TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-12-39-2018Influence of temperature fluctuations on equilibrium ice sheet volumeMikkelsenTroels Bøgeholmhttps://orcid.org/0000-0003-3904-1048GrinstedAslakhttps://orcid.org/0000-0003-1634-6009DitlevsenPeterhttps://orcid.org/0000-0003-2120-7732Centre for Ice and Climate, Niels Bohr Institute, Juliane Maries Vej 30, 2100 Copenhagen Ø, DenmarkTroels Mikkelsen (bogeholm@nbi.ku.dk)8January2018121394727March201720November20174November201724April2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/12/39/2018/tc-12-39-2018.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/12/39/2018/tc-12-39-2018.pdf
Forecasting the future sea level relies on accurate modeling of the response
of the Greenland and Antarctic ice sheets to changing temperatures. The
surface mass balance (SMB) of the Greenland Ice Sheet (GrIS) has a nonlinear
response to warming. Cold and warm anomalies of equal size do not cancel out
and it is therefore important to consider the effect of interannual
fluctuations in temperature. We find that the steady-state volume of an ice
sheet is biased toward larger size if interannual temperature fluctuations
are not taken into account in numerical modeling of the ice sheet. We
illustrate this in a simple ice sheet model and find that the equilibrium ice
volume is approximately 1 mSLE (meters sea level equivalent) smaller when the
simple model is forced with fluctuating temperatures as opposed to a stable
climate. It is therefore important to consider the effect of interannual
temperature fluctuations when designing long experiments such as
paleo-spin-ups. We show how the magnitude of the potential bias can be
quantified statistically. For recent simulations of the Greenland Ice Sheet,
we estimate the bias to be 30 Gtyr-1
(24–59 Gtyr-1, 95 % credibility) for a warming of
3 ∘C above preindustrial values, or 13 % (10–25,
95 % credibility) of the present-day rate of ice loss. Models of the
Greenland Ice Sheet show a collapse threshold beyond which the ice sheet
becomes unsustainable. The proximity of the threshold will be underestimated
if temperature fluctuations are not taken into account. We estimate the bias
to be 0.12 ∘C (0.10–0.18 ∘C, 95 %
credibility) for a recent estimate of the threshold. In light of our findings
it is important to gauge the extent to which this increased variability will
influence the mass balance of the ice sheets.
Introduction
Ice sheet mass balance has a nonlinear dependence on temperature. This
behavior is observed in simple ice sheet models
and in
regional climate modeling of Greenland surface mass balance (SMB)
, and the nonlinear effect of temperature on melt has
been observed in Greenland river discharge .
Ice sheets are characterized by a large interior plateau flanked by
comparatively steeper margins. A warming will shift the equilibrium line
altitude (ELA) to higher elevations, increasing the area exposed to melt. The
area exposed to melt will increase nonlinearly with ELA because of the
top-heavy hypsometry . This mechanism explains the
nonlinear dependence of mass balance on temperature for ice sheets where
runoff is a significant fraction of the total mass balance. This mechanism
is important for the mass balance of present-day Greenland but less so for
present-day Antarctica where mass loss is dominated by solid ice discharge
p. 1170. However, observations show that the response
of Antarctic melt to temperature is nonlinear , while the
potential for a large nonlinear response of Antarctic mass balance is
particularly evident in the simulations from .
The nonlinear relationship between mass balance and warming means that there
is an asymmetry in the response to cold vs. warm anomalies. Using a simple
ice sheet model we will show how, as a consequence of this nonlinearity, the
average mass balance will be different when forcing the model with a variable
climate compared to a constant average climate. Simulations using constant
climate will therefore be biased unless they make statistical corrections to
allow for variance. Constant climate forcing is sometimes used to trace the
long-term equilibrium response of ice sheet models as a function of
temperature (e.g., ).
Ice sheet modeling and evidence from paleoclimatic records indicate that ice
sheets display a hysteresis response to climate forcing, indicating
a critical threshold in temperature, a tipping point, beyond which an ice
sheet becomes unsustainable . This is
a generic saddle-node bifurcation point, estimated by
to be reached for the Greenland Ice Sheet (GrIS) at a global warming of
+1.6 ∘C (0.8–3.2 ∘C) above the
preindustrial value.
The stability of ice sheets is typically investigated by imposing a constant
climate forcing and then letting the ice sheet model reach equilibrium
. The
hysteresis curve and collapse thresholds are then traced out by
repeating these experiments for a range of temperatures and starting from
ice-free conditions. However, this approach disregards the effects of interannual
temperature variability.
That the surface mass balance of an ice sheet model is nonlinear with respect to temperature
has previously been investigated in several studies. In a simplified model of
continental ice sheets, show that the total annual
ablation scales with the cube of temperature at the ice sheet margin.
specifically avoid using average monthly temperature and
precipitation climatologies and instead use time series from individual
months in order to include the effect of interannual variability in their
study. see Fig. 6h investigate the GrIS SMB simulated
by regional climate models (RCMs) as a function of mean surface temperature
from general circulation models (GCMs). Our contribution is a quantification
of this effect and an estimate of the necessary bias correction to surface
temperature needed to account for temperature fluctuations in long-term ice
sheet simulations.
Previous studies of natural variability in the context of ice sheets include
, who found that the variability in the GrIS surface mass
balance will increase in a warmer climate due to increased ablation area, and
, who found that large fluctuations in glacier extent can
be driven by natural, fast fluctuations in climate. Sub-annual temperature
variability in the context of positive-degree-day (PDD) models is
investigated in, for example, , ,
, , , and .
PDD models connect surface melting and air temperature and are used
extensively due to their simplicity and wide availability of air temperature
data . compares the Greenland SMB
calculated from four different annual PDD formulations with a reference SMB
calculated from a PDD scheme using a monthly air temperature and
precipitation climatology and deviations from a long-term interannual mean.
On the scale of sub-annual climatology, there are large uncertainties as the
estimates of the SMB differ significantly depending on the simplifying
assumptions used in the PDD formulation, highlighting the need to accurately
model both spatial and temporal variability. These findings are built upon by
, who found that the standard deviation of monthly average
temperature may be represented as a quadratic function of monthly average
temperature. In the present study we are concerned with interannual
variability and expect our results to apply independently of the chosen SMB
model.
We investigate how climate variability influences the mass balance of ice
sheets with a nonlinear response to climate forcing. We derive a simple
statistical relationship which can be used to quantify the effect and
illustrate why it matters on a minimal ice sheet model. We then proceed to
show how this may be applied to published results from a coupled ice sheet
model. In Sect. we derive an analytical relationship
between the magnitude of temperature fluctuations and ice sheet volume,
assuming a simple relationship between the mass balance, temperature and ice
sheet volume. This relationship is shown to hold using a simple ice sheet
model (that includes a surface mass balance model) in
Sect. , and in Sect. we estimate
the consequences of temperature fluctuations on a recent long-term ice sheet
study , assuming the effect of temperature fluctuation
presented here is not already accounted for. The limitations of this
approach, as well as further possible applications, are discussed in
Sect. .
The mass balance of an ice sheetA minimal ice sheet model
In order to investigate the influence of temperature fluctuations on the mass
balance we consider a simple ice sheet model introduced by
hereafter denoted Oer03. This model describes
the essential dynamics of an ice sheet assumed to be axially symmetric and
resting on a bed that slopes linearly downwards from the center. The ice is
modeled as a perfectly plastic material, and the ice sheet is coupled to the
surrounding climate by adjusting the height of the equilibrium line
hEq:
hEq=hE,0+(T-T‾)⋅1000/6.5.
Specific balance B for T‾=0 from
Eq. () using the parameters in Table 1 and
Eqs. (S3)–(S4) of the Supplement. hEq denotes the equilibrium
line. The runoff line hr specifies the simplified climatic
conditions, as the specific balance is constant above hr (see also
Supplement, Eq. S4) and the balance gradient is constant below hr.
Equation () represents an increase in the equilibrium
line altitude of roughly 154 m∘C-1. The influence of
hEq on the specific balance B is illustrated in
Fig. . It should be noted that the simple relationship
described by Eq. () does not capture situations where the
SMB may increase with increasing temperature, as discussed in
Sect. . Further details of the Oer03 model allowing the
formulation in Eq. () below are described in the
Supplement.
The model is chosen for its simplicity; thus, it is not accurately modeling
a specific ice sheet. The two main reasons for choosing it for our analysis
are: (1) the simplicity of Oer03 allows the analytical approach detailed
below and (2) the Oer03 model shows the same functional relationship between
surface mass balance and temperature as has been found for regional
climate models for a range of temperature scenarios
. The change in volume or mass of the ice sheet depends
on the balance between accumulation, ablation and ice sheet discharge which
in turn depends on both the interplay between the fluctuating temperature and
the state of the ice sheet itself.
Before proceeding with the simple model, we investigate the effect of
interannual temperature fluctuations by considering the ice sheet as a simple
dynamical system. We assume that the temporal change in volume of the ice
sheet depends only on the volume V itself and a single time-varying mean
temperature over the ice sheet, T. The mass balance (change in ice sheet
volume) is denoted as dV/dt,
dV/dt=f(T,V),
where f(T,V) is some nonlinear function. The (stable) fixed point, f(T,V)=0, corresponds to a balance between loss and gain in the ice volume.
This is in general an implicit equation to determine the steady-state volume
V0(T) as a function of temperature, such that f(T,V0(T))=0.
(Left) Simulations of the Oer03 model for T‾=-1.5, 0,
1.5 and 3. The black curves denote a constant temperature and the grey curves
denote fluctuating temperatures generated with Eq. ().
(Right) The mass balance Eq. () for the Oer03 model in
the (T,V) plane. The black contour is the steady-state f=dV/dt=0. The markers represent the average of the
numerical simulation with constant (+) and fluctuating (∘)
temperature seen on the left. Finally the yellow contour shows the
approximation derived in Eq. ().
However, the fixed point is not identical to the statistically steady-state
volume with a temporally fluctuating temperature Tt=T(t) with
expectation value 〈Tt〉=T‾. A numerical
integration to equilibrium of an ice sheet model with and without interannual
fluctuating temperature shows that in steady state the ice sheet volume Vt
will fluctuate around 〈Vt〉=V‾,
where V‾ is systematically smaller than the corresponding
V0(T) (Fig. ). In Fig. ,
T‾ is shown on the horizontal axis in the right panel, and the
corresponding V‾ is shown on the vertical axis (both panels).
Since the temperature Tt – and thus the ice sheet volume Vt – is
a stochastic variable, the following will characterize an equilibrium state:
〈f(Tt,Vt)〉=0.
To calculate V‾ we perform a Taylor expansion of
Eq. () around the – presently unknown – steady state
(T‾,V‾) and calculate the mean volume V‾.
We use the notation fT:=∂f∂T, fTV:=∂2f∂T∂V, etc. Furthermore, f0:=f(T‾,V‾), fT0:=∂f∂T(T,V)|(T‾,V‾), etc. We then get
〈f(Tt,Vt)〉=f0+〈Tt-T‾〉fT0+〈Vt-V‾〉fV0+12〈(Tt-T‾)2〉fTT0+12〈(Vt-V‾)2〉fVV0+〈(Tt-T‾)(Vt-V‾)〉fTV0+O(3),
where O(3) represents higher order terms. We can simplify
Eq. () considerably: first note that since T‾
is the expectation value of Tt we have 〈Tt-T‾〉=〈Tt〉-T‾=T‾-T‾=0 and
with the same argument 〈Vt-V‾〉=0. The quantity
〈(Tt-T‾)2〉 is the variance of the fluctuating
temperature – we will assume this is known in simulations and substitute
〈(Tt-T‾)2〉=σT2. Since the temperature
variations are small with respect to the mean and have a symmetric
distribution, we may neglect higher order terms in Eq. ()
. We are left with
〈f(Tt,Vt)〉≈f0+σT22fTT0+12〈(Vt-V‾)2〉fVV0+〈(Tt-T‾)(Vt-V‾)〉fTV0.
We have evaluated the last two terms in Eq. ()
numerically for the Oer03 model and found that 〈(Vt-V‾)2〉 and 〈(Tt-T‾)(Vt-V‾)〉 tend to zero as the ice sheet approaches equilibrium
volume (Fig. S3, Supplement) – neglecting the last two terms,
Eq. () reduces to
〈f(Tt,Vt)〉≈f0+σT22fTT0.
Equation () is the main observation in this work. We shall
in the following estimate the implications of this result on realistic
asynchronously coupled state-of-the-art ice sheet climate model simulations.
As 〈f(Tt,Vt)〉=0 at the steady state it can be seen from
Eq. () that
0=f0+σT22fTT0⇒f0=-σT22fTT0>0
since fTT0<0 – this negative curvature of f0 is the
nonlinear effect causing the bias. V0(T) is the stable fixed point;
f(T,V0(T))=0 – thus f(T,V)>0 for V<V0 and f(T,V)<0 for V>V0.
This together with Eq. () implies that
V‾<V0; that is, a positive temperature anomaly increases the
mass loss more than what can be compensated for by an equally large negative
anomaly .
Ice sheet simulationsFluctuating temperatures
To generate an ensemble of volume simulations we use time series Tt
comparable to the observed temperatures over Greenland between year 1851 and
2011. For this we use the AR(1) process
:
Tt+1=T‾+a×(Tt-T‾)+σARWt,
where Wt,t=1,2,… are independent, random draws from a standard
normal distribution. The exact form of the model used for generating
temperature time series Tt is of less importance than the variance of the
resulting Tt as only the variance enters into Eq. ().
The parameters (a,σAR2) were obtained by fitting
Eq. () to the observed annual mean temperatures over
Greenland between years 1851 and 2011 (Supplement). We obtain (a,σAR2)=(0.67,0.85); thus, the AR(1) process
Eq. () has variance σT2=σAR2/(1-a2)=1.54∘C2.
As we quantify the effect of interannual stochastic variability we use
annually averaged temperatures, consistent with the formulation of the Oer03
model (see Table S1 of the Supplement). We find time step size of 1 year to
be sufficient for integrating the Oer03 model (Fig. S1); thus, Tt+1 in
Eq. () represents the temperature 1 year after Tt.
To find the steady-state volume we run the Oer03 model forward long enough
for the ice sheet to reach equilibrium, with and without fluctuating
temperatures. The results of this procedure are shown in
Fig. (left) where it is clearly seen that the
steady-state volume is smaller for simulations with fluctuating temperatures than
with constant temperature. We emphasize that the fluctuating temperature time
series {Tt} have as mean the constant temperature, 〈Tt〉=T‾, so that the differences are due only to the annual
temperature fluctuation.
In Fig. (right) the effect of temperature fluctuations
is shown in the (T,V) plane: the markers “+” are steady states of
numerical simulations with constant temperature, while the circles represent
ensemble averages of simulations with fluctuating temperatures. It is evident
that temperature fluctuations decrease the steady-state ice volume. The
yellow curve in Fig. (right) was calculated using
Eq. () and gives a good agreement with the results from
ensemble simulations.
In order to illustrate the physics behind Eq. (), consider
values of the mass budget function f for different ice sheet volumes V
(shown in Fig. ). The insert shows, for a particular value
of V, how the steady state is influenced by fluctuating temperatures: the
average mass budget of a colder year and a warmer year is less than the mass
budget of a year with a temperature corresponding to the average of “cold”
and “warm”; to put it another way, the increased SMB of a single
anomalously cold year cannot balance the increased melt from an equally
anomalously warm year . In particular let Tc=T‾-σ and Th=T‾+σ:
f(Tc,V)+f(Th,V)2<fTc+Th2,V,
which is consistent with fTT0<0 as shown in
Eq. ().
(a) Mass balance dV/dt of the ice sheet
for different values of the total ice sheet ice volume V in the
Oer03 model. Similar to Fig. but here we show
dV/dt as a function of T‾ for different total
volumes V. (c) The curvature of
dV(T‾)/dt influences the steady-state
behavior – a cold year does not cancel out the effect of an equally warm
year as shown in Eq. (). The value of σT is used for
illustration and is given as the square root of the temperature variance,
σT=1.54∘C2=1.24∘C. Note
the similarity of the dV(T‾)/dt found here to
Fig. 6h in . (b) Estimating the effect of
fluctuating temperatures on GrIS projections. The full curve is obtained by
fitting a third-degree polynomial f̃(T) to an SMB(T) from
. The dotted line shows the effect of temperature
fluctuations obtained by applying Eq. (). For a warming of
4 ∘C the green circle shows the SMB. ΔSMB is obtained
by applying Eq. () and represents the change in mass balance
resulting from the temperature fluctuations. -ΔT is the temperature
change required to negate this effect and is obtained implicitly from
Eq. ().
Maximum likelihood estimates of ΔT (effective temperature
change) and ΔSMB (effective SMB change where positive values
correspond to SMB loss, red curves) resulting from a given temperature
increase. ΔT and ΔSMB defined as in
Fig. b. The grey curves are estimates from
individual simulations and the blue shaded area denotes 95 % credibility
regions.
Consequences for long-term ice sheet simulations
Here we investigate the effect of accounting for fluctuating temperatures
when running long timescale climate simulations. These can be either
transient runs, scenarios with specified changing CO2 forcing or
equilibrium runs with specified constant forcing. Specifically, we analyze
the results of where the long-term stability of the
GrIS is investigated. In that study, an ice sheet model is forced by the
output of a regional climate model driven by the ERA40 climatology with
a constant temperature anomaly applied (see , and
the Supplement).
As parameters in ice sheet models are often tuned to reproduce an observed
ice sheet history from a time series of forcing observations (e.g.,
), the ice sheet volume bias we describe may already
be implicitly compensated for. To estimate the size of the temperature
fluctuation bias, we assume that this bias has not already been
accounted for by parameter tuning.
compare the output of RCMs forced with multiple future
climate scenarios and show that the effect of rising temperature on the GrIS
SMB is well described by a third-degree polynomial, which is consistent with the
aforementioned findings of . The reader may note the
qualitative similarities between Fig. in the present
article and Fig. 6h in . We follow
, and to the ensemble of simulations in
we fit third-degree polynomials to the SMB as a function
of temperature at time t= 200 years (see also the Supplement)
and obtain third-degree polynomials in T:
f̃ij(T)|f̃ij(T)=AijT3+BijT2+CijT+Dij,
where the indices i and j run over two separate parameters in the model
that take 9 and 11 values, respectively , so in total
we have 99 unique polynomial fits. These polynomials are then used as
a simple description of the mass balance function as a function of
temperature, SMBij(T)=f̃ij(T). Differentiating twice
we obtain f̃TT(T)=6AT+2B (suppressing indices i and j
for clarity).
For all parameter pairs (i,j) we evaluate f̃(T) and f̃(T)+(σT2/2)f̃TT(T) – this is shown in
Fig. b as the full and dotted lines, respectively.
To illustrate this approach we pick a specific temperature T0.
f̃(T0) is thus the SMB for a constant temperature and
f̃(T0)+(σT2/2)f̃TT(T0) represents the
effect of letting the temperatures fluctuate. This procedure gives us an
expression for ΔSMB:
ΔSMB=f̃(T0)-f̃(T0)+σT22f̃TT(T0)=-σT22f̃TT(T0),
where ΔSMB is positive in accordance with
Eq. (). Next we find the temperature
difference ΔT such that
f̃(T0-ΔT)+σT22f̃TT(T0-ΔT)=f̃(T0).
In this way ΔT is the effective temperature change resulting
from considering fluctuating temperatures.
The results of applying the steps outlined above on the data from
are shown in Fig. . The red
curves in Fig. show the most likely ΔT and
ΔSMB; the grey curves are estimates for the 9×11 individual
parameter values and the blue shaded area represents the 95 % credibility
region.
The warmings quoted in are relative to the
preindustrial period, whereas the reported warming from the preindustrial
period to the present day is estimated to 1 ∘Cp.
78. Following the RCP45 scenario it is more likely than
not that Earth will experience a further warming of 2.0 ∘Cp. 21 from today to the year 2100. Combining these
numbers we arrive at a warming of 3.0 ∘C in the year 2100
relative to the preindustrial period when considering the RCP45 scenario. For
this value it can be seen on Fig. (top) that an
additional 0.12 ∘C (0.10–0.18 ∘C, 95 %
credibility) should be added to any constant warming term when considering
simulations of the Greenland Ice Sheet, assuming the same temperature
variance as in Sect. . We note that this bias correction
is small compared to the spread in temperature projections. Nevertheless this
is a known bias that should be accounted for. The threshold for GrIS ice loss
has been estimated to be at +1.6 ∘C
(0.8–3.2 ∘C) . Applying the bias
correction above indicates that the threshold for GrIS may be
0.12 ∘C (0.1–0.18 ∘C) colder
(Fig. , top). This is not a large adjustment considering
other uncertainties, but it places additional constraints on the maximum
temperature increase admissible to avoid passing this threshold and the
corresponding multi-millennial sea level commitment.
Figure (bottom) shows the most likely ΔSMB
resulting from temperature fluctuations at a 3 ∘C warming to
be 30 Gtyr-1 (24–59 Gtyr-1, 95 % credibility)
or – for context – 30 Gtyr-1 (24–59 Gtyr-1,
95 % credibility) of the average GrIS SMB of
-234 ± 20 Gtyr-1 reported for the period 2003–2011
.
Observe in Fig. that ΔT goes to zero for low
temperature anomalies and appears to reach a constant value for higher
temperature anomalies. In the framework presented here this can be explained
by considering the SMB(T) curves shown in Fig. a. For
low temperature anomalies the SMB(T) curve in Fig. a is
close to flat so the second derivative is small; this gives a small
contribution to ΔSMB from Eq. (). On the other hand,
as the SMB(T) curve in Fig. a becomes progressively
steeper, a correspondingly smaller ΔT in Eq. () is
required to compensate for ΔSMB.
The results above highlight that interannual temperature variability cannot
be neglected in long-term studies involving ice sheet models. The
straightforward approach would be to simply include the expected temperature
variability in a number of simulations followed by calculating the ensemble
average. Conversely, one could calculate the effect of temperature
variability for a range of climate scenarios as a starting point for
a following bias adjustment.
Conclusions
From a theoretical argument and by considering a minimal model of an ice
sheet we have shown that fluctuating temperatures forcing the ice sheet have
an effect on the mass balance and thus on the steady-state volume of the ice
sheet (Eq. and Fig. ). The effect is
explained by the curvature, or second derivative, of the mass balance as
a function of temperature.
Temperature fluctuations can be accounted for in ice sheet modeling studies,
either explicitly (e.g., ) or implicitly,
as happens when tuning the ice sheet model to reproduce an observed ice sheet
history with observed forcing as input (e.g., ).
Temperature fluctuations may also be explicitly accounted for by forcing the
ice sheet model with climate model output that reproduces the magnitude of
the observed interannual temperature variability. Our results show the importance
of considering temperature fluctuations in the mass balance schemes before
bias correcting for other possible model deficiencies.
We find that the steady-state ice sheet volume in Oer03 is
0.5–1 mSLE (meters sea level equivalent) smaller when the minimal
model is forced with fluctuating temperatures compared to constant
temperature (Fig. ). It is therefore necessary to
consider the impact of temperature variability when designing long-term model
experiments such as paleo-spin-ups
, especially when
downsampling the paleo-forcing series. Though differences between ice sheet
models may be larger than the effect of temperature fluctuations estimated
here, we expect the effect to be in the same direction and of similar
magnitude for all models. Furthermore, models of sub-shelf melting, grounding
line migration and ice discharge have the potential to respond nonlinearly to
changes in ocean temperatures ; thus, it
is critical to take variability into account for quantitative assessments.
The response of a real ice sheet to temperature increase is naturally much
more complex than what can be described in a simple study such as the present
paper. In a model study, observe mass loss
acceleration of the northeastern GrIS as a response to warming. This part of
the GrIS experiences comparatively little precipitation and thus increasing
melt is not compensated for by increasing accumulation. However, the opposite has
been shown to be the case for Antarctica. show that
increasing temperatures will increase Antarctic SMB on continental
scales due to increasing precipitation. This is a case of
accumulation-dominated mass balance where the curvature term in Eq. ()
has the opposite sign; thus, an underestimated temperature fluctuation would
lead to an underestimation of the growth of the ice sheet.
When calculating the f̃'s in Eqs. ()
and () we assume a constant volume in the data from
, but in reality the relative variations are as large
as 9.5% when considering all the warming temperatures shown in
Fig. (Fig. S4). However, to draw the conclusion about
the consequences of a 3 ∘C warming it is adequate to consider
warmings less than 4 ∘C, and here the volume variation was
less than 3 % of the average (Fig. S5). Neglecting variations in volume
does add uncertainty to our results, and it is not immediately clear to us
how to quantify that uncertainty. Additionally, at time
t= 200 years where we extracted the SMB data from the
simulations in , the ice sheet model simulations had
not yet reached steady state; thus, expanding the analysis using a data set
from ice sheet simulations in steady state would be desirable.
We have evaluated the consequences of the temperature fluctuation bias on
long-term GrIS simulations and found that, if the full effects are taken into
account with no further modifications, a significant effective temperature
change would be required for an unbiased estimation of the equilibrium ice
volume.
The code is available online
. Data used in this study were obtained from the
authors of .
The Supplement related to this article is available online at https://doi.org/10.5194/tc-12-39-2018-supplement.
TBM, AG and PD designed the study. TBM performed the data analysis. TBM, AG and PD wrote the article.
The authors declare that they have no conflict of
interest.
Acknowledgements
We would like to thank Johannes Oerlemans for providing the original code for
his model and Alexander Robinson for being very helpful in providing the data
from their study. We are very grateful to Xavier Fettweis, Gerard Roe and an
anonymous reviewer for valuable comments and suggestions that helped improve
this paper. This work is part of the Dynamical Systems Interdisciplinary
Network (DSIN) – Troels Bøgeholm Mikkelsen was financially supported by
the Centre for Ice and Climate and the DSIN, both from the University of
Copenhagen. Edited by: Olivier
Gagliardini Reviewed by: Xavier Fettweis, Gerard Roe, and one
anonymous referee
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