solve for the subglacial water pressure in
Eq. (

To link Coulomb friction to basal drag, the formulation proposed by

Alternatively, both the power-law sliding law Eq. (

Previous studies have indicated that it is necessary to resolve the
transition zone/boundary layer at sufficiently fine resolution in order to
capture grounding-line migration accurately

This yields the vertically averaged velocity

Equation (

The grounding-line parametrization based on the boundary layer theory by

The TGL flux condition can be used in conjunction with power-law basal
sliding. Indeed,

Ice sheet thickness evolution is based on mass conservation, leading to the
continuity equation. For the general ice sheet–ice shelf system, this is
written as

It is also ensured that thinning due to grounding-line retreat does not
exceed the maximum permissible rate, using theoretical knowledge of maximum
possible stresses at the grounding line that is called the “maximum strain
check”. Similar to

Ice-front calving is obtained from the large-scale stress field

Given the relatively low spatial resolution of a large-scale ice sheet model,
small pinning points underneath ice shelves due to small bathymetric rises
scraping the bottom of the ice and exerting an extra back pressure on the ice
shelf

The diffusion–advection equation for an ice sheet is given by

The basal boundary condition is given by

In ice shelves, a simple temperature model is adopted, considering the
accumulation at the surface balanced by basal melting underneath an ice shelf
and with only vertical diffusion and advection in play

The mean column temperature

The response of the bedrock to changing ice and ocean loads is solved through
a combined time-lagged asthenospheric relaxation and elastic lithospheric
response due to the applied load

Staggered grids used in the model: the basic grid is the
ice-thickness grid (shown in open circles).

The ice sheet-shelf model uses a finite-difference staggered grid, where
horizontal velocities

The SSA velocity field (Eqs.

The f.ETISh model is implemented in MATLAB^{®}.
Computational improvements involved the omission of all “for” loops by
using circular shifts (with exception of the time loop), thereby optimizing
the use of matrix operations. The bulk of computational time is devoted to
the solution of the sparse matrix systems, which are natively optimized in
MATLAB^{®} using multi-threading. A
preconditioned conjugate gradient method is used for solving the ice
sheet–ice shelf continuity equation. The velocity field in the hybrid model
is solved using a stabilized bi-conjugate gradients method, which is also
preconditioned and further initialized by the velocity field solution from
the previous time step. Both numerical solvers are iterative and the
preconditioning limits the number of iterations to reach convergence. They
are considerably faster compared to the direct solution.

The f.ETISh model is compared to other ice sheet models via a series of
benchmarks, such as the EISMINT-I benchmark for isothermal ice sheet models

For modelling the Antarctic ice sheet, the bedrock topography is based on the
Bedmap2 data

For geothermal heat flux we employ a recent update of

All datasets are resampled on the spatial resolution used for the experiments. The experiments shown in this paper employ a grid spacing of 25 (and in a few cases 40 or 16) km.

Atmospheric forcing is applied in a parametrized way, based on the observed
fields of precipitation (accumulation rate) and surface temperature. For a
change in background (forcing) temperature

Surface melt is parametrized using a PDD model

Melting underneath the floating ice shelves is often based on
parametrizations that relate sub-shelf melting to ocean temperature and
ice shelf depth

Bedrock topography

Model initialization to the modern Antarctic ice sheet geometry is based on
the method by

In addition to

For the Coulomb friction law, optimization starts with a constant field of

Values of

Top row: optimized basal sliding coefficients

Optimized basal sliding coefficients (Fig.

The lower row of Fig.

The basal temperature fields (Fig.

Observed

Modelled velocities form an independent check of the model performance, since
the optimized basal sliding coefficients are obtained solely from the
observed surface topography. The modelled flow field of the Antarctic ice
sheet (Fig.

A direct comparison between the present-day velocity field

Top: grounded ice sheet surface elevation (m a.s.l.) 500 years after sudden removal of all ice shelves. Bottom: grounding-line position in time according to the same experiment (colour scale is non-linear and represents time in years) for the Weertman sliding law with SGL condition (left), Coulomb friction law with TGL condition (centre), and Weertman sliding law with TGL condition (right). SLR denotes the contribution to sea-level rise after 500 years.

Ice shelves are the prime gatekeepers of Antarctic continental ice discharge.
The breakup of the Larsen B ice shelf (Fig.

Since ice shelf buttressing is a key element in the stability of the
Antarctic ice sheet, a useful experiment to understand underlying model
buttressing physics is the sudden removal of all floating ice shelves,
starting from the initialized model state, and to let the model evolve over
time. Over this period ice shelves were not allowed to regrow, which is
equivalent to removing all floating ice at each time step. This experiment is
carried out for three cases: (i) power-law sliding with the flux
condition according to

For all experiments, grounding-line retreat starts in the marine sections
discharging in the Ronne and Ross ice shelves. For the SGL experiment, the
retreat from Ellsworth Land leads to thinning in the inland sectors of the
Pine Island basin, which after

The higher TGL grounding-line sensitivity must be sought in its underlying
physics: at the grounding line the basal shear stress vanishes in a smooth
way to reach zero exactly at the grounding line. As shown by

Evolution of sea-level contribution

Antarctic ice sheet sensitivity to sub-shelf melting is investigated with a
multi-parameter/multi-resolution forcing ensemble over a period of 500 years.
Atmospheric forcing includes changes in background temperature

Sea-level contribution according to the forcing experiments and rate of
change of sea level for the

The major differences in sea-level response are due to the treatment of
grounding-line fluxes. As shown above, the TGL flux condition systematically
leads to significant higher mass losses, making grounding-line migration a
more sensitive process (Sect.

Only the higher melt-rate scenarios (

Comparison of sea-level contribution after 500 years as a function
of model resolution (25 vs. 40 km). Colours denote sub-shelf melt rates;
shapes represent background temperature forcing: 0

The effect of spatial resolution on model result is summarized in
Fig.

In order to validate this claim, two more experiments were carried out to
make comparison with an existing experimental result at high resolution
possible

In

It limits the melt rate between zero (for ice shelves thinner than 100 m)
and 400 m a

In terms of model complexity, the f.ETISh model is comparable to the

Given the differences in approach with continental-scale ice sheet models,
such as AISM-VUB

An important experiment for marine ice sheet models is a test of steady-state
grounding-line positions in absence of buttressing

The main advantage of using a grounding-line flux parametrization based on a
heuristic rule (Sect.

A major finding in this paper is the increased sensitivity of the grounding
line based on a Coulomb friction law

Direct comparison is not possible with recent studies of Antarctic ice mass
loss that are forced by atmosphere–ocean models following so-called RCPs
(Representative Concentration Pathways). Direct comparison with the SeaRISE
experiments

However, the TGL model is less sensitive than the PSU-ISM model including
cliff failure and hydrofracturing

Finally, computational time of f.ETISh largely depends on the spatial resolution, which also governs time steps needed under the CFL condition. A hybrid model 5000-year run with a grid size of 40 km and a time step of 0.2 years takes approximately 10 000 CPU seconds on a single AMD Opteron 2378 2.4 GHz core of the Hydra cluster (VUB-ULB) and 20 000 CPU seconds for a 500-year run with a grid size of 16 km and time step of 0.02 years on a multicore. Future developments will focus on improving the numerical solution schemes in order to reduce the calculation time (larger time steps), especially at higher spatial resolutions.

I developed a new marine ice sheet model, based on common descriptions of ice physics (combined shallow-ice and shallow-shelf approximation) and novel implementation of parametrizations of thermodynamics and grounding-line migration. The model has been extensively tested against existing benchmarks and has been shown to be scale independent, with the exception of grounding zones with small-scale bedrock variability, where grounding-line response to atmospheric and oceanic forcing is sensitive to spatial resolution. This makes the model extremely attractive to couple within Earth system models.

The model has been initialized to the present-day Antarctic ice sheet conditions in order to obtain initial steady-state conditions as close as possible to the observed ice sheet. Independent validation has been obtained through comparison with observed surface velocities that are not utilised during the optimization phase.

Two forcing experiments over a period of 500 years are carried out, one
during which all floating ice shelves are removed and one during which
sudden atmospheric and oceanic forcing is applied. Both experiments show a
very high sensitivity to grounding-line conditions, as Coulomb friction in
the grounding-line transition zone leads to significantly higher mass loss in
both West and East Antarctica, compared to commonly used power-law sliding
laws (such as Weertman type). For the ice shelf removal experiment this leads
to 5 and 16 m SLR for the power-law basal sliding and Coulomb friction
conditions at the grounding line, respectively. This high-end response is of
the same order of magnitude as obtained by

The atmospheric–oceanic forcing experiments clearly show the dominance of ocean forcing in sea-level response, where significant MISIs occur under relatively mild sub-shelf melt scenarios over centennial timescales (500 years).

All datasets used in this paper are publicly available,
such as Bedmap2

The EISMINT-I benchmark is the first series of ice sheet model
intercomparisons aiming at benchmarking large-scale ice sheet models under
idealized and controlled conditions

The f.ETISh model is a 3D Type I model according to the classification scheme
in EISMINT-I; i.e. diffusion coefficients for the grounded ice sheet are
calculated on a staggered Arakawa B grid. Table

The moving margin experiment includes ice ablation, hence the presence of an
equilibrium line on the ice sheet. This is obtained by defining the climatic
conditions by

Basic characteristics of the experiment are listed in Table

Comparison of f.ETISh with the EISMINT-I fixed (FM) and moving
margin (MM) experiment benchmark based on an ensemble of two to three models

Homologous basal temperatures along the central line according to the EISMINT-I experiment calculated with f.ETISh (circles) and according to the EISMINT-I benchmark (crosses) for the fixed margin (blue) and moving margin (red) experiment.

Temporal changes in ice thickness/volume and basal temperature are analysed
with a forcing experiment, where the surface temperature and mass balance
perturbations are defined as follows

Comparison of f.ETISh with the EISMINT-I fixed (FM) and moving margin (MM) experiment benchmark based
on an ensemble of two to three models

Ice thickness and basal temperature variations for the EISMINT-I fixed margin experiment with a 20 ka (black) and a 40 ka (blue) forcing.

Ice thickness and basal temperature variations for the EISMINT-I moving margin experiment with a 20 ka (black) and a 40 ka (blue) forcing.

All ice thickness changes (amplitude and phase) as well as the phase in
temperature according to the two forcing scenarios are in close agreement
with the benchmark. However, amplitude differences for the basal temperatures
deviate, but the EISMINT I data sample is rather limited for comparison. The
phase of the basal temperature response is in agreement with the benchmark.
All other parameters are within the bounds of the benchmark
(Table

The EISMINT-II benchmark

Six further experiments were carried out: B, C, D, F, G and
H

Results for experiments A–H are summarized in Table

Predicted basal temperatures (corrected for pressure-dependence) according to EISMINT-II experiment H.

Comparison of f.ETISh with the EISMINT-II experiments

The emblematic experiments F and H in

The capacity of an ice sheet model to cope with the marine boundary, and more
specifically migration of the grounding line, is essential in Antarctic
ice sheet modelling. Since grounding-line dynamics were elucidated
mathematically based on boundary layer theory

Steady-state ice sheet–ice shelf profiles for the MISMIP experiments
corresponding to different values of the flow parameter

Median (upper panel) and standard deviation (centre panel) of
steady-state grounding-line positions according to the MISMIP experiments for
a circular ice sheet as a function of the flow parameter

The initial ice sheet is obtained for a constant value of the flow parameter

Errors on the advance and retreat grounding-line positions are displayed in
the bottom panel of Fig.

The author declares that he has no conflict of interest.

I should like to thank Lionel Favier and Heiko Goelzer for the numerous discussions that helped in developing and improving the f.ETISh model and their helpful comments on an earlier version of the manuscript. I am also indebted to my “guinea pigs” Thomas Bogaert, Violaine Coulon and Sainan Sun for revealing a few coding errors as well as for their patience while struggling with initial and non-optimized versions of the model. Finally, I would like to thank the two anonymous referees for their very helpful comments that improved the manuscript and model significantly.Edited by: G. Hilmar Gudmundsson Reviewed by: two anonymous referees