In this paper, we propose a new sub-grid scale parameterization for the ice discharge into the ocean through outlet glaciers and inspect the role of different observational and palaeo constraints for the choice of an optimal set of model parameters. This parameterization was introduced into the polythermal ice-sheet model SICOPOLIS, which is coupled to the regional climate model of intermediate complexity REMBO. Using the coupled model, we performed large ensemble simulations over the last two glacial cycles by varying two major parameters: a melt parameter in the surface melt scheme of REMBO and a discharge scaling parameter in our parameterization of ice discharge. Our empirical constraints are the present-day Greenland ice sheet surface elevation, the surface mass balance partition (ratio between total ice discharge and total precipitation) and the Eemian interglacial elevation drop relative to present day in the vicinity of the NEEM ice core. We show that the ice discharge parameterization enables us to simulate both the correct ice-sheet shape and mass balance partition at the same time without explicitly resolving the Greenland outlet glaciers. For model verification, we compare the simulated total and sectoral ice discharge with other estimates. For the model versions that are consistent with the range of observational and palaeo constraints, our simulated Greenland ice sheet contribution to Eemian sea-level rise relative to present-day amounts to 1.4 m on average (in the range of 0.6 and 2.5 m).

Modelling the response of the Greenland ice sheet (GrIS) to anthropogenic
warming has already been undertaken for more than 2 decades

Observational data indicate that during the past decade mass loss by the
GrIS, both through surface melt and enhanced ice discharge, has contributed
appreciably to global sea level rise

Models of the GrIS contain a number of parameters that can be used for tuning
the model using observational constraints. Present-day extent and surface
elevation of the GrIS are accurately known and it is natural to use them as
such constraints

However, the observed shape of the GrIS is not the only characteristic that
can serve as a constraint for the GrIS models. Recently,

Since standard coarse-resolution GrIS ice sheet models cannot simulate a
realistic present-day surface orography of the GrIS and at the same time have
the correct mass balance partition, we developed a novel approach, which
allows us to circumvent this problem without resolving individual ice
streams and outlet glaciers – and without an increase in computational cost.
This approach is in the spirit of our previous modelling work

In addition to the present-day constraints on the ice-sheet shape and the
MBP, we use the Eemian as a palaeo constraint. Eemian conditions have already
been recognized earlier as an important palaeo constraint for GrIS model
parameters

The paper is summarized as follows. First, we give a short description of the
ice-sheet model SICOPOLIS and the regional energy-moisture balance model
REMBO (Sect.

For this study, we used the three-dimensional polythermal ice-sheet model
SICOPOLIS (version 2.9) coupled to the regional energy-moisture balance model
(REMBO). SICOPOLIS treats the evolution of ice thickness, ice temperature and
water content

The regional energy-moisture balance model REMBO is a climate model of
intermediate complexity and it is described in detail by

The coupling between the models is bi-directional, i.e. SICOPOLIS provides the climate model with information about surface elevation and spatial extent of the ice sheet. In turn, REMBO provides SICOPOLIS with surface mass balance and mean annual surface temperature.

Principle sketch of the discharge parameterization over a part of
the horizontal computational domain. The gray shading shows the ice-covered
cells, while the dark gray shaded area

Most ice discharge of the GrIS is brought into the ocean via fast-flowing
narrow outlet glaciers

Distance field over the entire Greenland area in km. It is
determined by the minimal distance of every land grid point (ice-free and ice-covered ones) to the coast (first ocean grid point, see
Fig.

The divergence of the explicitly non-resolved fast lateral ice flow

Additionally, we assume that ice discharge into the ocean via fast moving ice
streams only occurs from the area where ice surface is descending toward the
coast. This is enforced by setting a maximum value of

Simulated parameterized divergence of fast ice flow in m yr

The discharge parameterization is applied only to the ice-covered grid cells
that are located not more than

The value of parameter

As seen in Fig.

In the model, the ice-marginal ring

There are numerous possibilities to define a measure of the performance of a
model based on the comparison of simulated geometrical characteristics of an
ice sheet with observational data. The simplest is arguably to use the error
in simulated total ice area and ice volume, which we define as

Following

We fix the powers

Figure

Error measures for a modelled ice sheet in the

Figure

In addition to the relative error in present-day ice thickness, we use the
following as further empirical constraints on the ensemble of the model
realizations: the present-day surface mass balance partition and the Eemian
drop in surface elevation relative to present day at the upstream position of
the NEEM ice borehole. Figure

As mentioned in the introduction, the mass balance partition is the amount of
total ice discharge compared to total precipitation. In our work, we always
refer to MBP as a characteristic of the ice sheet defined in its present-day
state. Its practical definition is the total ice discharge divided by the
total precipitation for the simulated present-day ice sheet. In

Estimated constraints on the parameters

From measurement of air content in the NEEM borehole samples, the

Figure

Figure

Simulated geographical position of present-day ice margins for
simulations with the discharge parameterization (gray areas) compared to
observations (red line). The gray shaded areas cover the range of simulated
ice margins determined by different constraints.

Present-day (

Figure

Figure

For all valid parameter sets, our simulated reduction in Eemian ice volume is
accompanied by a strong retreat of ice in Greenland in particular in its
northern part, see Fig.

Interestingly, the NEEM location almost becomes ice free at 121 kyr BP in our
most sensitive model version (see Fig.

Figure

Time series of the simulated Greenland ice sheet evolution during
the last two glacial cycles. Blue shading represents the range of valid model
versions including our discharge parameterization. Black and red lines show
simulations without the discharge parameterization (

Additionally, two extreme and unrealistic simulations, depicted by the red
lines, were set up in order to demonstrate, what happens when a shape-only
tuning applies in a coarse-resolution model that disregards fast sub-grid
processes of small outlet glaciers. Technically, we restrict the parameter
space by setting

Moreover, the strong drop in Eemian sea-level and NEEMup elevation hints at
very different stability properties of the model version without ice
discharge parameterization and shape-only tuning compared to all our valid
model versions which contain the sub-grid scale discharge parameterization.
Even more, the models with shape-only tuning are much less stable with
respect to applied positive temperature anomalies than all the model versions
that are constrained using the MBP and palaeo data, whether they include
discharge parameterization or not. In other words, the models with shape-only
tuning of the melt parameter are less stable than both the valid model
versions of our former approach without the discharge parameterization

Temperature threshold of the stability of the Greenland ice sheet for a number of valid model parameters.

To achieve more detailed information about the stability of the GrIS, we
performed an analysis based on many steady state runs as in

We applied the procedure to three representative valid simulations with the
discharge parameterization. From these simulations, we obtain thresholds of
decay of the GrIS between 1.25 and 2.5

Present-day Greenland ice sheet topography.

In summary, if we optimize the melt parameter in the coarse resolution model
without the sub-grid scale ice discharge parameterization for only
err(

One major advantage of our simple parameterization is that it applies easily
for climates far away from present day – a fully explicit modelling of
present-day outlet glaciers could fail for the Eemian, because many
present-day outlet glaciers just vanish in the Eemian.
Figure

Simulated ice discharge (open bars) versus observations and findings
by others (horizontal lines) at present day (i.e. pre-industrial
conditions). The heights of the open rectangles indicate the range of our
simulated discharge. Acronyms are as follows: Re94:

As stated in Sect.

Furthermore, we demonstrate that the regions of fast flow can be reduced drastically for the Eemian time period compared to the present-day state. For the Eemian, there is practically no ice discharge from regions far away from the coast. In particular, the land bridge between the large ice sheet in the north and the smaller ice cap in the south of Greenland shows diminishing ice discharge. In general, our model results suggest that during the Eemian more ice calves into the ocean from the eastern coast of Greenland than from its western coast. In particular, the Kangerlussuaq Glacier region delivers ice into the ocean during the Eemian in all our valid model versions.

A direct comparison of our simulated Greenland surface elevation with the
observed elevation by

Figure

Over the sectors N and SW, our simulated range of ice discharge compares well
with previous estimates. While our simulated ice discharge range is somewhat
low over sector NW, it is certainly too high over sector NE. The latter can
be explained by the overestimation of our simulated present-day accumulation
over sector NE by some 10 Gt yr

Our range of valid model versions is 326–479 Gt yr

In spite of significant improvements in the simulated GrIS topography with
our discharge parameterization, for all of our simulations it was impossible
to yield an error in ice thickness smaller than about 18 %. These rather
large errors partly underline the limits of our ice discharge
parameterization and modelling approach in general. The errors can be reduced
by incorporating additional parameters, in particular such parameters which
affect the interior of the ice sheet, like the basal sliding parameter

We designed this parameterization as a workaround until a more comprehensive whole-Greenland glacial system model becomes available. Of course, additional improvements are possible, like introducing physically based models for individual outlet glaciers and fjords. Nevertheless, note that the relative high error in ice thickness (up to 20 %) also results from the fact that this is a stronger measure of the error in ice-sheet shape than the error in total ice area or in ice volume.

Although our model enables us to reduce the cumulative error in ice thickness
from about 30 to 18 %, there is still room for further improvement. For
example, higher-order models can play an important role in capturing the GrIS
dynamics

The term

We restricted our simulations to the spatial horizontal resolution of 20 km and have not inspected a possible dependence of our ice discharge parameterization on resolution. We cannot rule out that a recalibration of the parameters will be necessary for a different resolution. For a better understanding, we plan to investigate the potential of resolution dependency in the future. At the same time, we regard such a parameterization as an important tool for use in a fixed resolution model.

While the model agrees reasonably well with observations overall, there are
some significant biases in simulated ice discharge at the regional scale. For
example, we have too much ice discharge in the north-eastern and too little in
the north-western sector. The disagreements can be partly attributed to
regional biases of simulated precipitation by REMBO and to difficulties in
interpretation of the data used for comparison. When designing our
constraints, we took the reduction in Eemian surface elevation upstream of
the NEEM ice core from the

We introduced a new sub-grid scale ice discharge parameterization aimed at mimicking Greenland's fast outlet glaciers in a coarse resolution ice-sheet model. Our simulated ice discharge compares reasonably well with observations and other model estimates. The ice discharge parameterization enables us to simulate an ice sheet, whose shape is in good agreement with observations and whose partition between total ice discharge and total surface melt is in good agreement with state-of-the-art regional climate models.

We used various constraints to reduce the range of valid melt and discharge
parameters of the REMBO-SICOPOLIS model: a shape constraint, a constraint on
the mass balance partition

The NEEM constraint proved to be a complementary constraint to the other two present-day constraints. It was the strongest constraint in controlling the upper end of the range of valid melt parameter values and thereby the highest Greenland's contribution to Eemian sea-level rise. Taken individually, this constraint was also comparable to the shape constraint in determining the range of simulated present-day GrIS margins. This demonstrates the importance of palaeoclimate information for determining the range of model parameters applicable for future prediction of the contribution of the GrIS to sea level.

We can satisfy all constraints if our sub-grid scale ice discharge
parameterization is included in a coarse resolution ice sheet model in order
to mimic small-scale fast processes. When using a shape-only constraints in a
coarse resolution model without the parameterization of fast processes, we
obtained a very unstable ice sheet – i.e. a regional temperature rise of as
low as 0.25

The inclusion of our ice discharge parameterization along with the
above-described constraints leads to similar results concerning long-term
stability as

We stated that our ice discharge parameterization locally represents the divergence of fast flow. Here we cannot give a complete proof of that, however we can present a plausibility argument.

We start with rewriting Eq. (

We would like to thank Ellyn Enderlin for providing us with ice discharge data, as well as Roger Bales and Qinghua Guo, who provided accumulation and precipitation data. We are grateful to Fuyuki Saito and two anonymous reviewers for their constructive comments. Edited by: E. Larour