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# A simple equation for the melt elevation feedback of ice sheets

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## Abstract and Figures

In recent decades, the Greenland Ice Sheet has been losing mass and has thereby contributed to global sea-level rise. The rate of ice loss is highly relevant for coastal protection worldwide. The ice loss is likely to increase under future warming. Beyond a critical temperature threshold, a meltdown of the Greenland Ice Sheet is induced by the self-enforcing feedback between its lowering surface elevation and its increasing surface mass loss: the more ice that is lost, the lower the ice surface and the warmer the surface air temperature, which fosters further melting and ice loss. The computation of this rate so far relies on complex numerical models which are the appropriate tools for capturing the complexity of the problem. By contrast we aim here at gaining a conceptual understanding by deriving a purposefully simple equation for the self-enforcing feedback which is then used to estimate the melt time for different levels of warming using three observable characteristics of the ice sheet itself and its surroundings. The analysis is purely conceptual in nature. It is missing important processes like ice dynamics for it to be useful for applications to sea-level rise on centennial timescales, but if the volume loss is dominated by the feedback, the resulting logarithmic equation unifies existing numerical simulations and shows that the melt time depends strongly on the level of warming with a critical slowdown near the threshold: the median time to lose 10 % of the present-day ice volume varies between about 3500 years for a temperature level of 0.5 °C above the threshold and 500 years for 5 °C. Unless future observations show a significantly higher melting sensitivity than currently observed, a complete meltdown is unlikely within the next 2000 years without significant ice-dynamical contributions.
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The Cryosphere, 10, 1799–1807, 2016
www.the-cryosphere.net/10/1799/2016/
doi:10.5194/tc-10-1799-2016
A simple equation for the melt elevation feedback of ice sheets
Anders Levermann1,2,3 and Ricarda Winkelmann1,3
1Potsdam Institute for Climate Impact Research, Potsdam, Germany
2LDEO, Columbia University, NY, USA
3Institute of Physics, Potsdam University, Potsdam, Germany
Correspondence to: Anders Levermann (anders.levermann@pik-potsdam.de)
Received: 7 March 2016 – Published in The Cryosphere Discuss.: 12 April 2016
Revised: 15 July 2016 – Published: 18 August 2016
Abstract. In recent decades, the Greenland Ice Sheet has
been losing mass and has thereby contributed to global sea-
level rise. The rate of ice loss is highly relevant for coastal
protection worldwide. The ice loss is likely to increase un-
der future warming. Beyond a critical temperature thresh-
old, a meltdown of the Greenland Ice Sheet is induced by
the self-enforcing feedback between its lowering surface ele-
vation and its increasing surface mass loss: the more ice that
is lost, the lower the ice surface and the warmer the surface
air temperature, which fosters further melting and ice loss.
The computation of this rate so far relies on complex nu-
merical models which are the appropriate tools for capturing
the complexity of the problem. By contrast we aim here at
gaining a conceptual understanding by deriving a purpose-
fully simple equation for the self-enforcing feedback which
is then used to estimate the melt time for different levels
of warming using three observable characteristics of the ice
sheet itself and its surroundings. The analysis is purely con-
ceptual in nature. It is missing important processes like ice
dynamics for it to be useful for applications to sea-level rise
on centennial timescales, but if the volume loss is dominated
by the feedback, the resulting logarithmic equation uniﬁes
existing numerical simulations and shows that the melt time
depends strongly on the level of warming with a critical slow-
down near the threshold: the median time to lose 10 % of the
present-day ice volume varies between about 3500 years for
a temperature level of 0.5 C above the threshold and 500
years for 5 C. Unless future observations show a signiﬁ-
cantly higher melting sensitivity than currently observed, a
complete meltdown is unlikely within the next 2000 years
without signiﬁcant ice-dynamical contributions.
1 Introduction
In past decades global mean sea level has been rising, mainly
by expansion of ocean waters and melting of ice on land
(Church et al., 2013). Over the past 2 decades, the Greenland
Ice Sheet has lost mass at an accelerating pace (Bamber et al.,
2000; Box et al., 2012; van den Broeke et al., 2009; Fettweis
et al., 2013; Mernild et al., 2011; Nick et al., 2009; Rignot
et al., 2008, 2011; Shepherd and Wingham, 2007; Thomas et
al., 2011). The ice loss is likely to increase under unabated
greenhouse gas emissions (Clark et al., 2016; Fettweis et al.,
2013; Goelzer et al., 2012; Graversen et al., 2011; Harper et
al., 2012; Huybrechts et al., 2011; Levermann et al., 2013;
Nowicki et al., 2013; Price et al., 2011).
Numerical simulations suggest that a decline of the Green-
land Ice Sheet is inevitable once its surface temperature per-
manently exceeds a certain threshold (Charbit et al., 2008;
Greve, 2000; Huybrechts and De Wolde, 1999; Huybrechts
et al., 2011; Ridley et al., 2005, 2010; Robinson et al., 2012;
Solgaard and Langen, 2012). If and when this temperature
threshold is passed depends critically on past and future
greenhouse gas emissions (Fettweis et al., 2013; Goelzer
et al., 2013; Gregory et al., 2004a; Rae et al., 2012). Even
if emissions were reduced to zero, temperatures would not
drop signiﬁcantly for thousands of years because of the long
lifetime of anthropogenic CO2in the atmosphere and re-
duced oceanic heat uptake if oceanic convection is extenu-
ated (Allen et al., 2009; Solomon et al., 2009; Zickfeld et al.,
2013). This implies a possible commitment of a melt down
of the Greenland Ice Sheet in the near future, which would
eventually raise global sea-level by more than 7 m (Gregory
et al., 2004a). Whether this occurs on a multi-centennial or
1800 A. Levermann and R. Winkelmann: A simple equation for the melt elevation feedback of ice sheets
rather a multi-millennial timescale is of relevance for coastal
planning.
simple representation of the melt elevation feedback towards
a governing equation for a steady-state ice sheet in 1 dimen-
sion, then we derive the critical warming threshold for the
existence of an ice sheet in this simple model (Sect. 2). In
Sect. 3 we derive a simple time evolution equation for the de-
cay of the ice sheet after surface temperatures have exceeded
the threshold. Finally we use observational estimates of the
three characteristics that enter the model to estimate the de-
cay time of the ice sheet under melting above the threshold
(Sect. 4). Here solid ice discharge is neglected as well as
any other ice sheet dynamics (Andresen et al., 2012; Howat
and Eddy, 2012; Moon et al., 2012; Nick et al., 2009; Price
et al., 2011; Straneo et al., 2011; Walsh et al., 2012). The
framework that we introduce here can be used to include new
physical processes that might be discovered in the future, e.g.
potential changes in surface albedo through melting (Box et
al., 2012) or aerosol-induced surface melt or the lack thereof
(Polashenski et al., 2015).
2 Governing equation for shallow-ice steady states
under melt elevation feedback
A non-linear threshold behaviour is generally associated with
a fundamental self-enforcing feedback and thereby an as-
sociated system memory (Levermann et al., 2012). For the
Greenland Ice Sheet, such a feedback is given by the interac-
tion between surface elevation and surface melting (Weert-
man, 1961). For illustration, we include this feedback in a
well-established highly idealized ice proﬁle of an ice sheet in
1 dimension, the so-called Vialov proﬁle (Vialov, 1958). We
introduce the melt elevation feedback in the simplest possi-
ble way by assuming that the surface melt rate depends lin-
early on the surface temperature and that the temperature de-
creases linearly with the height of the ice surface following a
constant atmospheric lapse rate.
2.1 Governing equation
We consider a highly simpliﬁed ﬂow line model for an
isothermal ice sheet grounded on a ﬂat and rigid bed. The
solution of the shallow-ice approximation in 1 dimension for
the ice sheet elevation under these simplifying assumptions
is the Vialov proﬁle:
e
h(x) =hm1(x /L)(n+1)/nn/(2n+2)
,(1)
where hmis the maximum surface elevation and nis Glen’s
ﬂow law exponent (Glen, 1955). xdenotes the horizontal po-
sition and Lthe horizontal limit of the ice sheet. The inher-
ent assumption of isothermal ice is a strong simpliﬁcation,
which needs to be kept in mind when interpreting the results.
The aim of this derivation is purposefully not a comprehen-
sive representation of the ice ﬂow but to derive a measure
of the average height of the ice sheet and its dependence on
changes in the surface mass balance. The surface mass bal-
ance is considered to be spatially and temporally constant at
a value, a, which will later be considered to be dependent on
the surface elevation and thereby temporally variable. The
overall horizontal extension of the ice sheet is set to L, and
it is thereby assumed that any ice ﬂow across this point is
calved off into icebergs. This situation represents a conﬁned
ice-bearing bedrock topography as in most of Greenland’s
interior (Howat et al., 2014). The mean surface elevation can
then be computed to be
h=L1
L
Z
0
dxh(x ) =ω·hm.(2)
It is proportional to the maximum surface elevation hmwith
a proportionality factor
ω
1
Z
0
dξ1ξ(n+1)/n n/(2n+2)
,(3)
which only depends on the ﬂow law exponent.
The maximum surface elevation is determined by the sur-
face mass balance e
aand the ice softness e
A:
hm=2(n1)/(2n+2)·L1/2·(n+2)e
a
(ρg)ne
A1/(2n+2)
,(4)
with ρbeing the ice density and gthe gravity constant.
We normalize all three quantities by deﬁning hω·hm/h0,
ae
a/a0and Ae
A/A0, where a0is the accumulation rate
on the ground, i.e. in the absence of an ice sheet, and
A0=a0/(ρg)n( ·L)(n+1)with =h0/L being the typ-
ical height-to-width ratio. h0is the equilibrium line altitude
of the considered ice sheet in the initial equilibrium situa-
tion. Values for a0,h0and Lare later chosen to resemble the
conditions of the Greenland Ice Sheet.
The non-dimensional surface elevation, h, of the ice sheet
can then be expressed as
h=a
A1/m
.(5)
For the Vialov proﬁle, m=2(n +1)where the Glen ﬂow
law exponent is commonly chosen to be around n=3, which
yields m=8.
We introduce the melt elevation feedback in its simplest
form through a dependency of the surface melt rate on the
surface elevation:
a=a0+γ 0 ·h, (6)
with the atmospheric lapse rate 0 > 0. γdenotes the melting
sensitivity of the ice surface, i.e. the increase in surface melt
The Cryosphere, 10, 1799–1807, 2016 www.the-cryosphere.net/10/1799/2016/
A. Levermann and R. Winkelmann: A simple equation for the melt elevation feedback of ice sheets 1801
0 Tc
0
hc
Ice thickness
Ground temperature
Unstable
Stable
Stable
hm - γ · Г · h + T = 0
Figure 1. Ice sheet hysteresis. If the ice sheet is in an unstable con-
ﬁguration (dashed black branch), a slight perturbation will either
cause it to converge into the stable state (upper red branch) or to
melt completely. For a given temperature, the dotted line gives the
critical surface elevation (Sect. 3). If the surface elevation is lower
than hc, a complete meltdown of the ice sheet is inevitable. Once
the temperature threshold, Tc, is crossed, the time for a collapse of
a certain fraction of the ice sheet can be estimated via Eq. (17).
rate per degree of warming, which is regularly measured and
comprises a large number of physical processes (e.g. Box,
2013). For simplicity we rescale the surface mass balance by
the constant ice softness parameter, A, to obtain h=(a0+
γ 0 ·h)1/m. The steady-state solution for the surface elevation
of the ice sheet is thus governed by the following equation:
hmγ 0 ·ha0=0,(7)
which has two positive solutions for has long as the surface
mass balance on the ground is negative, i.e. a0<0. Note that
the surface mass balance can be positive even if a0<0. If the
ice sheet is in an unstable conﬁguration, a slight perturbation
will either cause it to converge into the stable state with a
positive surface mass balance or to melt completely.
Our simple approach qualitatively captures the basic hys-
teresis behaviour of the Greenland Ice Sheet caused by the
melt elevation feedback (Fig. 1, in which we have assumed
the surface mass balance to depend linearly on temperature):
For a given surface temperature, a stable state of the ice sheet
(red line) annihilates an external perturbation in surface el-
evation by changes in surface mass balance (grey arrows).
The unstable solution branch deﬁnes the basin of attraction
for the stable state. A surface elevation that is lower than the
unstable solution branch cannot be sustained. In that case the
melting reduces the surface elevation to practically zero even
without further external perturbation (grey arrows). Beyond
a certain surface temperature threshold (vertical dotted line),
no ice sheet can be sustained.
2.2 Critical surface mass balance in steady state
As illustrated in Fig. 1, there is a critical temperature above
which the ice sheet is not sustainable. Let us denote the cor-
responding surface elevation by hc. The critical point (Tc,hc)
has to fulﬁll two conditions, i.e. being a solution of the gov-
erning Eq. (7) and minimum of the function
F (h) =hmγ 0 ·ha0,(8)
which we can determine by setting the derivative of Fto
zero. Consequently,
hc=0·γ
m1/(m1)
.(9)
Inserting this into the governing equation yields the critical
surface mass balance at the ground:
a0c= −(m 1)·0·γ
mm/(m1)
.(10)
For illustrative purposes we have assumed a0to decline
linearly with the surrounding temperature and plotted the so-
lution of Eq. (7) against that temperature with an arbitrary
offset in Fig. 1.
3 A simple temporal equation for the melt elevation
feedback
Once the critical surface mass balance and surface elevation
threshold (as derived in the previous Sect. 2) is transgressed,
a meltdown of the ice sheet is inevitable in our conceptual
model. Let us deﬁne the time ταas the time it takes to melt
a fraction αof the initial ice volume and the threshold tem-
perature Tcas the temperature above the pre-industrial level
at which the surface mass balance becomes negative. Robin-
son et al. (2012) ﬁnd a range of 0.8–3.2 C for the thresh-
old warming beyond which no ice sheet can be sustained
on Greenland. Their best estimate for the threshold is 1.6 C
above pre-industrial level. The study uses a regional climate
model of intermediate complexity (Robinson et al., 2010)
coupled to the SICOPOLIS (SImulation COde for POLyther-
mal Ice Sheets) ice sheet model (Greve, 1997). Using a dif-
ferent model combination, Ridley et al. (2010) ﬁnd that in
their model the ice sheet cannot be sustained for a warm-
ing of 2 C. They combine the HadCM3 atmosphere–ocean
general circulation model (Gordon et al., 2000) with an at-
mospheric resolution of 2.5×3.75(Pope et al., 2000) to an
ice sheet model of 20 km horizontal resolution (Huybrechts
and De Wolde, 1999).
Some studies assume that the threshold is associated with
a mean negative surface mass balance (Gregory et al., 2004b;
Ridley et al., 2005; Toniazzo et al., 2004). In Fig. 2 we use
1.6 C as a threshold value for both models because this value
is given by Robinson et al. (2012) and consistent with Ridley
et al. (2010) and is thus a simple and transparent choice. This
number can be easily adjusted if new estimates are obtained.
For the translation from percentage ice thickness change to
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1802 A. Levermann and R. Winkelmann: A simple equation for the melt elevation feedback of ice sheets
Figure 2. Decay time of the Greenland Ice Sheet. The decay time
depends critically on the level of warming above the temperature
threshold. Shown are the median (black line), likely (18–83 % quan-
tiles, dark blue shading) and very likely (5–95 % quantiles, light
blue shading) ranges for the time to melt 10 % of the present-day
ice volume, estimated via Eq. (17). The red circles and crosses in-
dicate the results from process-based model simulations by Ridley
et al. (2010) and Robinson et al. (2012) respectively.
percentage ice volume change, a constant horizontal ice sur-
face area was assumed which renders the analysis concep-
tual in nature. Thus the quantitative interpretation of the melt
times are subject to this additional simpliﬁcation.
For a ﬁxed anomalous melt rate 1a0= −γ·1T in re-
sponse to an anomalous temperature increase 1T =TTc
above this threshold temperature, Tc, the decay time without
any feedbacks would be
τ0= − h0
1a0
=h0
γ·1T .(11)
Since the surface temperature increases with decreasing ele-
vation, this zero-order estimate for the decay time is higher
than the actual value. As a ﬁrst-order correction to the situa-
tion of ﬁxed melting, let us assume that the anomalous sur-
face mass balance behaves as
1a =1a0+1
τγ
·(h h0), (12)
where τγ=1/(γ ·0). From the relation dh/dt=1a, we then
obtain
d1h
dt= −1a0+1h
τγ
,(13)
if 1h h0his deﬁned as the reduction in height. For a
time-dependent melting induced by surface warming 1a0=
γ·1T , the general solution of Eq. (13) is
1h(t) =γ·
t
Z
0
dt01T (t0)·e(tt0)/τγ.(14)
This equation corresponds to a linear response theory with
the melting γ·1T as forcing and an exponential response
function
R(t 0)=et0γ.(15)
Linear response theory states that the convolution of Eq. (14)
yields the linear response of the system (Good et al., 2011;
Winkelmann and Levermann, 2013). Note that linear re-
sponse theory is generally used as an approximation of a
non-linear system to relatively weak forcing. In these cir-
cumstances the response function has to decline with time
because it represents the history of the system’s response to
past perturbation. For example, if the response function was
a declining exponential R(t 0)=et0, this would mean that
the effect of forcing that occurred in the past, i.e. prior to the
time tthat is considered, becomes exponentially less relevant
for the current system response. Here, however, the response
function is increasing with time, which means that the past
deviation from the steady state is ampliﬁed as expected near
an unstable ﬁxed point. The exponent 1γcan be considered
the Lyaponov exponent of the system.
Given the boundary condition 1h(t =0)=0 for a con-
stant temperature increase 1T , Eq. (14) becomes
1h(t) =h0·τγ
τ0
τγ
τ0
·et/τγh0
τ0
h0
τγ
.(16)
The decay time for a relative volume reduction of αis then
given by
τα=1
γ 0 ·log1+α·0·h0
1T ,(17)
where log denotes the natural logarithm. Equation (17) is de-
noted the decay time equation hereafter.
4 Estimating the melt time of the Greenland Ice Sheet
from observables
In this simpliﬁed approach, the collapse time is thus a func-
tion of three observable quantities: the equilibrium line alti-
tude, h0, the atmospheric lapse rate, 0, and the melting sen-
sitivity to temperature, γ. The average equilibrium line alti-
tude of the Greenland Ice Sheet is at about 1150 m (Box and
Steffen, 2001). The observed range for the atmospheric lapse
rate is estimated to be between 5 ±2C km1(Fausto et al.,
2009; Gardner and Sharp, 2009) and current estimates for
the melting sensitivity scatter around 4.4±2 cm year1C1
(Box, 2013). In order to obtain an estimate of the decay time
and the uncertainty around this estimate we use Eq. (17) and
choose the lapse rate and melting sensitivity uniformly ran-
domly from these observed intervals (Table 1, Figs. 2–4).
Following the decay time equation (Eq. 17), the observa-
tional constraints for the atmospheric lapse rate, 0, and the
melting sensitivity, γ, translate into an uncertainty range for
The Cryosphere, 10, 1799–1807, 2016 www.the-cryosphere.net/10/1799/2016/
A. Levermann and R. Winkelmann: A simple equation for the melt elevation feedback of ice sheets 1803
Table 1. Decay time. Time period after which different percentages of volume loss have occurred at different warming levels. Provided are
the median values of the distributions from Figs. 2 and 3 together with the lower and upper limits that are derived respectively from the upper
and lower limits of the uncertainty range of the observed melting sensitivity and atmospheric lapse rate. The simple decay time equation
(Eq. 17) does not take any ice dynamic effects into account and its translation to ice volume assumes a constant horizontal ice sheet area.
Thus the values provided here best ﬁt the complex model simulations only when these assumptions are reasonably well justiﬁed, which is
most likely not the case for high ice loss such as 50 or 100 % of the original ice volume.
Volume loss 0.5 C 1 C 2 C 3 C 4 C 5 C
10 % Lower 2140 years 1320 years 760 years 530 years 410 years 330 years
Median 3430 years 2040 years 1140 years 790 years 610 years 500 years
Upper 7290 years 4120 years 2210 years 1520 years 1150 years 930 years
50 % Lower 4920 years 3600 years 2460 years 1900 years 1550 years 1320 years
Median 8740 years 6170 years 4040 years 3040 years 2450 years 2090 years
Upper 20 740 years 13 920 years 8640 years 6310 years 4980 years 4120 years
100 % Lower 6340 years 4920 years 3600 years 2910 years 2460 years 2140 years
Median 11 610 years 8730 years 6160 years 4840 years 4020 years 3500 years
Upper 28 710 years 20 740 years 13 920 years 10 630 years 8640 years 7290 years
the melt time of the Greenland Ice Sheet, assuming uniform
probability distributions for both 0and γwithin the above
intervals. Figure 2 shows the histograms of the time until
10 % of its present-day ice volume (corresponding to 0.7 m
global sea-level rise) is melted for different warming scenar-
ios. The melt time strongly depends on the level of warming
beyond the temperature threshold: the median estimate varies
from more than 2000 years for a warming of +1C to less
than 500 years for a warming of +5C.
Existing numerical simulations of a decay of the Green-
land Ice Sheet (Ridley et al., 2010; Robinson et al., 2012)
differ in their trajectories for the total ice volume, but exhibit
a characteristic functional form when the relative ice volume
is expressed as a function of the temperature anomaly above
the critical temperature threshold (Fig. 2). This characteristic
relation is captured by our ﬁrst-order equation for the de-
cay time, embedding the results from process-based models
into a simple analytical framework. This approach provides
a good approximation if, on the one hand, the volume loss is
large enough for the melt elevation feedback to become rel-
evant and, on the other hand, the melting dominates the ice
loss in contrast to the dynamic ice discharge.
Since the simple equation provided here does not account
for any dynamic discharge or even ice motion, the results
from Eq. (17) strongly deviate from numerical simulations
when the ice has time to adjust dynamically to the volume
loss. This can be seen for a stronger ice loss of 50 % of the
initial volume where the functional dependence between the
decay time and the temperature anomaly clearly follows a
different functional form than predicted by Eq. (17)(Fig. 3).
Since the melt time is a monotonically decreasing func-
tion of both the lapse rate and the melting sensitivity, the
upper and lower limits of the estimates can be directly com-
puted from the observed uncertainty interval of these quan-
tities. However, the functional form of Eq. (17) introduces a
Figure 3. Time until 50% of the Greenland Ice Sheet is melted.
Shown are the median (black line) and the likely (18–83 % per-
centiles, dark blue shading) and very likely (5–95 % percentiles,
light blue shading) ranges for the time to melt 50 % of the present-
day ice volume, estimated via the equation for the decay time τα.
The red crosses indicate the results from process-based model sim-
ulations by Robinson et al. (2012).
speciﬁc structure into the histogram of the melt time which
is highly skewed towards the low end (Table 1 and Fig. 4).
For increasing warming levels the histogram shifts towards
lower decay times. At the same time the histogram narrows
and higher decay times become less frequent within the cho-
sen parameter range (see description above).
5 Discussion and conclusion
Our estimate for the decay time captures the characteristic
slowdown near the critical threshold as can be seen from
the divergence of the decay time, τα, in the limit of vanish-
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1804 A. Levermann and R. Winkelmann: A simple equation for the melt elevation feedback of ice sheets
   












Probability density
#
#
#
#
(a)
(b)
(c)
(d)
Median
Likely range
Very likely range
Probability density
Probability density
Probability density
Time to lose 10 % of ice [years]
Figure 4. Likelihood for 10 % decay of Greenland Ice Sheet. Shown
are the probabilities for the ice sheet to lose 10 % of its initial ice
volume in a certain time period for surface warming of (a) +1C,
(b) +2C, (c) +3C and (d) +4C above the threshold. The me-
dian is indicated by the black line, and the likely and very likely
ranges are shaded in dark and light blue respectively.
ing warming above the threshold (Eq. 17). The simple equa-
tion of the decay time quantitatively reproduces the range
given by simulations with process-based models. The rela-
tive speed-up of ice loss due to the melt elevation feedback
(Fig. 5) is estimated, using the central values of the param-
eter ranges, i.e. equilibrium line altitude h0=1150 m, at-
Figure 5. Role of melt elevation feedback in melting of Greenland
Ice Sheet declines with increasing temperature. Shown is the ratio of
melt time with melt elevation feedback over melt time without the
feedback τα0. Each line represents the ratio for a loss of different
percent of the initial ice volume. The red line shows the ratio of the
decay time with feedback over the decay time without feedback for
a 10 % ice loss (corresponding to Figs. 2 and 4). The inﬂuence of the
feedback becomes less dominant with stronger warming above the
critical threshold (xaxis). Near the threshold the melt time without
feedback diverges stronger (1/1T ) than the melt time with feed-
back which declines logarithmically.
mospheric lapse rate 0=5C km1and melting sensitivity
γ=4.4 cm year1C1. The feedback becomes more dom-
inant near the threshold compared to larger temperature in-
creases for which the external climatic forcing is more rele-
vant.
The simple equation provided here is clearly limited in its
applicability. The role of the ice material properties is com-
prised into one parameter, the melting sensitivity of the ice
to a temperature increase at the surface. This sensitivity will
in general vary not only with time but also spatially and due
to the melting itself. Similarly, the feedback role of the sur-
rounding climate is represented by only one parameter, the
atmospheric lapse rate which will again vary spatially but
also with time as the ice surface declines.
Ice dynamics are deliberately excluded in our simple con-
ceptual approach in order to separate and characterize the
melt elevation feedback. In reality, ice dynamics of course
play an important role in the ice sheet mass balance: radar
(ERS-2) and laser (ICESat) altimetry observations show that
mass changes in Greenland were dominated by changes in
the surface mass balance (SMB) between 1995 and 2001,
and both SMB and dynamics contributed equally to mass
loss from the Greenland Ice Sheet between 2001 and 2009
(Hurkmans et al., 2014). Fürst et al. (2015) estimate that
40 % of the recent loss (2000–2010) is due to an increase
in ice dynamic discharge, 60 % due to changes in the surface
mass balance. Their results suggest that the future volume
loss from the Greenland Ice Sheet might be predominantly
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A. Levermann and R. Winkelmann: A simple equation for the melt elevation feedback of ice sheets 1805
caused by surface melting and dynamic discharge is limited
by margin thinning and retreat.
Some studies suggest (Graversen et al., 2010; Price et al.,
2011) that the dynamic discharge from Greenland is strongly
limited by the ice sheet’s bottom topography, for which esti-
mates yield an upper bound of approximately 5–13 cm dur-
ing the next century. Over a period during which the ice loss
is dominated by the feedback and the ice-dynamic effect is
limited, our approach provides a quantitative estimate of the
melt time based on observable quantities. Equation (17) can
thus be used if new observations suggest an altered melting
sensitivity or changes in the atmospheric response to Green-
land ice loss.
For a temperature increase of 5 C, which could be reached
within this century (IPCC, 2013), the median rate of sea-level
that of its current contribution of about 0.4 mm year1(Rig-
not et al., 2011; Shepherd et al., 2012). Even for extremely
high temperatures, however, the Greenland Ice Sheet cannot
melt inﬁnitely fast – our results show that a complete disinte-
gration within the next 2 millennia is highly unlikely unless
ice dynamics effects become dominant or the melting sen-
sitivity is signiﬁcantly higher than currently observed. For
a global mean temperature increase below 2 C, as agreed
upon during the 2015 Paris UNFCCC climate summit, the
threshold temperature would only be exceeded mildly and
the decay time of the Greenland Ice Sheet would be multi-
millennial.
funding from the European Union Seventh Framework Programme
FP7/2007-2013 under Grant Agreement no. 603864.
Edited by: X. Fettweis
Reviewed by: two anonymous referees
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... In this way, the GrIS has been a key component in the emergence of glacial cycles and their implications for overall Earth system stability, as can also be analyzed from a dynamical systems point of view (Crucifix, 2012). Simple models also allow to study the "deep future", i.e., the future on timescales beyond the ethical time horizon as defined by Lenton et al. (2019), for example, of the Greenland Ice Sheet and the Earth system and reveal that anthropogenic CO 2 emissions affect the climate evolution for up to 500 kyr and can postpone the next glaciation (Talento and Ganopolski, 2021). ...
... The instability of the melt-elevation feedback, as studied by Levermann and Winkelmann (2016), assumes a static bed, so that changes in ice thickness equal changes in ice surface altitude. GIA can mitigate this feedback: due to bedrock deformation, changes in ice thickness do not directly translate to changes in surface elevation. ...
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The stability of the Greenland Ice Sheet under global warming is governed by a number of dynamic processes and interacting feedback mechanisms in the ice sheet, atmosphere and solid Earth. Here we study the long-term effects due to the interplay of the competing melt–elevation and glacial isostatic adjustment (GIA) feedbacks for different temperature step forcing experiments with a coupled ice-sheet and solid-Earth model. Our model results show that for warming levels above 2 ∘C, Greenland could become essentially ice-free within several millennia, mainly as a result of surface melting and acceleration of ice flow. These ice losses are mitigated, however, in some cases with strong GIA feedback even promoting an incomplete recovery of the Greenland ice volume. We further explore the full-factorial parameter space determining the relative strengths of the two feedbacks: our findings suggest distinct dynamic regimes of the Greenland Ice Sheets on the route to destabilization under global warming – from incomplete recovery, via quasi-periodic oscillations in ice volume to ice-sheet collapse. In the incomplete recovery regime, the initial ice loss due to warming is essentially reversed within 50 000 years, and the ice volume stabilizes at 61 %–93 % of the present-day volume. For certain combinations of temperature increase, atmospheric lapse rate and mantle viscosity, the interaction of the GIA feedback and the melt–elevation feedback leads to self-sustained, long-term oscillations in ice-sheet volume with oscillation periods between 74 000 and over 300 000 years and oscillation amplitudes between 15 %–70 % of present-day ice volume. This oscillatory regime reveals a possible mode of internal climatic variability in the Earth system on timescales on the order of 100 000 years that may be excited by or synchronized with orbital forcing or interact with glacial cycles and other slow modes of variability. Our findings are not meant as scenario-based near-term projections of ice losses but rather providing insight into of the feedback loops governing the “deep future” and, thus, long-term resilience of the Greenland Ice Sheet.
... The most important positive-and thus potentially destabilizing-feedback mechanism relevant for the dynamics of ice sheets is the melt-elevation feedback (Weertman 1961, Källén et al 1979, Ghil and Childress 1987, Levermann and Winkelmann 2016: an initial melting effectively reduces the ice sheet surface, which exposes it to relatively warmer temperatures due to the atmospheric lapse rate; in turn, this leads to enhanced melting, further height reductions, and so on. This idea goes back to Weertman (1961); see also section 11.2 and figure 11.10 in Ghil and Childress (1987). ...
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Specific components of the Earth system may abruptly change their state in response to gradual changes in forcing. This possibility has attracted great scientific interest in recent years, and has been recognized as one of the greatest threats associated with anthropogenic climate change. Examples of such components, called tipping elements, include the Atlantic Meridional Overturning Circulation, the polar ice sheets, the Amazon rainforest, as well as the tropical monsoon systems. The mathematical language to describe abrupt climatic transitions is mainly based on the theory of nonlinear dynamical systems and, in particular, on their bifurcations. Applications of this theory to nonautonomous and stochastically forced systems are a very active field of climate research. The empirical evidence that abrupt transitions have indeed occurred in the past stems exclusively from paleoclimate proxy records. In this review, we explain the basic theory needed to describe critical transitions, summarize the proxy evidence for past abrupt climate transitions in different parts of the Earth system, and examine some candidates for future abrupt transitions in response to ongoing anthropogenic forcing. Predicting such transitions remains difficult and is subject to large uncertainties. Substantial improvements in our understanding of the nonlinear mechanisms underlying abrupt transitions of Earth system components are needed. We argue that such an improved understanding requires combining insights from (a) paleoclimatic records; (b) simulations using a hierarchy of models, from conceptual to comprehensive ones; and (c) time series analysis of recent observation-based data that encode the dynamics of the present-day Earth system components that are potentially prone to tipping.
... If we initialize the system with the upper equilibrium and gradually decrease u, then the system jumps to the lower equilibrium at u = u c,ℓ , implying a hysteresis. Similar models have been used for representing alternative stable states and hysteresis in ecosystems [46][47][48][49], thermohaline circulation [50], and ice sheets [51]. ...
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... While several studies 14,15,18 have considered the effects of a feedback between the surface elevation and behavior of landbased ice sheets, all existing studies of stability of marine ice sheets 5,10-12,19-21 did not take into account feedbacks between the external conditions and the ice-sheet characteristics. It is, thus, not a priori clear whether the results of these studies remain valid in the presence of feedbacks. ...
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The “marine ice-sheet instability” hypothesis continues to be used to interpret the observed mass loss from the Antarctic and Greenland ice sheets. This hypothesis has been developed for conditions that do not account for feedbacks between ice sheets and environmental conditions. However, snow accumulation and the ice-sheet surface melting depend on the surface temperature, which is a strong function of elevation. Consequently, there is a feedback between precipitation, atmospheric surface temperature and ice-sheet surface elevation. Here, we investigate stability conditions of a marine-based ice sheet in the presence of such a feedback. Our results show that no general stability condition similar to one associated with the “marine ice-sheet instability” hypothesis can be determined. Stability of individual configurations can be established only on a case-by-case basis. These results apply to a wide range of feedbacks between marine ice sheets and atmosphere, ocean and lithosphere. Using theoretical, numerical and data analyses, this study finds that there are no general stability conditions for marine ice sheets if feedbacks caused by interactions of ice sheets with atmosphere, ocean and lithosphere are taken into account.
... As such the so-called 'surface-melt-elevation feedback' (e.g. Levermann and Winkelmann, 2016) is absent from these simulations. This effect should be most significant where surface temperatures rise above freezing in confined areas around the edges of the ice sheet that progress inwards as the WAIS collapses in the high-emissions scenarios. ...
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Ice-sheet simulations of Antarctica extending to the year 3000 are analysed to investigate the long-term impacts of 21st-century warming. Climate projections are used as forcing until 2100 and afterwards no climate trend is applied. Fourteen experiments are for the ‘unabated warming’ pathway, and three are for the ‘reduced emissions’ pathway. For the unabated warming path simulations, West Antarctica suffers a much more severe ice loss than East Antarctica. In these cases, the mass loss amounts to an ensemble average of ~3.5 m sea-level equivalent (SLE) by the year 3000 and ~5.3 m for the most sensitive experiment. Four phases of mass loss occur during the collapse of the West Antarctic ice sheet. For the reduced emissions pathway, the mean mass loss is ~0.24 m SLE. By demonstrating that the consequences of the 21st century unabated warming path forcing are large and long term, the results present a different perspective to ISMIP6 (Ice Sheet Model Intercomparison Project for CMIP6). Extended ABUMIP (Antarctic BUttressing Model Intercomparison Project) simulations, assuming sudden and sustained ice-shelf collapse, with and without bedrock rebound, corroborate a negative feedback for ice loss found in previous studies, where bedrock rebound acts to slow the rate of ice loss.
... Here, approximately 35 % can be attributed to a decrease in climatic mass balance and 65 % are due to an increase in ice discharge. While it has been 5 suggested that the Greenland Ice Sheet could become unstable beyond temperature anomalies of 1.6 − 3.2 • C due to the selfamplifying melt-elevation feedback (Levermann and Winkelmann, 2016), recent studies debate whether a tipping point might have already been crossed (Robinson et al., 2012;Winkelmann et al., 2011;Boers and Rypdal, 2021). Understanding the feedback mechanisms and involved time scales at play in GrIS mass loss dynamics is necessary to understanding its stability under climatic changes. ...
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The stability of the Greenland Ice Sheet under global warming is governed by a number of dynamic processes and interacting feedback mechanisms in the ice sheet, atmosphere and solid Earth. Here we study the long-term effects due to the interplay of the competing melt-elevation and glacial isostatic adjustment (GIA) feedbacks for different temperature step forcing experiments with a coupled ice-sheet and solid-Earth model. Our model results show that for warming levels above 2 °C, Greenland could become essentially ice-free on the long-term, mainly as a result of surface melting and acceleration of ice flow. These ice losses can be mitigated, however, in some cases with strong GIA feedback even promoting the partial recovery of the Greenland ice volume. We further explore the full-factorial parameter space determining the relative strengths of the two feedbacks: Our findings suggest distinct dynamic regimes of the Greenland Ice Sheets on the route to destabilization under global warming – from recovery, via quasi-periodic oscillations in ice volume to ice-sheet collapse. In the recovery regime, the initial ice loss due to warming is essentially reversed within 50,000 years and the ice volume stabilizes at 61–93 % of the present-day volume. For certain combinations of temperature increase, atmospheric lapse rate and mantle viscosity, the interaction of the GIA feedback and the melt-elevation feedback leads to self-sustained, long-term oscillations in ice-sheet volume with oscillation periods of tens to hundreds of thousands of years and oscillation amplitudes between 15–70 % of present-day ice volume. This oscillatory regime reveals a possible mode of internal climatic variability in the Earth system on time scales on the order of 100,000 years that may be excited by or synchronized with orbital forcing or interact with glacial cycles and other slow modes of variability. Our findings are not meant as scenario-based near-term projections of ice losses but rather providing insight into of the feedback loops governing the "deep future" and, thus, long-term resilience of the Greenland Ice Sheet.
... Finally, it is assumed that the long-term behaviour of many real-world systems in terms of the system's state such as the overturning strength of the Atlantic Meridional Overturning Circulation [23,97], the ice volume of the Greenland Ice Sheet [98] and the algae density in shallow lakes [19,20] can be qualitatively captured by the studied idealized tipping elements featuring a fold bifurcation as tipping mechanism. However, biogeophysical and biogeochemical processes involved in the behaviour of these real-world systems and included in some more complex climate models may either give rise to further types of cascading tipping or may dampen the overall possibilities of tipping behavior [47,99]. ...
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Based on suggested interactions of potential tipping elements in the Earth’s climate and in ecological systems, tipping cascades as possible dynamics are increasingly discussed and studied as their activation would impose a considerable risk for human societies and biosphere integrity. However, there are ambiguities in the description of tipping cascades within the literature so far. Here we illustrate how different patterns of multiple tipping dynamics emerge from a very simple coupling of two previously studied idealized tipping elements. In particular, we distinguish between a two phase cascade, a domino cascade and a joint cascade. While a mitigation of an unfolding two phase cascade may be possible and common early warning indicators are sensitive to upcoming critical transitions to a certain degree, the domino cascade may hardly be stopped once initiated and critical slowing down–based indicators fail to indicate tipping of the following element. These different potentials for intervention and anticipation across the distinct patterns of multiple tipping dynamics should be seen as a call to be more precise in future analyses on cascading dynamics arising from tipping element interactions in the Earth system.
... Mountain glacier melt, sea ice decline, and permafrost thaw are all reversible as well, although there may be a lag of years to decades after the climate first stabilises, due to the current state of disequilibrium. Hysteresis in the system can also make it more difficult to recover from the decline or loss of some elements of the cryosphere (e.g., Ridley et al., 2010;Levermann and Winkelmann, 2016). ...
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Glaciers and ice sheets are experiencing dramatic changes in response to recent climate change. This is true in both mountain and polar regions, where the extreme sensitivity of the cryosphere to warming temperatures may be exacerbated by amplification of global climate change. For glaciers and ice sheets, this sensitivity is due to a number of non-linear and threshold processes within glacier mass balance and glacier dynamics. Some of this is simply tied to the freezing point of water; snow and ice are no longer viable above 0°C, so a gradual warming that crosses this threshold triggers the onset of melting or gives rise to an abrupt regime shift between snowfall and rainfall. Other non-linear, temperature-dependent processes are more subtle, such as the evolution from polythermal to temperate ice, which supports faster ice flow, a shift from meltwater retention to runoff in temperate or ice-rich (i.e., heavily melt-affected) firn, and transitions from sublimation to melting under warmer and more humid atmospheric conditions. As melt seasons lengthen, there is also a longer snow-free season and an expansion of glacier ablation area, with the increased exposure of low-albedo ice non-linearly increasing melt rates and meltwater runoff. This can be accentuated by increased concentration of particulate matter associated with algal activity, dust loading from adjacent deglaciated terrain, and deposition of impurities from industrial and wildfire activity. The loss of ice and darkening of glaciers represent an effective transition from white to grey in the world's mountain regions. This article discusses these transitions and regime shifts in the context of challenges to model and project glacier and ice sheet response to climate change.
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Many complex dynamical systems in the real world, including ecological, climate, financial and power-grid systems, often show critical transitions, or tipping points, in which the system’s dynamics suddenly transit into a qualitatively different state. In mathematical models, tipping points happen as a control parameter gradually changes and crosses a certain threshold. Tipping elements in such systems may interact with each other as a network, and understanding the behaviour of interacting tipping elements is a challenge because of the high dimensionality originating from the network. Here, we develop a degree-based mean-field theory for a prototypical double-well system coupled on a network with the aim of understanding coupled tipping dynamics with a low-dimensional description. The method approximates both the onset of the tipping point and the position of equilibria with a reasonable accuracy. Based on the developed theory and numerical simulations, we also provide evidence for multistage tipping point transitions in networks of double-well systems.
Thesis
With ongoing anthropogenic global warming, some of the most vulnerable components of the Earth system might become unstable and undergo a critical transition. These subsystems are the so-called tipping elements. They are believed to exhibit threshold behaviour and would, if triggered, result in severe consequences for the biosphere and human societies. Furthermore, it has been shown that climate tipping elements are not isolated entities, but interact across the entire Earth system. Therefore, this thesis aims at mapping out the potential for tipping events and feedbacks in the Earth system mainly by the use of complex dynamical systems and network science approaches, but partially also by more detailed process-based models of the Earth system. In the first part of this thesis, the theoretical foundations are laid by the investigation of networks of interacting tipping elements. For this purpose, the conditions for the emergence of global cascades are analysed against the structure of paradigmatic network types such as Erdös-Rényi, Barabási-Albert, Watts-Strogatz and explicitly spatially embedded networks. Furthermore, micro-scale structures are detected that are decisive for the transition of local to global cascades. These so-called motifs link the micro- to the macro-scale in the network of tipping elements. Alongside a model description paper, all these results are entered into the Python software package PyCascades, which is publicly available on github. In the second part of this dissertation, the tipping element framework is first applied to components of the Earth system such as the cryosphere and to parts of the biosphere. Afterwards it is applied to a set of interacting climate tipping elements on a global scale. Using the Earth system Model of Intermediate Complexity (EMIC) CLIMBER-2, the temperature feedbacks are quantified, which would arise if some of the large cryosphere elements disintegrate over a long span of time. The cryosphere components that are investigated are the Arctic summer sea ice, the mountain glaciers, the Greenland and the West Antarctic Ice Sheets. The committed temperature increase, in case the ice masses disintegrate, is on the order of an additional half a degree on a global average (0.39-0.46 °C), while local to regional additional temperature increases can exceed 5 °C. This means that, once tipping has begun, additional reinforcing feedbacks are able to increase global warming and with that the risk of further tipping events. This is also the case in the Amazon rainforest, whose parts are dependent on each other via the so-called moisture-recycling feedback. In this thesis, the importance of drought-induced tipping events in the Amazon rainforest is investigated in detail. Despite the Amazon rainforest is assumed to be adapted to past environmental conditions, it is found that tipping events sharply increase if the drought conditions become too intense in a too short amount of time, outpacing the adaptive capacity of the Amazon rainforest. In these cases, the frequency of tipping cascades also increases to 50% (or above) of all tipping events. In the model that was developed in this study, the southeastern region of the Amazon basin is hit hardest by the simulated drought patterns. This is also the region that already nowadays suffers a lot from extensive human-induced changes due to large-scale deforestation, cattle ranching or infrastructure projects. Moreover, on the larger Earth system wide scale, a network of conceptualised climate tipping elements is constructed in this dissertation making use of a large literature review, expert knowledge and topological properties of the tipping elements. In global warming scenarios, tipping cascades are detected even under modest scenarios of climate change, limiting global warming to 2 °C above pre-industrial levels. In addition, the structural roles of the climate tipping elements in the network are revealed. While the large ice sheets on Greenland and Antarctica are the initiators of tipping cascades, the Atlantic Meridional Overturning Circulation (AMOC) acts as the transmitter of cascades. Furthermore, in our conceptual climate tipping element model, it is found that the ice sheets are of particular importance for the stability of the entire system of investigated climate tipping elements. In the last part of this thesis, the results from the temperature feedback study with the EMIC CLIMBER-2 are combined with the conceptual model of climate tipping elements. There, it is observed that the likelihood of further tipping events slightly increases due to the temperature feedbacks even if no further CO$_2$ would be added to the atmosphere. Although the developed network model is of conceptual nature, it is possible with this work for the first time to quantify the risk of tipping events between interacting components of the Earth system under global warming scenarios, by allowing for dynamic temperature feedbacks at the same time.
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We discuss potential transitions of six climatic subsystems with large-scale impact on Europe, sometimes denoted as tipping elements. These are the ice sheets on Greenland and West Antarctica, the Atlantic thermohaline circulation, Arctic sea ice, Alpine glaciers and northern hemisphere stratospheric ozone. Each system is represented by co-authors actively publishing in the corresponding field. For each subsystem we summarize the mechanism of a potential transition in a warmer climate along with its impact on Europe and assess the likelihood for such a transition based on published scientific literature. As a summary, the ‘tipping’ potential for each system is provided as a function of global mean temperature increase which required some subjective interpretation of scientific facts by the authors and should be considered as a snapshot of our current understanding.
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Mass changes of the Greenland Ice Sheet may be estimated by the input–output method (IOM), satellite gravimetry, or via surface elevation change rates (dH/dt). Whereas the first two have been shown to agree well in reconstructing ice-sheet wide mass changes over the last decade, there are few decadal estimates from satellite altimetry and none that provide a time-evolving trend that can be readily compared with the other methods. Here, we interpolate radar and laser altimetry data between 1995 and 2009 in both space and time to reconstruct the evolving volume changes. A firn densification model forced by the output of a regional climate model is used to convert volume to mass. We consider and investigate the potential sources of error in our reconstruction of mass trends, including geophysical biases in the altimetry, and the resulting mass change rates are compared to other published estimates. We find that mass changes are dominated by surface mass balance (SMB) until about 2001, when mass loss rapidly accelerates. The onset of this acceleration is somewhat later, and less gradual, compared to the IOM. Our time-averaged mass changes agree well with recently published estimates based on gravimetry, IOM, laser altimetry, and with radar altimetry when merged with airborne data over outlet glaciers. We demonstrate that, with appropriate treatment, satellite radar altimetry can provide reliable estimates of mass trends for the Greenland Ice Sheet. With the inclusion of data from CryoSat-2, this provides the possibility of producing a continuous time series of regional mass trends from 1992 onward.
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Most of the policy debate surrounding the actions needed to mitigate and adapt to anthropogenic climate change has been framed by observations of the past 150 years as well as climate and sea-level projections for the twenty-first century. The focus on this 250-year window, however, obscures some of the most profound problems associated with climate change. Here, we argue that the twentieth and twenty-first centuries, a period during which the overwhelming majority of human-caused carbon emissions are likely to occur, need to be placed into a long-term context that includes the past 20 millennia, when the last Ice Age ended and human civilization developed, and the next ten millennia, over which time the projected impacts of anthropogenic climate change will grow and persist. This long-term perspective illustrates that policy decisions made in the next few years to decades will have profound impacts on global climate, ecosystems and human societies-not just for this century, but for the next ten millennia and beyond.
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Continuing global warming will have a strong impact on the Greenland ice sheet in the coming centuries. During the last decade (2000–2010), both increased melt-water runoff and enhanced ice discharge from calving glaciers have contributed 0.6 ± 0.1 mm yr−1 to global sea-level rise, with a relative contribution of 60 and 40% respectively. Here we use a higher-order ice flow model, spun up to present day, to simulate future ice volume changes driven by both atmospheric and oceanic temperature changes. For these projections, the flow model accounts for runoff-induced basal lubrication and ocean warming-induced discharge increase at the marine margins. For a suite of 10 atmosphere and ocean general circulation models and four representative concentration pathway scenarios, the projected sea-level rise between 2000 and 2100 lies in the range of +1.4 to +16.6 cm. For two low emission scenarios, the projections are conducted up to 2300. Ice loss rates are found to abate for the most favourable scenario where the warming peaks in this century, allowing the ice sheet to maintain a geometry close to the present-day state. For the other moderate scenario, loss rates remain at a constant level over 300 years. In any scenario, volume loss is predominantly caused by increased surface melting as the contribution from enhanced ice discharge decreases over time and is self-limited by thinning and retreat of the marine margin, reducing the ice–ocean contact area. As confirmed by other studies, we find that the effect of enhanced basal lubrication on the volume evolution is negligible on centennial timescales. Our projections show that the observed rates of volume change over the last decades cannot simply be extrapolated over the 21st century on account of a different balance of processes causing ice loss over time. Our results also indicate that the largest source of uncertainty arises from the surface mass balance and the underlying climate change projections, not from ice dynamics.
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Remote sensing observations suggest Greenland ice sheet (GrIS) albedo has declined since 2001, even in the dry snow zone. We seek to explain the apparent dry snow albedo decline. We analyze samples representing 2012–2014 snowfall across NW Greenland for black carbon and dust light absorbing impurities (LAI) and model their impacts on snow albedo. Albedo reductions due to LAI are small, averaging 0.003, with episodic enhancements resulting in reductions of 0.01-0.02. No significant increase in black carbon or dust concentrations relative to recent decades is found. Enhanced deposition of LAI is not, therefore, causing significant dry snow albedo reduction or driving melt events. Analysis of C5 MODIS surface reflectance data indicates that the decline and spectral shift in dry snow albedo contains important contributions from uncorrected Terra sensor degradation. Though discrepancies are mostly below the stated accuracy of MODIS products, they will require revisiting some prior conclusions with C6 data.
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Meteorological station records, ice cores, and regional climate model output are combined to develop a continuous 171-yr (1840–2010) reconstruction of Greenland ice sheet climatic surface mass balance (Bclim) and its subcomponents including near-surface air temperature (SAT) since the end of the Little Ice Age. Independent observations are used to assess and compensate errors. Melt water production is computed using separate degree-day factors for snow and bare ice surfaces. A simple meltwater retention scheme yields the time variation of internal accumulation, runoff, and bare ice area. At decadal time scales over the 1840–2010 time span, summer (June–August) SAT increased by 1.6°C, driving a 59% surface meltwater production increase. Winter warming was +2.0°C. Substantial interdecadal variability linked with episodic volcanism and atmospheric circulation anomalies is also evident. Increasing accumulation and melt rates, bare ice area, and meltwater retention are driven by increasing SAT. As a consequence of increasing accumulation and melt rates, calculated meltwater retention by firn increased 51% over the period, nearly compensating a 63% runoff increase. Calculated ice sheet end of melt season bare ice area increased more than 5%. Multiple regression of interannual SAT and precipitation anomalies suggests a dominance of melting on Bclim and a positive SAT precipitation sensitivity (+32 Gt yr−1 K−1 or 6.8% K−1). The Bclim component magnitudes from this study are compared with results from Hanna et al. Periods of shared interannual variability are evident. However, the long-term trend in accumulation differs in sign.