In this paper, we present the GRISLI
(Grenoble ice sheet and land ice) model in its newest revision (version 2.0).
Whilst GRISLI is applicable to any given ice sheet, we focus here on the
Antarctic ice sheet because it highlights the importance of grounding line
dynamics. Important improvements have been implemented in the model since its
original version

Continental ice sheets are a major climatic component for Earth system
dynamics. They operate on a variety of timescales, from diurnal to
multi-millennial, through multiple feedbacks such as temperature – surface
albedo, gravity waves and oceanic circulation changes related to freshwater
flux release. Over the last decades, observations of the Greenland and
Antarctic ice sheets (e.g. altimetry, gravimetry, echo sounding) have shown
important changes such as an increase in surface and sub-shelf melt, glacier
speed-up, dynamical thinning and drastic calving events

Continental ice sheets are difficult to model because they include processes
operating on a variety of temporal, from diurnal to multi-millennial, and
spatial scales, from a few metres to thousand of kilometres, but also because
we lack crucial observations (e.g. basal conditions and internal
thermomechanics). Most numerical models consider ice sheets as an
incompressible fluid, where motion can be described with the Navier–Stokes
equations. Even if some processes generally have to be parameterised (e.g.
ice anisotropy), the complete set of equations can be solved explicitly and
does not require the use of any approximation. The most comprehensive
continental ice sheet models, namely the full Stokes models, solve explicitly
all the terms in the stress tensor

In order to decrease the degree of complexity, simpler models were
historically developed that make use of the small aspect ratio of ice sheets
(vertical- to horizontal-scale ratio) to derive approximations for the
Navier–Stokes equations

The aim of our current study is to provide a technical description of the
GRISLI model in its current version (GRISLI version 2.0, hereafter GRISLI),
including several additional features from

In Sect. 2, we describe the fundamental equations of the GRISLI ice sheet
model with a particular emphasis on the model developments departing from

GRISLI model parameters used in this study.

Schematic representation of the different types of flows in GRISLI and their associated velocity profiles. The red arrows stand for the sliding velocity, which is non-zero for temperate-based grounded regions.

GRISLI constitutive equations were presented in

Ice deformation and mass conservation in GRISLI version 2.0 are mostly treated
as in

GRISLI considers the ice sheet as solely formed of pure ice with a constant
and homogeneous density (

The vertically integrated expression of the mass conservation equation
(Eq.

The quasi-static approximation is used for the velocity field, in which the
inertial terms of the momentum conservation equation are ignored. With the gravity
force being the sole external force acting on an infinitesimal cube of ice,
we have

Assuming isotropy, the deviatoric stress and the deformation rate

Like most ice sheet models, GRISLI considers the ice as a non-Newtonian
viscous fluid following a Norton–Hoff constitutive law (commonly named Glen
flow law):

To account for the fact that the activation energy increases close to the
melting point

The Glen flow law is an empirical formulation, derived from laboratory
experiments. However, laboratory experiments cannot cover the full range of
deviatoric stress operating in real ice sheets. The timescale over which this
stress is applied in real ice sheets is also not reproducible in
laboratories. Most modelling studies use

Like other large-scale ice sheet models, GRISLI does not explicitly take into
account anisotropy, which tends to facilitate deformation due to vertical
shear, but reduces deformation due to longitudinal stress. The role of the
flow enhancement factor

Differing from

The SIA

The basal velocity can be computed with a sliding law

For fast-flowing regions, the vertical stresses are much smaller than the
longitudinal shear stresses. In this case, the velocity fields with the SSA

The condition at the front of the ice shelf is given by the balance between
the water pressure and the horizontal longitudinal stress (see also
Sect.

The code section relative to the elliptic equation is available in the Supplement.

For floating ice shelves, the basal drag,

In some recent applications of GRISLI, the basal drag coefficient has been
inferred with an inverse method in order to match present-day ice sheet
geometry

Inverse methods are especially suited to produce an ice sheet state (e.g.
geometry and/or velocity) close to observations. However, by construction,
such methods do not provide information where no ice is present in
observations. As such, they are difficult to apply for palaeo reconstructions
of the American or Eurasian ice sheets. More generally, inverse methods are
no longer appropriate for long-term integrations, either palaeo or future,
when ice thickness is very different from its present state and especially if
the ice margin migrates from its present-day position. This motivates the use
of a basal drag coefficient computed from GRISLI internal variables. We
generally assume that its value is modulated by the effective water pressure,

In our approach, any temperate-based grounded point will have a non-zero
sliding velocity, depending on the

In

The flux at the grounding line following

Conversely,

Horizontal staggered grids used in the model. The blue arrows stand
for the staggered velocity grid, while the green circles represent the
standard centred grid (for, e.g. ice thickness, temperature, effective
pressure). The plain brown line is an illustration of the grounding line
position with an example of the flux (

In GRISLI, from the last grounded point in the direction of the flow, we
compute the sub-grid position of the grounding line in the

To evaluate the back force coefficient

The code for the implementation of the flux at the grounding line in GRISLI is available in the Supplement.

Iceberg calving is not modelled explicitly. Instead, we used a simple ice thickness threshold criterion. Because this simple scheme can prevent ice shelf extension, we also maintain downstream ice shelf grid points neighbouring the last grid points meeting the criterion. The cut-off threshold may vary in space (e.g. oceanic depth dependency) and time. In the following, we use a constant and homogeneous thickness criterion (set to 250 m, roughly corresponding to the observed present-day Antarctic ice shelves' front).

The ice temperature calculation has remained identical to

Horizontal diffusion is assumed to be negligible relative to the vertical diffusion.

The heat production is given by

A geothermal heat flux

The heat equation is solved in the bedrock similarly to
Eq. (

The ice–bedrock interface heat flux is used differently for cold- and temperate-based points:

For cold-based points, the heat at the ice–bedrock interface is
transferred to the ice via a Neumann boundary condition:

For temperate points, a Dirichlet boundary condition is applied as the temperature is kept at
the pressure melting point. The excess heat in this case is used to compute basal melting:

Basal melting for oceanic points is usually imposed. For specific applications, we have different values for deep-ocean and continental shelves, or a geographical distribution depending on the oceanic basin.

Using a Darcy law, the water produced by melting at the base of the ice sheet is routed outside glaciated areas following the highest gradient in the total water potential.

Such a gradient can be written as in

In GRISLI, we assume that the basal water flows within a sub-glacial till
following a Darcy-type flow law:

The till is assumed to be present everywhere below the ice sheet with a
constant and homogeneous thickness (

In GRISLI, we assume that the flow of water within the till can be described
with a diffusivity equation for the hydraulic head:

From the hydraulic head,

Fortran modules for the basal hydrology are available in the Supplement.

As in

GRISLI includes a passive tracer model that allows for the computation of
vertical ice stratigraphy, i.e. time and location of ice deposition for the
vertical model grid points. The model is the one of

The GRISLI code section related to the passive tracers is available in the Supplement.

The model uses finite differences computed on a staggered Arakawa C grid in
the horizontal plane (Fig.

The mass balance equation is solved as an advection-only equation with an
upwind scheme in space and a semi-implicit scheme in time (velocities at the
previous time step are used). The numerical resolution is performed with a
point-relaxation method with a variable time step. The value of this time
step is chosen to ensure that the matrix becomes strongly diagonal dominant
to achieve convergence of the point-relaxation method. The criterion is thus a
threshold that is inversely proportional to the fastest velocity on the whole
grid. Note that this smaller time step is solely used for the mass
conservation equation (Eq.

To solve the ice shelf/ice stream equation, Eq. (14) needs to be
linearised. The viscosity is computed using an iterative method starting from
the viscosity calculated from strain rates from the previous time step. As
this equation is the most expensive part of the model, the iteration mode is
not always used depending on the type of experiment (for instance, not crucial
when the objective is to reach a steady state). In this case, the viscosity
of the previous time step is used. The linear system is solved with a direct
method (Gaussian elimination, sgbsv in the Lapack library;

The resolution of the elliptic system (Eq.

For the temperature equation (Eq.

The model has been recently partially parallelised with OpenMP
(^{®} Xeon^{®}
CPU at 3.47 GHz).

Over the years, several GRISLI internal parameters have been shown to be of
importance to appropriately simulate the flow and mass balance of the
Antarctic ice sheet. Values for these parameters have been so far derived
from non-systematic tests and expert knowledge. To systematically investigate
the role of those parameters and find the best fitting set for the simulated
Antarctic ice sheet with respect to the observed one, a calibration
methodology with systematic exploration of the different values is performed
in the following. The best fitting set will be considered as plausible models
within the chosen parameter space. Given its degree of complexity, GRISLI is
mostly designed for multi-millennial integrations. Due to long-term diffusive
response to SMB and temperature changes, an accurate methodology to select
unknown parameters of the model would be to run long transient simulations
with a climate forcing as close as possible to past climate states, ideally
with a synchronous coupling between the ice sheet and the atmosphere.
However, climate models generally fail at reproducing the regional climate
changes during the last glacial–interglacial cycle as recorded by proxy data

Antarctic ice shelf sectors

In the following, we use the 27 km grid atmospheric outputs, namely annual
mean temperature and SMB, from the regional climate model RACMO2.3

We choose to restrict this study to a coarse horizontal resolution, namely
40 km, as it allows for large ensembles of multi-millennial simulations.
Whilst 6.7 h on one thread of an Intel^{®}
Xeon^{®} CPU at 3.47 GHz (4 h on four threads)
are needed to perform 100 000 years of simulation over Antarctica on a
40 km grid (19 881 horizontal grid points), this time goes up significantly
on a 16 km grid (145 161 points) for which we need 25 h to perform
2000 years (17 h on four threads). In addition, the 40 km resolution
corresponds to the one used in the coupled version within the

Selected parameters included in the Latin hypercube sampling (LHS) ensemble with their associated ranges.

Bedmap2 ice thickness

From our experience with GRISLI, we identified four unknown independent
parameters that have a crucial role for ice dynamics:

The SIA flow enhancement factor

The basal drag coefficient

The till conductivity

An ice shelf basal melting rate coefficient

Simulated total ice volume for each ensemble members as a function
of parameter values when using the

Same as Fig.

The initial ice sheet geometry, bedrock and ice thickness, is taken from the
Bedmap2 dataset

Ice thickness difference with the observations in metres (simulated
minus observed) from the 12 ensemble members showing the lowest RMSE when
using the

Same as Fig.

In the following, individual member performance is assessed with the root
mean square error (RMSE) computed from simulated and observed ice thickness

Figure

The general model response is not fundamentally different when the flux at
the grounding line is computed from

Ice thickness root mean square error respective to observations in
the parameter space for the 300 model members using the

In Fig.

Same as Fig.

Observed velocity

Map of observed

Figure

Although our quality metric is based on the ice sheet thickness, we show in
Figs.

Parameter values for the ensemble members that yield the lowest RMSE with respect to observations at the end of the 100 kyr simulation under perpetual present-day climate forcing.

From each of the two ensembles (AN40S and AN40T), we keep the 12 ensemble
members out of 300 that have the lowest RMSE and use them in the next section
for transient simulations covering the last 400 kyr. Using these
24 plausible models on long-term transient integration provides insight on
the GRISLI result spread for models yielding a similar present-day ice sheet.
Indeed, while they have a similar RMSE, they have distinct parameter values
(Figs.

By construction, equilibrium simulations such as the ones shown in
Sect.

The near-surface air temperature, used in the model as a surface boundary
condition for the advection–diffusion temperature equation, is assumed to
follow the European Project for Ice Coring in Antarctica (EPICA) Dome C deuterium record (

We also account for the additional temperature perturbation due to topography
changes using a fixed and homogeneous lapse rate (

For a given near-surface air temperature change

In order to account for changes in basal melting rates below ice shelves,
there is the need to define a continuous proxy covering several
glacial–interglacial cycles for past sub-surface oceanic conditions around
Antarctica. To this end, and due to the lack of such a record in the Southern
Ocean, we used the temperature derived from a benthic foraminifer

The atmospheric and oceanic indexes,

Climatic perturbation used in the 400 kyr glacial–interglacial
simulations for the near-surface air temperature,

In the following, we discuss the model behaviour in response to the 400 kyr
forcing. We performed simulations using the 12 parameter combinations from
Sect.

Simulated total ice sheet volume evolution over the last 400 kyr
for the 12 ensemble members showing the lowest RMSE in
Sect.

In Fig.

The uncertainty related to the choice of the internal parameters within our
subset leads generally to up to

Simulated surface elevation at selected snapshots for the two
ensemble members that produce the minimal RMSE at 0 ka BP in the transient
simulations: AN40S252

Ice thickness difference with the observations (simulated minus
observed) at 0 ka BP for the two ensemble members that produce the minimal
RMSE at 0 ka BP in the transient simulations: AN40S252

Simulated ice sheet surface elevations at selected snapshots for the two
ensemble members with the lowest RMSE at 0 ka BP after the transient
simulations are presented in Fig.

The RMSE computed at 0 ka BP for the 24 members used for the transient
simulations ranges from 372 to 467 m within AN40S and 326 to 376 m within
AN40T. These numbers are only slightly greater than the ones obtained using a
constant forcing (Sect.

We have presented results from the updated version of the GRISLI model.
Whilst the model is able to reproduce present-day Greenland

We used a basal drag coefficient computed from an internal model parameter,
namely the basal effective pressure. For long-term multi-millennial
integrations, this is preferred to deducing the basal drag coefficient from
inversion using present-day geometry since it is fully consistent with the
model physics and, in principle, remains valid for large ice sheet geometry
change. However, by design, the fit with observations is systematically
poorer compared to model results that make use of an inverse basal drag
coefficient. A step forward would be to use the basal drag computed from
inversion in order to deduce a formulation based solely on internal
parameters. Amongst these parameters, along with the basal effective
pressure, the large-scale bedrock curvature and/or sub-grid roughness could
be used, as in

Although widely used for ice sheet model spin-up or calibration, long-term
integrations under present-day forcing induce a warm bias in the vertical
temperature profile because they discard the diffusion of
glacial–interglacial changes in surface temperature. Calibrated parameters
obtained with such a methodology tend to compensate for the underestimated
viscosity and are in theory not suitable for palaeo-reconstructions. Whilst a
parameter calibration based on glacial–interglacial simulations is ideally
preferred, the determination of a realistic climate forcing is a considerable
challenge given the many degrees of freedom. Here, we presented a very
simplified climate reconstructions for the last 400 kyr based on a minimal
parameter set (proxy for atmospheric temperatures and oceanic conditions) in
order to illustrate the possible model behaviour for long-term integrations.
Using the parameters calibrated under perpetual modern climate, the model is
nonetheless able to reproduce ice geometry changes compatible with
palaeo-constraints. Further work will consist in the determination of more
realistic climate reconstruction using general circulation model snapshots.
We also aim to expand the work of

The implementation of an explicit flux computation at the grounding line
following

In the current version of the model, some important processes are still
largely simplified. In particular, further developments will consist in the
implementation of a new basal hydrology model relying on an explicit routing
scheme

We have presented the GRISLI (version 2.0) model along with the significant
improvements from the previous version of

The developments on the GRISLI source code are hosted at

The supplement related to this article is available online at:

AQ, CD, CR and VP have made significant recent contributions to the GRISLI version 2.0 model. AQ, CD and CR designed the project. AQ and CD performed and analysed the simulations with inputs from DMR. AQ wrote the paper with contributions from all co-authors.

The authors declare that they have no conflict of interest.

We thank Michiel van den Broeke (IMAU, Utrecht University) for providing the RACMO2.3 model outputs. We also warmly thank Claire Waelbroeck for fruitful discussions on the construction of the index for sub-shelf melting rates. This is a contribution to ERC project ACCLIMATE; the research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 339108. Edited by: Julia Hargreaves Reviewed by: Fuyuki Saito and two anonymous referees