TCThe CryosphereTCThe Cryosphere1994-0424Copernicus GmbHGöttingen, Germany10.5194/tc-9-179-2015Simulating the Greenland ice sheet under present-day and palaeo constraints including a new discharge parameterizationCalovR.calov@pik-potsdam.deRobinsonA.https://orcid.org/0000-0003-3519-5293PerretteM.https://orcid.org/0000-0002-6309-4863GanopolskiA.Potsdam Institute for Climate Impact Research, Potsdam, GermanyUniversidad Complutense Madrid, 28040 Madrid, SpainInstituto de Geociencias, UCM-CSIC, 28040 Madrid, SpainR. Calov (calov@pik-potsdam.de)5February20159117919622January201417February201422November20146January2015This 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://www.the-cryosphere.net/9/179/2015/tc-9-179-2015.htmlThe full text article is available as a PDF file from https://www.the-cryosphere.net/9/179/2015/tc-9-179-2015.pdf
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).
Introduction
Modelling the response of the Greenland ice sheet (GrIS) to anthropogenic
warming has already been undertaken for more than 2 decades
and attracted considerable attention in recent years
,
including higher-order and full Stokes modelling approaches .
The recent SeaRISE ice sheet modelling project
highlighted the importance of treatment of processes
at the ice-ocean interface for the response of models of the Greenland ice
sheet to future climate change.
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 . The latest
projections suggest that the GrIS will contribute notably to sea level rise
during the next century . In the longer-term
perspective, the GrIS can become even more important, because even if the
global temperature is stabilized at the level of 2 ∘C above
preindustrial, the GrIS would still continue to melt and, in the long-term
perspective, can lose a significant fraction of its mass even for moderate
warming .
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 e.g.. At the same time, it is known that coarse-resolution
ice-sheet models have problems simulating the correct margins of the GrIS and
they systematically overestimate its volume. One reason is that under
present-day conditions, most ice discharge into the ocean occurs through
relatively narrow outlet glaciers. As a result, although ice discharge into
the ocean currently accounts for more than half of surface accumulation, the majority of Greenland the ice sheet margin is located several tens of kilometres
away from the ocean. Since current ice-sheet models do not resolve outlet
glaciers and their interaction with the ocean in the fjords, the modelled
GrIS needs too much contact with the ocean to produce realistic discharge.
This leads to systematic overestimation of the ice area and volume and makes
observational “geometrical” constraints difficult to apply.
However, the observed shape of the GrIS is not the only characteristic that
can serve as a constraint for the GrIS models. Recently,
introduced the mass balance partition (defined as
the ratio between total ice discharge into the ocean and total precipitation
over the GrIS, MBP) as another constraint on the GrIS. The MBP is an
important characteristic for short-term as well as for long-term (future)
behaviour of the GrIS, because it determines the GrIS mass balance
sensitivity to climate change. In particular, the present-day MBP is related
to the long-term stability properties of the GrIS ,
i.e. for low MBP values (large surface melt) the modelled GrIS is more
susceptible to warming than for high ones. For the long-term stability of the
GrIS, the MBP is a more important characteristic than the present-day shape
of the GrIS. However for short-term (centennial time scale) future global
warming simulations, such an ice sheet would be an unfavourable initial
condition. This is because during a considerable portion of time of such a future
simulation, the modelled ice would melt in areas where in reality no ice
exists .
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
and is based on
a rather simple semi-empirical parameterization of ice discharge through the
outlet glaciers. We propose the usage of this approach until a more complete
representation of fast processes is available. Considering the above
concerns, this approach is feasible for short-term as well as for long-term simulations.
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 and have been applied more recently in
several studies .
While the Greenland Summit position might be located too far in the interior
of the GrIS to serve as a strong palaeo constraint, the position of the new
NEEM ice core appears more promising, because it is located rather near the
ice margin, a location which is very sensitive to the climate changes during
the Eemian . In the present paper, we make use of the
recently published estimate of the Eemian elevation drop at a position of
about 200 km upstream of NEEM , where the borehole ice sampled
today was deposited during the Eemian.
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. ). Our new discharge parameterization is
comprehensively explained in Sect. . Different metrics of the
ice-sheet shape are introduced in Sect. and the the
model setup is given in Section . We discuss the behaviour of
the shape metrics in Sect. , while we present constraints
on our model parameters in Section . Further
simulations inspect Eemian climate and GrIS stability
(Sect. ) and compare our findings with those of others
(Sect. ). We close with a discussion and finally our conclusions.
Model description
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 based on the shallow ice approximation
. The dependence of the ice velocities on the ice
temperature and water is introduced via the rate factor. SICOPOLIS enables a
free and easy choice of several parameters including resolution. In our
paper, Greenland is mapped onto a stereographic plane with 76 × 141
grid points (20 km grid spacing) using the topographic data set by
. The vertical is resolved by 90 layers with decreasing
layers thickness towards the bed of the ice sheet. A 10-layer thermal rock
bed is coupled to the overlying ice sheet via heat fluxes. The geothermal
heat flux is prescribed at the lower border of the thermal bedrock. We use
the Weertman-type sliding law with powers as described in
. The bedrock adjusts to the load caused by the ice
sheet's weight using a local lithosphere relaxing asthenosphere model with a
time delay of 3000 years.
The regional energy-moisture balance model REMBO is a climate model of
intermediate complexity and it is described in detail by
. REMBO uses diffusion-type equations for surface air
temperature and atmospheric water content. For temperature, the well-known
Budyko-Sellers energy balance approach is implemented. Planetary albedo is
related to surface albedo via a linear parameterization based on empirical
data. The lateral boundary conditions for temperature and relative humidity
are taken from climatology for the 1958–2001 period,
which in this paper is referred to as “present day”. REMBO includes a
1-layer snowpack model with a simple parameterization of refreezing. Surface
albedo depends on snow thickness and the melt rate. Surface melt is computed
using a simple parameterization of and depends
on both temperature and insolation. The formula for surface melt contains a
free parameter cm (melt parameter), which is one of the major parameters
determining the sensitivity of the ice sheet to climate change . It reads
m=1ρwLmτs1-αsS+cm+λT,
which relates the surface melt m with the free melt parameter cm. The
remaining variables ρw, Lm, τs, αs, S, λ and
T, are the density of water, the latent heat of ice melting, the total
transmissivity, the surface albedo, the insolation at the top of the
atmosphere, the long wave radiation coefficient and the surface temperature,
respectively. It is important to note that REMBO resolves the seasonal
cycle of temperature and insolation, which are both important for properly
modelling surface melt . Please, refer to
for more details.
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 R indicates the region over the ice
sheet where the discharge parameterization applies. The (half) circle
illustrates how that band R with the width of about ΔR is
determined in our scheme. Namely, the centre of such a circle is applied to
every land point (open circles over the brown area). The smallest distance to
the ocean lij is depicted here for one example ice grid point. It is
determined for every grid point inside the band R. For the discharge
parameterization, the ice thickness hij and the smallest distance to the
ocean lij are evaluated at grid points (i,j), see
Eq. ().
Ice discharge parameterization
Most ice discharge of the GrIS is brought into the ocean via fast-flowing
narrow outlet glaciers , most of which cannot be
resolved by the coarse-resolution GrIS models used for palaeoclimate
simulations and long-term sea level change projections. This motivates us to
include this portion of ice discharge in the mass balance of the ice sheet,
which reads
∂h∂t=-∇⋅q-d+b+b*,
where h is the ice thickness, ∇⋅q is
the divergence of large-scale lateral ice flow explicitly resolved by the
model, d is the divergence of the ice flow associated with non-resolved ice
streams/outlet glaciers which is parameterized (see below), b is the
surface mass balance and b* is the basal mass balance, which is rather
small. Note that, although b and d have the same units (m s-1) and are
treated as surface fluxes, they represent completely different physical
processes. The integral of d over the entire area of the GrIS represents
the total solid ice discharge into the ocean through outlet glaciers. Total
solid ice discharge into the ocean is equal to the sum of parameterized ice
discharge and the lateral ice discharge into the ocean explicitly computed by
SICOPOLIS. Under present day climate conditions the latter is much smaller
than the parameterized solid discharge. This is explained by the fact that
simulated modern GrIS has rather limited direct contact with the ocean. Note
that Eq. () per definition conserves mass.
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. ). It defines the length l in the discharge
parameterization (Eq. ). The yellow line indicates the ice
margin of the present-day Greenland ice sheet.
The divergence of the explicitly non-resolved fast lateral ice flow d in
each grid cell (i, j) is parameterized in a simple heuristic statistical
approach via the local thickness of ice h and the distance from the actual
grid cell to the nearest ocean grid cell l as
d=chplqinside areaR,0outside areaR,
where c, p and q are constant model parameters. R is defined as the
area which consists of grid points located within a distance not more than
ΔR from the nearest ice-free land surface point or the nearest ocean
point in case if the ice sheet has direct contact with the ocean (see
Fig. ). The rationale for this parameterization is the
following. The ice thickness near the ice margins represents the amount of
ice available to the outlet glaciers, while the distance to the coast can be
regarded as a statistical measure of the outlet glacier density: if the ice
margin is far away from the coast, it is very unlikely that any outlet
glacier has contact with the ocean and there is only minor calving flux into
the ocean, while one would expect a large calving flux for small distances to
the coast. It is assumed that significant influence of outlet glaciers does
not penetrate inside of the ice sheet (distance more than ΔR).
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 α0= 60∘ to
the angle between the gradients of surface elevation ∇zs
and distance field ∇l (see
Fig. for the characteristics of the field l). Using
the definition of the scalar product for the angle between above two
gradients, this reads
∇zs⋅∇l|∇zs||∇l|≥cosα0,evaluated at(i,j).
This prevents our parameterization from simulating ice discharge into the
ocean from an ice margin, which is oriented towards the interior of the
Greenland island, which could happen when the GrIS is retreating under warm
climates, and when only a few small ice sheets remain; see the small land bridge
between the two ice caps in Fig. b for an example.
Simulated parameterized divergence of fast ice flow in m yr-1
(Eq. ) at (a) present day and, (b) at Eemian from the
model version with the most reduced Eemian ice sheet.
The discharge parameterization is applied only to the ice-covered grid cells
that are located not more than ΔR (here ΔR= 120 km) from the
ice margin (see Fig. ). The resulting belt encloses the
regions of ice with rather high velocities as found by satellite measurements
. This reflects that our
parameterization primarily accounts for the ice discharge via the near margin
fast-flow and outlet glaciers.
The value of parameter c depends strongly on the choice of the powers p
and q to maintain total ice discharge known form observational data. For
convenience, we normalize c as
c=c0cd,
i.e. for any fixed value of p and q we selected c0 such that cd
has a value of about one. In practice, after selecting p and q we chose
c0 so that the parameterization applied to observed Greenland elevation
matched observed total discharge, for which we used 350 Gt yr-1. The latter
value is just about the average of the totals of ice discharge as found by
and . Although the discharge
for the modelled present-day GrIS will not be precisely 350 Gt yr-1, such an
approach guarantees that all valid values of the parameter cd keep the
order of magnitude of about one for any power p and q. We thus have three
free parameters cd, p and q in our ice discharge parameterization
(Eqs. and ).
As seen in Fig. , the minimal distance to the coast l
can reach up to about 400 km in the centre of Greenland. In our
parameterization, the inverse dependence of the ice discharge on the distance
to the coast results in high ice discharge over regions very near to the
coast (dark brown colour) and low discharge further inland (lighter brown
colours). For a distance of around 400 km, the parameterized divergence of
fast flow would nearly vanish. In our discharge parameterization, this is
only relevant for an ice sheet under warmer climates, which can be much
smaller than the present-day GrIS. In fact, because our parameterization only
applies for ΔR< 120 km around the ice margins, most
parameterized divergence of fast flow occurs at the near-margin ice area for
present-day Greenland (Fig. 3a). These are the regions where a high
present-day ice discharge due to many outlet glaciers is observed
e.g., e.g. over the north-western region of the GrIS.
For regions with fewer marine outlet glaciers, e.g. in the south-west, the
observed ice margin resides rather in regions further from the coast.
In the model, the ice-marginal ring ΔR and the distance field are
computed every time step. The former is necessary, because the ice sheet
changes its shape in time and because the coastline can potentially change
its shape due to sea level change.
Measures of geometrical characteristics of an ice sheet
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
err(A)=|Amod-Aobs|Aobs⋅100,err(V)=|Vmod-Vobs|Vobs⋅100,
where Aobs, Amod, Vobs and
Vmod are the observed ice area, the modelled ice area, the
observed ice volume and the modelled ice volume, respectively. These errors
can approach zero in principle, but this does not guarantee accurate
simulation of the ice-sheet geometry, since regional errors can compensate
each other. Therefore, we choose a stronger constraint based on the error in
ice thickness expressed relatively to the total ice thickness. This reads
err(H)=∑ij|Hijmod-Hijobs|∑ijHijobs⋅100,
where Hijobs and Hijmod are the observed and
modelled ice thickness at the horizontal grid position (i, j),
respectively. The indices i, j run over the entire domain of the
computational area and assume that the ice thickness is zero outside the ice-covered area. This error only approaches zero when the ice-sheet thickness is
correctly simulated in each grid cell. Simulations assessed with the
different error measures are presented in Sect. .
Simulation setup
Following , we run the coupled REMBO-SICOPOLIS model
through two glacial cycles starting at 250 kyr BP. These simulations serve a
dual purpose: to perform a model spin-up necessary to simulate the
present-day state of the GrIS and to apply palaeoclimate constraints (see
below) to additionally reduce the range of model parameters. To drive the
model through two glacial cycles, we apply variations in insolation due to
changes in orbital parameters, equivalent CO2 concentration and regional
temperature anomaly obtained from the CLIMBER-2 model
. We took these
anomalies from the standard simulation as in . The
applied forcing is illustrated in Fig. 1 by . To
generate an ensemble of model realisations, we vary two parameters: the
discharge parameter cd (Sect. ) and the melt parameter
cm (Sect. ). The discharge parameter cd is varied in
steps of 0.2 and the melt parameter cm in steps of
5 W m-2. The geothermal heat flux is set to
50 m W m-2 and the sliding coefficient to
15 m (yr Pa)-2. All other parameters are the same as in .
We fix the powers p and q by minimizing the relative error in present-day
ice thickness (see Sect. ). Based on ensemble
simulations over the parameter space (cd, cm) for different powers p and
q (not explicitly displayed here), we found that decreasing p and
increasing q reduce the error in ice thickness err(H). For simplicity, we
chose the integers p= 1 and q= 3. To normalize the parameter cd (see
Sect. and Eq. ()), we set the
dependent parameter c0= 2.61 × 104 m3 s-1. All
ensemble simulations presented in this paper are performed with these values
for the powers and for the parameter c0 in the ice discharge parameterization.
Simulations of the GrIS in the entire parameter space
Figure compares the three error measures introduced in
Sect. for the ice-sheet shape in the phase space of
the melt, cm, and discharge, cd, parameters. At first sight, the error
fields in the three panels look similar. The smallest errors appear
approximately along the descending diagonal, which is characterized by
decreasing values of cm and increasing values of cd. One can also see
that the parameter combinations with small errors are not limited to our
sampled space: these regions expand in the direction of the descending
diagonal. This underlines the need for more constraints (Sect. ).
Error measures for a modelled ice sheet in the {cd}×{cm}
parameter space. Relative errors in (a) ice area, (b) ice
volume and (c) ice thickness, all in percent.
Figure illustrates that our discharge parameterization
allows us to reduce the errors in total area and volume practically to zero.
However, we found no parameter combination for which the error in ice
thickness was much lower then 20 %. Still, these are considerable
improvements compared to the standard version of the model without ice discharge
parameterization constrained only by the mass balance partition
, which overestimates total ice volume and area by
ca. 20 % and has a relative thickness error of ca. 30 %. When the mass
balance partition constraint is ignored, one can improve model performance in
simulation of ice sheet shape by increasing surface melt. By choosing
cm=-40 W m-2, one can make all three errors comparable
with those of the best model version with ice discharge parameterization.
However, such a high melt factor practically eliminates ice discharge into
the ocean and, as shown below, drastically affects the GrIS stability. In
fact, it causes the GrIS to be unstable even under near-present-day climate conditions.
Constraining the model parameters
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 a–c illustrates all
constraints used in this paper. We accept a value of 20 % for the relative
error in ice thickness. This choice is not totally arbitrary, because a
closer inspection of the error field shows a minimum error in ice thickness
of 18.2 %, i.e. there is indeed a plateau defined by ice thickness error
values ≤ 20 %, as illustrated in Fig. a by the medium
green shading. Within the parameter space, the error in ice thickness varies
much more strongly for values higher than 20 %. This plateau structure of the
error field motivated us to choose 20 % as the error limit.
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
, the MBP was diagnosed by REMBO from simulations
with prescribed observed present-day ice-sheet topography. This was done
because of systematic (and regionally significant) deviations of the
simulated present-day GrIS from observational data. With the ice discharge
parameterization, we can now safely operate with the simulated present day
ice-sheet topography for determining MBP, because of the better match between
simulated and observed topography. This means that our new MBP values
simulated with ice discharge parameterization slightly differ from our former
approach (cd=0), and the valid MBP range (Fig. b)
corresponds to somewhat lower values of cm compared to those by
. However, the MBP has large inherent uncertainty,
which we derived from results of regional climate models, following
and yielding a range of 45 to 65 %.
Estimated constraints on the parameters cd (ice discharge) and
cm (surface melt) illustrated with our ensemble simulations together with
our estimate of the GrIS contribution to Eemian sea-level rise.
(a) Relative error in ice thickness (%). (b) Mass balance partition (%).
(c) Maximum elevation reduction during the Eemian compared to
present day, 200 km upstream from NEEM (m). Here, regions where no Eemian ice
is simulated at the upstream position of NEEM are displayed in white.
(d) Simulated contribution of the Greenland ice sheet to sea-level rise
between the Eemian and present day under our constraints. The black lines in
(a)–(c) indicate our constraints: error values < 20 % for ice thickness,
a 45 to 65 % range for mass balance partition and an Eemian to present-day
surface elevation reduction of <= 430 m at the upstream position of NEEM.
From measurement of air content in the NEEM borehole samples, the
found that the surface elevation of the source area was
130 ± 300 m lower during the Eemian than at present day, which we exploit as our
third constraint. Following these findings, we assume the maximal surface
elevation drop at this location during the Eemian (130 to 115 kyr BP) did not
exceed 430 m (compared to present day). Accounting for the trajectory tracing
results by , we defined the deposition position of Eemian ice
in the NEEM ice core at the location about 200 km upstream from the NEEM
drilling site at (45∘ W, 76∘ N) (see
Fig. d), denoted NEEMup hereafter. We use the
Eemian elevation drop at the NEEM source location as additional
empirical constraint.
Figure illustrates that none of the constraints is
redundant, because the regions of valid simulations for all three constraints
intersect each other and there is plenty of space without a common crossover
for every constraint. In particular, the error in ice thickness excludes low
values of the ice discharge parameter, while the mass balance partition
constrains the range of the melt parameter, the upper bound of which is then
further constrained by the NEEMup elevation data. While the valid region of
all three constraints cover an approximately equally large part of the parameter
space (Fig. a–c), only a relatively small subset of model
parameters (Fig. d) is consistent with all of these
constraints simultaneously.
Figure shows the range of ice margins which are consistent
with the different constraints shown in Fig. . It can be
seen that the MBP constraint alone (Fig. a) gives a rather
broad band of valid ice margins, while the NEEMup constraint
(Fig. b) alone results in an ice margin range, which is quite
comparable with that of the err(H) constraint
(Fig. c). Finally, all three constraints together give a
pronounced reduction of spread of ice margins (Fig. d)
compared to the single-constraint cases. Simulated ice margins in the south,
mid-west and north-east of Greenland compare well with observations. However,
there are regions with rather strong mismatch in the south-west and in the
north. Parts of this mismatch can be attributed to our model biases in
precipitation. For example, REMBO simulates too much snow accumulation in the
north-east and south-west of Greenland compared to the compilation by .
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. (a)–(c) Single
constraint applies: (a) mass balance partition, (b) elevation
reduction during the Eemian referenced to present day at the upstream
position of NEEM, and (c) error in ice thickness. (d) All three
constraints apply.
Present-day (a–c) and Eemian (d–f) surface topography for
varying melt and discharge parameters. Simulations correspond to those giving
high medium and low contributions to Eemian sea-level rise (2.5, 1.5 and
0.6 m), respectively. The Eemian snapshots correspond to times with the
simulated minimum ice volume during the Eemian for the respective simulation.
NEEM locations are marked in magenta (square for borehole and circle for
upstream).
Figure d depicts the simulated difference in GrIS volume
between Eemian and present day expressed in units of global sea level
equivalent. Compared to the other figure panels, we show here results from
simulations with refined melt parameter spacing of
1 W m-2. We enhanced the resolution of the melt
parameter sampling, because the region of valid simulations appears rather
elongated in the parameter space. The estimated Eemian sea-level contribution
increases with increasing (less negative) cm. This is understandable,
because surface melting increases with a higher (less negative) melt
parameter. Nevertheless, there is also an increase of the GrIS contribution
to the Eemian sea-level highstand for increasing discharge parameter values.
Obviously, there is an interplay between ice discharge and surface melt,
because the ice discharge removes ice from the ice sheet and brings the ice
surface into lower regions of the atmosphere, where stronger surface melt can
occur. Averaged over the parameter space of valid simulations, we have a
contribution of the GrIS to Eemian sea-level rise (above present-day value)
of 1.4 m. The minimum contribution of the GrIS sea level rise among all valid
simulations is 0.6 m and the corresponding maximum is 2.5 m.
Eemian versus present day and GrIS stability
Figure shows the simulated present-day and Eemian ice
sheet distributions from model versions with high, medium and low sea-level
rise contributions of the Eemian compared to present day. While all fields
look rather similar for the present day, there is a considerable difference
between the corresponding Eemian fields. However, the present-day surface
elevations for the different valid parameters sets still show slight
differences. Naturally, these differences appear mainly near the ice margin,
while the interior of the ice sheet remains almost unchanged for any valid
parameter set. As is often the case in such optimization problems, there is a
trade-off concerning agreement with observations in certain regions (see
Fig. a for the observed surface elevation). While the
simulation with cd= 0.8, cm=-53 W m-2
(Fig. a) better resembles the ice-free south-western
region, the northern region around Petermann Glacier matches the
observation less well. This situation is opposite for the simulation with
cd= 1.2, cm=-66 W m-2 (Fig. c).
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. d–f, which spans the simulated
lowest and highest Eemian to present-day GrIS contribution to sea level rise.
For model versions with high sensitivity to climate forcing, the GrIS splits
into two parts: a small ice cap in southern Greenland and a larger ice sheet
in central Greenland (Fig. d). For the intermediate and
low sensitivity model versions, the GrIS remains in one piece
(Fig. e and f). In all valid model versions, there is a
strong retreat of ice, mainly in western and northern Greenland. Our
estimates showing a strong retreat of the GrIS during the Eemian rather
correspond to the simulations by , while the
medium to modest retreat of the Eemian GrIS was found in simulations by
and .
Interestingly, the NEEM location almost becomes ice free at 121 kyr BP in our
most sensitive model version (see Fig. d). Nonetheless,
an ice-free NEEM position during the Eemian would not contradict the existence of
Eemian ice and most probably pre-Eemian ice in the NEEM ice core at
present day, as reported by , since the Eemian ice was
accumulated farther upstream of NEEM. Similar argumentation would hold for
Camp Century as well.
Figure shows time series of ice volume and the NEEMup
surface elevation for simulations over the last two glacial cycles from
previous work with the same model and our present
approach, which includes the sub-grid scale discharge parameterization. At
all times, the valid model versions with the discharge parameterization
simulate less ice volume than that without the discharge parameterization
(Fig. a). This has two reasons: (i) previously, the model
was not tuned for agreement with present-day surface elevation (ice
volume). The present-day surface mass balance partition was (and is here)
regarded as the more adequate characteristic to capture the sensitivity of
the GrIS to long-term climate change. (ii) In our present approach, the
inclusion of the discharge parameterization enables our rather coarse
resolution model to mimic the calving of the small-scale outlet glaciers
(i.e. removal in ice into the ocean by ice discharge) without an
overestimation of contact regions of the ice sheet with the ocean, which
leads to a smaller ice sheet.
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 (cd= 0). Solid black
lines indicate the central run of a set of optimized simulations by
. The red lines are from simulations found via
shape-only tuning of the melt parameter (see main text for explanation). In
particular, a simulation with cm=-42 W m-2, found by
minimizing err(H) (solid red line) and, alternatively,
with cm=-40 W m-2, determined by minimizing
err(V) (dashed red line). (a) Ice volume of the
Greenland ice sheet. (b) Surface elevation at the NEEMup location.
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 cd= 0 (discharge parameterization off) and minimize the
error measure err(H) and, alternatively, the weaker error
measure err(V) to get the right present-day shape. The
former belongs to the parameter setting cd= 0,
cm=-42 W m-2 and the latter to cd= 0,
cm=-40 W m-2. Please, note that these melt parameter
values are outside the valid range of MBP as determined by
using observed present-day topography as well as
outside the valid cm values in MBP space of this work
(Fig. b) using simulated present-day topography to
determine MBP. Because we consider the present-day ice-sheet shape as the
only constraint (for demonstration), the model without the discharge
parameterization (cd= 0) requires a rather high value of the melt parameter
cm to minimized shape errors (Fig. ). As one can see
in Fig. a, around present-day the red line corresponding
to minimal err(H) is very close to the upper value of the
range of the simulations with sub-grid discharge parameterization (blue
shading), while the other red line (minimal err(V)) even
merges with that valid range. Simulated present-day elevations at NEEMup lie
rather close to each other in different model versions. However, during the
Eemian interglacial, the runs from the shape-only constraints show strong
downward excursions for ice volume as well as for the NEEMup elevation
(Fig. b). Whether such a small Eemian ice volume is still
realistic might be disputable but in any case the simulated Eemian reduction
in NEEMup elevation is by far larger than that estimated from the ice core
. The NEEMup position was even ice free during the Eemian in the
simulation without the ice discharge parameterization and minimized
err(V), which certainly contradicts observational data.
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
and our present ones with 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
, but in temperature space instead of insolation
space. Namely, we performed a suite of steady state simulations each 300 kyr
long imposing different spatially uniform and temporarily constant surface
air temperature anomalies to the lateral boundary conditions of the REMBO
model for each single simulation. We use the simulated present-day GrIS as an
initial condition. We sample with a temperature increment of
0.25 ∘C, i.e. we add a ΔT= 0.25 ∘C,
0.5 ∘C, …, 2.75 ∘C to the present-day
temperature at the lateral boundaries of the REMBO model domain and run the
model for each individual ΔT into a steady state. If the GrIS decays
for a certain ΔT but covers most of the Greenland land surface (see
) for ΔT- 0.25 ∘C, we define
ΔT as the temperature threshold of GrIS decay.
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 ∘C,
depending on values of model parameters (see also
Table ). We also applied our stability analysis with
the steady state method to a model version without the discharge
parameterization using the shape-only constraint. The threshold estimated
with this shape-only setting with err(V) minimisation
(cd= 0, cm=-40 W m-2) is much lower – only
0.25 ∘C. The higher values for GrIS decay between 1.25
and 2.5 ∘C from our new REMBO-SICOPOLIS
version with discharge parameterization are in the range of those that we
found previously without using the discharge parameterization
. This similarity clearly is an implication of the
use of the MBP as one common constraint in both approaches. In this work, the
shape and palaeo constraints are important as well, because they cover
different regions in parameters space as discussed in
Sect. . It should be noted that a complete
uncertainty analysis with our new model version is planned in the future,
which will likely widen the range of our estimates.
Present-day Greenland ice sheet topography. (a) Observed data
compilation by . (b) Simulation after
. (c) Simulation from present work, given the
parameter combination with smallest error in err(H),
i.e. cd= 1.4, cm=-60 W m-2.
In summary, if we optimize the melt parameter in the coarse resolution model
without the sub-grid scale ice discharge parameterization for only
err(H) or err(V), the resulting
models all overestimate surface melt and violate the MBP criterion. This
leads to a strong drop of the NEEM elevation far below the one reconstructed
from palaeo data, and the Greenland ice sheet becomes too sensitive to
temperature anomalies. This is why and
used the MBP criterion together with a palaeo
constraint for calibration of their coarse resolution model, ensuring the
correct long-term stability properties reported in this work. In our improved
model with sub-grid scale discharge parameterization, we found a stability
behaviour similar to that found by when using the
MBP and NEEMup constraints, but now – still in the coarse resolution
ice-sheet model – we can additionally fulfil a strong present-day
shape constraint and achieve err(H) < 20 %. We expect that
all of our constraints would play a similar role in a model of the Greenland
glacial system, which explicitly describes small-scale fast processes.
Development of such a model is in our future plans.
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 a and b compares simulated divergence of fast flow
during present-day and the Eemian.
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: , Ri08:
and En14: . Further on, SIM indicates our
simulations. (a)–(e) Sectoral Greenland ice discharge in Gt yr-1 for the
northern (N), north-western (NW), north-eastern (NE), south-western (SW), and
south-eastern (SE) parts (colours of the sectors are indicated in the inset).
(f) Total ice discharge in Gt yr-1. Note that the y axes have
different scales.
As stated in Sect. , our discharge parameterization is not
intended to resolve every small individual outlet glacier; it is rather
designed to capture their draining effect on the spatial and temporal average
in a sub-grid scale statistical approach. Consequently, the simulated ice
discharge of a certain region is the integral of the divergence of fast flow
(Eq. ) over this region. In general, our simulated
present-day ice discharge (Fig. a) is high over regions
with many observed outlet glaciers and low over regions with fewer observed
outlet glaciers, see Fig. 3 in for comparison.
However, in the south-western part of Greenland, the model overestimates the
(sectoral/regional) ice discharge, in part due to too high simulated
accumulation over that region (see 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.
Comparison with present-day observations and findings by others
A direct comparison of our simulated Greenland surface elevation with the
observed elevation by and the former approach of
is shown in Fig. . Overall, we
improved agreement with observations significantly. In particular, in the
simulation with the discharge parameterization several regions are now
ice-free, which look very similar to reality. The remaining deficiencies are
partly due to the simple discharge parameterization and the limitations in
the REMBO climatology, e.g. biases in representation of precipitation as
discussed earlier. In this context, we would like to stress, that our model
is a fully interactive one, where no observational data over Greenland are
prescribed. This ensures that REMBO is applicable to climates far away from
the present state, what is vital because the Eemian climate can deliver
additional constraints for the model.
Figure compares our simulated present-day sectoral and total
ice discharge with findings by others. The sectors (see inset in the upper
right of Fig. ) correspond with those of and
sub-divide the GrIS into northern (sector N), north-western (sector NW),
north-eastern (sector NE), south-western (sector SW), and south-eastern
(sector SE) parts. This subdivision is also adequate to the degree of
complexity of our model in its current stage of
development (a refinement of the sectors is planned for our later work).
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-1 compared to the compilation by
. In sector SE, our results are consistent with
and but are significantly lower than
those by . The reason for the mismatch of our
simulated ice discharge over sector SE with that by
could be because Enderlin's data, from the year 2000, already reflect the
response of the outlet glaciers to warming in the fjords due to subtropic
water transported by the ocean towards Greenland . The
SE sector may be particularly sensitive to this change, because in this
region several outlet glaciers have contact with the ocean via fjords.
Our range of valid model versions is 326–479 Gt yr-1 simulated total ice
discharge. The lower estimate matches that by , while the
upper end of this range is somewhat larger than the ice discharge by
. The relative small total ice discharge by
corresponds to the rather small accumulation estimate of the
compilation, which Reeh used together with the
assumption that the GrIS is in equilibrium to derive his discharge values.
The data by can be regarded as roughly similar to
pre-industrial, because it is based on accumulation, which contains several
old data points, certainly before the 1990s. Meanwhile, the data by
belongs to a more recent time, which corresponds to
the upper value of the time interval of the re-analysis data used to
prescribed the lateral boundary conditions of REMBO (1958–2001). Because by
the year 2000, Greenland already started to respond to global warming, it is
reasonable that the upper end of our simulated ice discharge range approximately
agrees with the finding by .
Discussion
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
e.g.. In this paper, we restricted
our analysis to the parameters for surface melt and ice discharge, which
rather affect the marginal regions of the ice sheet. Nonetheless, we use the
error metric of the entire ice sheet here in order to keep our approach
simple, and because we intend to use the metric in forthcoming work, which
will include a more comprehensive approach.
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 , as inclusion of membrane stress can reduce
ice volume change in short-term climate projections by up to 20 %
. In particular, the role of stress transfer might
become important in short-term climate projections .
Inclusion of membrane stress e.g. in our modelling
approach together with better resolved topography
is planned for the future. In addition, a comparison of our discharge
parameterization with the high-resolution and high-order models will help to
improve the parameterization, because it will still be useful for
large-ensemble simulations.
The term d in the thickness evolution equation (Eq. ) represents
the divergence of the ice flow associated with non-resolved ice
streams/outlet glaciers, see also Appendix . In this sense, d
reduces the ice thickness over areas with fast flow. This is why we only
consider total and sectoral ice discharge, which is equivalent to the
spatial integral of d over areas near the ice margin where fast flow
appears. Of course, our parameterization cannot explicitly simulate ice
calving and we remove some mass over inland regions where in reality no
calving appears. The distance parameter accounts for this, because it appears
in the denominator of our parameterization and, therefore, our method removes
the highest amount of mass near the ocean. One advantages of our method is
that it enables us to distinguish between the mass that is removed by surface
melt and by calving.
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 . In their statistics, the NEEM
community members gave a one σ error for this value. In principle, one
could have included the more uncertain values too by using the two σ range.
Nonetheless, all of our simulations with valid parameter sets show a
strong retreat of the ice in northern Greenland during Eemian times. Such a
retreat strongly influences the local climate and might lead to an additional
Eemian temperature rise over that region, although it is unlikely to be as
strong as that reported by the . These and other uncertainties
in Eemian temperature and precipitation will be examined in future work.
Conclusions
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 , and a palaeo constraint
on Greenland's surface elevation drop (upstream of the NEEM borehole) during
the Eemian interglacial compared to present. We favoured a measure of ice
thickness error at each grid point instead of just considering total
Greenland area or volume, since it is a stronger measure of the quality of
simulated ice-sheet shape.
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 ∘C was sufficient to melt the GrIS almost
completely on longer time scales. When applying the MBP constraint in a
coarse resolution model without the sub-grid scale ice discharge
parameterization, the model has about the same stability properties as with
the discharge parameterization.
The inclusion of our ice discharge parameterization along with the
above-described constraints leads to similar results concerning long-term
stability as , with a decay threshold between
1.25 and 2.5 ∘C. Note that although this range is
consistent with previous work , it does not result
from an exhaustive uncertainty analysis. An updated range comparable with
will be the provided in future work. Finally,
complying with all three constraints leads to a GrIS contribution to sea
level rise during the Eemian compared to present day in the range of 0.6–2.5 m,
with an average of 1.4 m. Again, this range could widen if further
uncertainties were included.
The ice discharge parameterization as divergence of fast flow
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. ()
d∝hl3.
If we assume that the basal topography has a small spatial gradient, then the
surface gradient equals approximately the gradient of ice thickness.
Furthermore, let us rename l, the distance from a point in the ice sheet to
the nearest ocean grid: l=Δx. Because the ice thickness at an ocean
grid point is zero in our coarse resolution model, we can write Δh=h.
With that we can very crudely define the surface gradient Δh/Δx
at the ice margin. We can then write
hl3=hΔx3=ΔhΔh2hΔx3=1ΔxhΔh2ΔhΔx2≈1Δxhh2ΔhΔx2=1ΔxhΔhΔx2h2.
Making Δx very small and using Eq. () yields
d∝1dxhdhdx2h2.
The fraction in the latter equation is a Weertman-type sliding law
ub(x)∝dhdx2h2.
As sliding contributes by far the most to the velocity of ice streams, we can
thus generalize to two dimensions
d=∇⋅qfast,
where qfast=hub. This we interpret as
the divergence of the fast flow of outlet glaciers. Of course, this only
corresponds to our rather simple approach. Nonetheless, we believe that the
similarity of our parameterization to the divergence of a Weertman-type
sliding velocity substantiates our approach.
The Supplement related to this article is available online at doi:10.5194/tc-9-179-2015-supplement.
Acknowledgements
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
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