During 1961–1990 the Greenland ice sheet had a roughly constant mass, in which snowfall was balanced by the sum of surface ablation (meaning all processes of mass loss, predominantly liquid run-off due to melting) and solid discharge of ice into the sea (forming icebergs). Over the last 30 years both ablation and discharge have increased significantly, while snowfall has not . In recent years, the mass loss from the Greenland ice sheet of 239±20 Gt yr-1 in 2012–2017;, or about 0.7 mm yr-1 sea level equivalent (SLE), accounts for about 20 % of global-mean sea level rise (GMSLR), most of which is due to thermal expansion of seawater (i.e. thermosteric) or mass loss from glaciers.

The increase in discharge is probably the ice-dynamical response of outlet glaciers to reduced buttressing by their ice tongues, which have thinned due to basal melting by warmer seawater . The increase in ablation causes 60 % of the mass loss . It has been partly due to anthropogenic climatic warming, which is amplified at high northern latitudes, and partly to recent unusual atmospheric circulation . In recent years, the surface mass balance S=P-R, where P is snowfall and R ablation, has fallen lower than during the warm period in Greenland in the early 20th century , and summer temperatures have risen higher . Some recent summers have seen surface melting over practically the entire ice sheet because of high air temperature, decreased cloudiness and reduction in albedo, the latter due to the increase in snow grain size and the exposure of bare ice, both caused by surface snow melting .

The future of the Greenland ice sheet is one of the large uncertainties in projections of GMSLR . Ice discharge is projected to increase in coming decades with rising water temperature, but it will decline on longer timescales as the ice sheet thins at the coast, and its outlet glacier termini retreat inland . On multicentennial timescales, surface mass balance (SMB) is dominant and the source of greater uncertainty .

Projections indicate that ablation will increase non-linearly with temperature and more rapidly than snowfall, meaning that SMB will continue to decline, and the rate of mass loss will grow, especially under scenarios of high CO2 emissions . Recent projections of the contribution of the Greenland ice sheet to GMSLR mostly lie within the likely ranges of the Fifth Assessment Report (AR5) of the Intergovernmental Panel on Climate Change , viz. 0.04–0.12 and 0.09–0.28 m by 2100 relative to 1986–2005 under scenarios RCP2.6 and RCP8.5, respectively. The range of uncertainty arises from the spread in global warming simulated by atmosphere–ocean general circulation models (AOGCMs) and in their amplification of warming in Greenland relative to global warming as well as the sensitivity of Greenland SMB to regional climate change .

Although substantial, the contribution from the Greenland ice sheet is only 10 %–30 % of projected GMSLR by 2100. Its importance is greater on multicentury timescales because its size (mass M=7.4 m SLE) implies a large commitment to GMSLR. Thinning of the ice sheet due to increasing ablation is affected by a positive feedback loop between SMB and elevation: as the surface elevation falls, the surface air temperature rises, and surface melting increases, magnifying the ablation increase. We refer to this as the local lapse-rate feedback. Another positive feedback on ablation is caused by the decrease in surface albedo due to melting, as in recent years . Despite these feedbacks, a steady state could be regained with an ice sheet of smaller mass but little loss of area if the reduction in SMB were compensated by the reduction in discharge resulting from thinning of outlet glaciers (see Sect. ). The reduction in mass would be mitigated if snowfall increases, which is projected by AOGCMs.

On the other hand, previous work indicates there may be a threshold Tc of global-mean surface air temperature change ΔSAT (relative to pre-industrial) beyond which the ice sheet will be greatly reduced or vanish entirely (see Sect. ). estimated Tc=0.8–2.2 K using the model of , constrained by information from the last interglacial . If this range is correct, limiting ΔSAT to 2.0 K in accordance with the Paris Agreement or to its aspiration of 1.5 K could make a critical difference to whether Tc is exceeded . Loss of the Greenland ice sheet would cause much greater GMSLR than from glacier mass loss or thermosteric sea level rise for similar degrees of warming , although, even in the most extreme scenarios, the complete removal of the ice sheet would take a least 1000 years e.g..

The rate of change of the mass of the ice sheet is dM/dt=S-D, where D is discharge. In the unperturbed steady state dM/dt=0⇒S=D; i.e. SMB is balanced by discharge. In a warmer climate, ablation R and snowfall P both increase, but ΔR>ΔP⇒ΔS=ΔP-ΔR<0 (see references in Sect. ), where Δ denotes the difference from the initial state. A new steady state can be achieved if ΔD=ΔS, i.e. if discharge reduces by as much as SMB so that D+ΔD=S+ΔS.

Let us suppose that raising the global-mean SAT by T initially perturbs the SMB by an amount ΔST(T)<0. Further suppose that a new steady state can be achieved with little change in ice sheet area, in which discharge is reduced by marginal thinning such that ΔD=ΔS=ΔST(T). The larger T is, the greater the reduction in discharge needs to be to balance ΔST(T). The threshold T=Tc is reached when there is just sufficient marginal thinning to cause all outlet glaciers to retreat from the coast, reducing discharge D+ΔD to zero. To attain a balance, SMB must also fall to zero, with ΔS=-S. Hence ΔST(Tc)=-S defines Tc.

Any T exceeding Tc will give negative SMB, but discharge cannot be further reduced (i.e. below zero) to compensate. The unbalanced negative SMB will reduce the thickness of the ice sheet and the altitude of the surface, making the SMB even more negative by the local lapse-rate feedback. If no other process is involved, the ice sheet will be completely eliminated for any T>Tc by this feedback loop, which is called the “small ice cap instability”. The threshold has been estimated as Tc=1.9–4.5 K relative to pre-industrial by evaluating the warming required to reduce SMB to zero with the present-day surface topography. The same method gave Tc=2.1–4.1 K in the AR5 Sect. 13.4.3.3.

showed that this method of calculation overestimates the actual Tc for onset of the small ice cap instability. One possible contribution to the difference is the reduction in SMB, neglected above, due to the local lapse-rate effect of the marginal thinning before the threshold is reached. Let us write this contribution to ΔS as ΔSL. For steady state at the threshold, we now require ΔST(Tc)+ΔSL=-S⇒ΔST(Tc)=-(S+ΔSL). Since ΔSL<0, the right-hand side is less negative than before, so Tc is smaller.

An ice sheet model is required to allow for ΔSL in quantifying Tc because the change in ice sheet topography is determined simultaneously by SMB change and ice-dynamical change. With their model, demonstrated that SMB may initially be positive but decline to zero as the topography changes, whereupon the instability is triggered, leading to the eventual loss of the ice sheet. They determined their lower Tc by finding the final steady state for various T and versions of their coupled climate–ice sheet model, and our approach is similar. The coupled system simulated by their and our models is considerably more complicated than this simplified conceptual treatment, which is intended to illustrate the idea. The actual outcome is affected by further feedbacks, both positive and negative, which we discuss later.

If the ice sheet were removed then, even after CO2 fell, and global climate returned to pre-industrial, because of greater ablation or reduced snowfall due to lower elevation and albedo in deglaciated regions it might not be possible to regenerate it . If the ice sheet did not regrow, it would imply that its pre-industrial steady state is a relict of a colder climate . Previous work shows there may be more than two steady states for pre-industrial climate . Stable states of intermediate size (between zero and present-day) are possible because of the interaction of the ice sheet with its own climate through atmospheric dynamics, whereby its surface topography affects regional precipitation and temperature, like mountains do. The existence of intermediate states means that partial loss of the ice sheet could be irreversible.

It is the possibility of threshold behaviour (i.e. “tipping points”) and irreversibility which makes the future of the Greenland ice sheet of particular concern . Precautionary action to mitigate the threat of irreversible damage is a principle of the Framework Convention of Climate Change (Article 3.3), even when there is not full scientific certainty. The serious implications of the uncertainty are the motivation for the work presented in this paper, in which we re-examine the future decline and possible recovery of the ice sheet. Our conclusions differ in some critical ways from those of previous work because of the greater complexity of the model, which we describe next.

For sufficient speed, we use the FAMOUS AGCM, which is the atmosphere component of the FAMOUS AOGCM , itself a low-resolution version, at 7.5∘ longitude by 5∘ latitude, of the HadCM3 AOGCM . For physical consistency, we calculate the SMB in the AGCM, but Greenland spans only seven grid boxes in longitude and five in latitude in the free atmosphere of the model, which is far from adequate for simulating the important effects of topographic gradients and snow hydrology for ice sheets. Therefore in this work we use “FAMOUS–ice”, a new version of FAMOUS version xotzb; incorporating a multilayer surface snow scheme, which calculates melting, refreezing of meltwater, run-off and SMB on “tiles” at a set of elevations within each AGCM grid box (each tile covering a fraction of the grid box area). This is similar to the method implemented for the Greenland ice sheet in the Community Earth System Model (CESM) , and we use the same 10 elevations. show the improvement in the cumulative distribution of area as a function of altitude (the hypsometry) that results from the sub-grid-scale treatment. Below we summarize the FAMOUS–ice SMB and coupling schemes, of which further details are given by .

For vertical interpolation of atmospheric variables from the AGCM grid box elevation to the tile elevations, we prescribe a lapse rate of 6 K km-1 for air temperature. This we obtained from the climate of 1980–1999 simulated by with the MAR RCM using sea surface BCs (sea surface temperature and sea ice) from MIROC5, the AOGCM which  found to give the most satisfactory SMB simulation. Downwelling longwave radiation and specific humidity are vertically interpolated in FAMOUS–ice using gradients consistent with the prescribed lapse rate, but precipitation is neither redistributed vertically nor modified in phase. The same uniform air temperature lapse rate for Greenland is used, e.g. by , and found by to give the most similar SMB gradient to RACMO in their CESM ice sheet coupling, which, like our scheme, does not downscale precipitation.

We have paid particular attention to the treatment of the surface albedo of the Greenland ice sheet, to which SMB is very sensitive. Bare ice has lower albedo than snow in FAMOUS–ice, and snow albedo has different values for visible and near-infrared, both dependent on the snow grain size, which is a prognostic that depends on the “ageing” of the surface snow by melting and refreezing following new snowfall. There is an uncertain parameter in the relationship between snow grain size and albedo. In our experiments, we use three alternative parameter values that are consistent with observations of albedo. For convenience we refer to these as low, medium and high albedo, but the reader should keep in mind that the albedo is variable in each case. More details are given by .

Instead of simulating sea surface conditions by using the FAMOUS AOGCM, we use the AGCM alone for both recent and future climate, with sea surface BCs derived from AOGCM experiments of the Coupled Model Intercomparison Project Phase 5 (CMIP5) and atmospheric CO2 concentration to give the corresponding radiative forcing (see Table  and the start of Sect. ). We use the AGCM for two reasons. First, the FAMOUS AOGCM has larger biases in its simulation of recent climate than MIROC5 and the three other AOGCMs we use (CanESM2, HadGEM2-ES and NorESM1-M), which have all previously been selected as satisfactory for Greenland regional climate simulation (see , and , for evaluation of their regional climate simulations). Second, this method allows us to investigate the uncertainty in Greenland ice sheet projections that arises from the spread of climate projections given by AOGCMs for any given scenario. The AGCM sea surface BCs are 20-year climatological monthly means, which lack interannual variability; we have checked that statistically indistinguishable results for the ice sheet are obtained with the AGCM, cycling through a 20-year series of monthly mean BCs for the same climate (Fig. a).

By prescribing sea surface conditions, we exclude any climate interaction between the ice sheet and the ocean, in particular possible cooling of regional climate due to weakening of the Atlantic meridional overturning circulation (AMOC) caused by meltwater from the ice sheet e.g.. There is wide uncertainty in this aspect of ocean climate change, whose implications for the ice sheet could possibly be explored in further work by modifying the sea surface temperatures in a range of ways to represent the effects of AMOC changes projected by AOGCMs .

We use the Glimmer community ice sheet model (CISM) with the shallow-ice approximation at 20 km grid spacing and no basal sliding. Consequently the model does not simulate ice streams or rapid ice sheet dynamics, and it will inevitably underestimate the rate of ice sheet mass loss, especially in coming decades. This is acceptable because our aim is not to make realistic time-dependent projections but to study the steady state obtained under constant climates.

Because the ice sheet model lacks sufficient resolution and physical processes to simulate calving into fjords, we instantly remove ice which flows beyond the present margin of the ice sheet. This BC prevents a tendency for the ice sheet to expand slightly, and thus it makes the modelled ice edge coincide with the observed one. It becomes irrelevant in most of our experiments, when the ice sheet contracts. For simplicity in the model we omit isostatic uplift, which in reality gives a negative feedback on ice sheet mass loss through the local lapse-rate feedback because it is not a large effect (e.g. 2 % over 1000 years; ) and does not seem necessary given that our scenarios are idealized in other ways as well.

The AGCM and the ice sheet model are coupled to make FAMOUS–ice–Glimmer FiG;. After each AGCM year, the SMB simulated by the AGCM is interpolated horizontally and vertically (with the AGCM tile elevation as the vertical coordinate) to the ice sheet surface topography, and the AGCM topography and land-surface properties are updated according to the ice sheet model. When the ice sheet retreats, the newly exposed land is assigned the properties of bare soil, including a low snow-free albedo; its properties do not subsequently change because vegetation dynamics are not included in the model.

FiG runs at about 220 simulated AGCM years per wall clock day on six cores, with the AGCM consuming the great majority of the CPU time. Although this is fast for an AGCM, it is not fast enough for multimillennial experiments. Therefore, after each AGCM year, the ice sheet model runs for 10 years with the resulting SMB field, depending on the assumption that the local lapse-rate feedback will be negligible for changes in topography that occur within that decade, before the AGCM runs again. We have verified that this 10:1 acceleration makes no significant difference to our results (Fig. a). Hereafter by “year” in FiG experiments we mean an ice sheet year except where otherwise stated.

Because our aim is to simulate ice sheet response to climate change over millennia, we have to start from a coupled steady state, with little long-term tendency in the ice sheet topography. We initiate the ice sheet model with observed topography and run FiG under the MIROC5 AOGCM climate of 1980–1999, during which period the ice sheet was near a steady state in reality . In the first millennium the ice sheet mass M increases by 0.1–0.2 m SLE. With medium and low albedo it subsequently decreases again more slowly, while with high albedo it continues to grow slowly and stabilizes after 4 kyr at 0.3 m SLE above present day (Fig. b). The states obtained after about 4 kyr of spin-up are used to initiate the experiments described in Sect. . In these states M is close to reality, and the topography similar to observed (Fig. a, b, c), with the summit and southern dome altitudes being a few hundred metres too low (see also Appendix  concerning the constraint implied on albedo by the requirement of realistic M).

Comparing the three choices of albedo in FAMOUS–ice with BCs for the MIROC5 1980–1999 climate, we find that lower albedo produces lower SMB (the first group of cases in Table  differ significantly at the 10 % level) because ablation is greater due to greater snowmelt, but snowfall is about the same (slightly larger with higher albedo because of greater ice sheet area). For the same albedo (medium), the SMB is significantly lower with the CanESM2 and NorESM1-M historical climates than with MIROC5 because the ablation is larger, whereas the SMB is about the same with HadGEM2-ES as with MIROC5 (the second group in the table). This shows the influence of the different climate simulations of the AOGCMs.

A similarly large spread in SMB arises from the choice of Greenland model (FAMOUS–ice, MAR or RACMO), both because they simulate somewhat different regional climate in the free atmosphere and over land when given climate BCs from the same AOGCM and because they have different SMB schemes. MAR has much larger SMB than FAMOUS–ice with the MIROC5 climate because of smaller ablation, while RACMO has larger ablation than FAMOUS–ice with the HadGEM2-ES climate (the third group in the table). Comparison with MAR and RACMO for ERA-Interim BCs (i.e. observationally derived, the fourth group in the table) suggests that FAMOUS–ice with high albedo is similar to both of them.

Regarding its geographical distribution, FAMOUS–ice SMB interpolated to the Glimmer grid compares favourably with the MAR simulation for the MIROC5 climate (Fig. d, e, f). It shows positive and negative values of realistic magnitude and reproduces the important geographical features, including the confinement of negative SMB to the margins, especially on the west coast; the decrease in positive SMB towards the north-east; and the occurrence of greatest positive SMB in the strip of maximum snowfall along the south-east coast. We presume that the latter is not sufficiently intense in FAMOUS–ice because of the low resolution of the AGCM. The equilibrium line (black contour) is generally a little higher and further inland in FAMOUS–ice (see , for details).

Our set of BCs produces a wide range of global mean surface air temperature change ΔSAT  of -1 to +5 K, relative to the MIROC5 historical climate. Some are negative because the historical climate is warmer in MIROC5 than in the other three AOGCMs. In warmer climates, snowfall and ablation are both increased (Table ; fifth group shows results with MIROC5 RCP climates, and the last group shows the mean over results for each of the available AOGCMs for each climate). In general, the greater the global warming is, the more negative the ΔSMB initially produced will be relative to the time mean MIROC5 historical state with the same albedo (Fig. a). For a given scenario, the AOGCMs give a range of ΔSAT, as is very well known e.g.. In our set of AOGCMs, NorESM1-M warms the least, and HadGEM2-ES warms the most (Fig. a). ΔSAT in FAMOUS–ice and that in the BCs are very highly correlated (Fig. b). The spread of FAMOUS–ice results with BCs from different AOGCMs for a given warming is due to their different relationships between global ΔSAT and Greenland regional climate change (shown by the grey lines in Fig. c).

Global warming under RCP2.6 is fairly small, leading to small ΔSMB, especially for MIROC5 (squares near 1.0 K in Fig. a), although MIROC5 is in the middle of the range for RCP8.5 (squares near 3.5 K). For mean over AOGCM climates under RCP2.6, RCP4.5 and RCP8.5, FAMOUS–ice with medium albedo gives SMB change of -95, -247 and -580 Gt yr-1, respectively, with respect to the mean over AOGCM historical climates.

The greatest global warming is given by HadGEM2-ES abrupt4xCO2. With low albedo, this climate produces the most negative SMB, of -756 Gt yr-1, in the time mean of the first 300 years, during which the topography change from the initial state is still quite small (Fig. a1, b1). It is also the most negative ΔSMB, of -1066 Gt yr-1, relative to the MIROC5 historical climate with low albedo. Although this is a large ΔSMB, that of for RCP8.5 is larger still, perhaps because they use a degree-day scheme and assume geographically uniform warming.

In our experiment, the specific SMB is strongly negative all around the margin and especially in the southern dome, where it has a local maximum in the historical climate (Fig. a2, b2). The snowfall on the ice sheet is ∼10 % larger in the abrupt4xCO2 climate (Fig. a4, b4). We note that the precipitation is ∼50 % larger, consistent with the warming in Greenland of 11 K and the increase of ∼5 % K-1 found by previous studies, e.g. , but the snow fraction declines from ∼90 % to ∼70 %. The downwelling surface shortwave radiation in summer (June–August) is smaller because cloudiness is greater (Fig. a3, b3). Both the increased snowfall and the reduced insolation tend to make ΔSMB positive, but ΔSMB is actually large and negative because of the overwhelming effect of increased downwelling surface longwave radiation, which is mainly due to the air above the ice sheet being warmer and partly to the increase in cloud cover.

We find that the relationship between ΔSAT and ΔSMB in the set of FiG experiments roughly follows the cubic formula (shown as the solid curve in Fig. a) derived by for MAR projections and used in the AR5 for the Greenland contribution to GMSLR. There is a small spread due to the choice of albedo parameter and a larger spread due to choice of AOGCM. The FiG ΔSMB mostly lies within the AR5 likely range (dashed curves). In the majority of cases SMB remains positive (Fig. b), but because the ice sheet was initially in balance, negative ΔSMB leads to loss of mass (Fig. a). In the most extreme case, rapid retreat of the ice sheet margin reduces the discharge by a third in the first century alone; this slightly offsets ΔSMB, giving ice sheet mass loss of 2.5 mm yr-1 SLE.

Because of the effect of lowering topography, the SMB becomes more negative in most cases during the early centuries (Fig. b). For the 21st century, this effect is omitted in our experiments since we instantaneously impose the climates from the end of the century on the initial state. This is an acceptable approximation because the effect is small on that timescale; e.g.  give a best estimate of 4.3 % for the consequent increment in the GMSLR contribution by 2100, but this increases with time, e.g. to 9.3 % by 2150 and 9.6 % by 2200 . In the 28 cases with ΔSMB<-100 Gt yr-1 in the time mean of the first century of our experiments, ΔSMB becomes about 20 % more negative on average during the second and third centuries due to the local lapse-rate feedback, about twice the size of the effect estimated by . Thereafter the SMB becomes gradually more positive again (Fig. b) because the area contracts, with the areas most prone to ablation being removed most quickly, as happens with a retreating mountain glacier.

The experiments continue until the ice sheet reaches a steady state (defined as |dM/dt|<0.02 mm yr-1 SLE over 2000 years). The longest experiments, which take 40 kyr (Fig. b), are for large climate change (RCP8.5), which entails a large loss of mass, with high albedo, which causes a relatively slow rate of mass loss. The shortest are the experiments in which a different historical climate from MIROC5 is applied because the effect on SMB of differences among AOGCMs in their simulations of late 20th-century climate is relatively minor.

There is a wide range of M in the final steady state (Figs. b and b), between slightly greater than present day (in some historical experiments) and almost zero (in abrupt4xCO2). With one exception, historical and RCP2.6 climates produce final M of 6 m SLE or more (implying GMSLR not exceeding 1.5 m), while RCP8.5 climates all produce final M of 3 m SLE or less (GMSLR exceeding 4 m). In all cases the SMB is finally positive (Fig. b) and must be balanced by ice discharge, meaning that the ice sheet does not retreat entirely inland.

There is a clear tendency for climates of greater ΔSAT to produce smaller ice sheets, but the final M has quite a wide range for any given initial global-mean annual-mean ΔSAT within 1–4 K (Fig. c). For given BCs, we have found in test experiments that the ice sheet evolution follows somewhat different trajectories from slightly different initial states but that they converge on very similar final states (Fig. a). Thus, the scatter in Fig. c is not random noise but arises from the detailed interaction of the evolving ice sheet topography with its regional climate, which depends on the choice of BCs. The final M depends on which AOGCM is used because of their different patterns of SST and sea ice change; this dependence is omitted if the warming is assumed to be uniform e.g.

The Pearson product-moment correlation coefficient between final M and initial ΔSAT is -0.89, and the Spearman rank correlation coefficient is -0.83. The correlation is similar if for ΔSAT we use Greenland area-mean summer-mean air temperature change, either at the surface or at 600 hPa the latter as (Fig. d, e, f). However, the relationship is better-defined using ΔSMB instead of ΔSAT (Fig. d), with both product-moment and rank correlation coefficients of 0.92. If the initial ΔSMB is near zero, the ice sheet changes little; the more negative the initial ΔSMB is, the smaller the final M will be. Excluding the case with the most negative ΔSMB, a linear relationship is a fairly good fit.

With any choice of albedo, for any T>3 K, the final steady-state M≲2 m SLE, meaning GMSLR exceeds 5 m. It is important to note, however, that the spread of final M does not suggest a sharply defined threshold in T beyond which a complete or nearly complete loss of the ice sheet ensues (Fig. c). For low and medium albedo, there is a fairly monotonic decline in size of the steady state from near present-day M≃7 m at T=0 K to M<1 m for T>3 K. For high albedo, there might be a transition from M≃6 m at T=2.0 K to M≃3 m at T=2.5 K; to obtain a clearer description of the behaviour, more experiments are needed in this part of the diagram. In any case, the interval between temperatures giving a “large” and a “small” final ice sheet is wider in our results, or alternatively the mass interval between “large” and “small” is narrower than in the results of (their Fig. 1c). All of the versions of their model have a sharp transition between M>6 m and M<1 m over a temperature interval which appears to be less than 0.1 K. Our model gives a qualitatively different impression of the transition.

If negative feedbacks were neglected, there would be no final ice sheet for negative initial SMB, as described in Sect. . Actually all final states have positive SMB and non-zero M, although some have initially negative SMB (Fig. b). In our model, if any climate warmer than historical is maintained indefinitely the ice sheet will contract to a new non-zero steady state, whose size depends on the magnitude of the warming and the consequent SMB perturbation.

Observational analyses indicate that recent ΔSMB (with respect to a steady state before the 1990s) is between -200 and -150 Gt yr-1, with substantial interannual variation (e.g. ; dotted lines in Fig. d). If a climate giving such a ΔSMB were maintained, it would eventually lead to GMSLR of 0.5–2.5 m according to the linear fit (Fig. d; allowing for the range of FiG initial M). On the basis of the MAR simulations of , the CMIP5-mean projection of ΔSMB for 2080–2099 climate is -242 Gt yr-1 under RCP4.5 and -710 Gt yr-1 under RCP8.5. According to the fit, the former implies eventual GMSLR of about 3 m, the latter about 7 m.

To demonstrate the important influence of the climate–ice sheet interaction, we repeat the HadGEM2-ES abrupt4xCO2 low-albedo experiment (the case of most negative ΔSMB) using Glimmer alone, uncoupled from the AGCM, forced by the AGCM SMB field (a function of geographical location and tile elevation) from the start of the FiG experiment. As the uncoupled experiment runs, the time-independent three-dimensional AGCM SMB field is continually interpolated onto the time-dependent ice topography using the same methods as in the FiG coupling. Thus the local lapse-rate feedback on SMB is included in the uncoupled experiment, but the regional climate feedbacks of topography and albedo change on the atmospheric state and circulation are excluded.

The uncoupled Glimmer and FiG experiments begin from the same initial state and have the same initial rate of mass loss but soon diverge (the dotted red line and the lowest solid red line in Fig. a). While the rate of mass loss continuously decreases in the FiG experiment, it remains almost constant (2.1–2.6 mm yr-1 SLE) in the uncoupled experiment for about 2.5 kyr, and the ice sheet is completely eliminated in 3.4 kyr (Fig. b).

To understand the different behaviour, as an example we compare the state when M=2.38 m SLE, which is reached after 3600 years in the coupled experiment and 2020 years in the uncoupled experiment. The coupled ice sheet has a high central region (Fig. c1), where specific SMB exceeds 0.25 m yr-1 liquid water equivalent (LWE) over about the same area as in the initial state (Fig. b2, c2), surrounded closely by steep narrow margins with large negative specific SMB, giving negative area-integral SMB which is ∼3 times smaller in magnitude than in the initial state. The regions where negative specific SMB appears were near equilibrium in the initial state, and the change is consistent with the local lapse-rate feedback due to the lowered surface in the contracted margins. The area-mean ratio of changes in surface air temperature and surface elevation is 7.1 and 6.6 K km-1 within the initial and contracted ice sheet extent, respectively, close to the value of 6 K km-1 assumed in the downscaling scheme. It is not uniform over the ice sheet (Fig. ), but it is within the range of 4–8 K km-1 over more than half of the ice sheet (considering either extent).

The uncoupled ice sheet is similarly located in the north of Greenland but has a larger area and lower altitude (Fig. d1). Its specific SMB is negative everywhere. Its area-integral SMB (-991 Gt yr-1; Fig. d2) is more negative than in the initial state (-756 Gt yr-1; Fig. b2) and ∼4 times more than for the coupled ice sheet of the same M (-262 Gt yr-1; Fig. c2). The much larger change exceeds the lapse-rate effect, and the area-mean specific SMB for any surface altitude above 1000 m is more negative in the uncoupled case than the coupled case. The main cause is greater downwelling shortwave radiation at the surface in the uncoupled case (Fig. c3, d3) due to lower cloud fraction. The region occupied by the contracted ice sheet coincides geographically with the high cold interior of the initial ice sheet, where cloudiness is comparatively low, but in the coupled case the cloudiness increases there as the ice sheet becomes smaller and lower, giving a powerful negative feedback on the mass loss.

In the coupled experiment, the precipitation from the south-west advances inland, following the margin of the contracting ice sheet (compare the grey contour line for 1 m yr-1 in Fig. b4, c4). Consequently the precipitation on the ice sheet is about 15 % greater in the coupled case. However, the snowfall is about 15 % less in the coupled case (colours and numbers in Fig. c4, d4) because its surface is lower than in the initial climate, making the surface climate warmer and reducing the snowfall fraction (to 64 %). The uncoupled SMB has a larger snowfall fraction (84 %) because the surface in the region it occupies was initially much higher. The phase change in precipitation with elevation is omitted from the downscaling in the coupling scheme (as mentioned earlier); including it in the uncoupled model would reduce the snowfall and make its SMB even more negative.

In summary, the uncoupled ice sheet is eliminated rapidly through the small ice cap instability (local lapse-rate feedbacks from surface energy fluxes and temperature), whereas in the coupled case the decline is decelerated, and the ice sheet is not completely eliminated, owing to negative feedbacks of topographic change on regional climate (changes in cloudiness and precipitation). The comparison demonstrates the critical role of ice sheet–climate interaction.

According to the topographic features present, the final states can be put in five categories (indicated by symbols at the ends of the trajectories in Fig. b). In cases with small change in M, the final state is similar to the present day (configuration labelled “EWNS”; e.g. Fig. a1). The northern portion (denoted “N”) is absent in some final states and the summit further south than in the present day, e.g. Fig. f1 (EWS). Ice in the south (“S”) may become a separate ice cap (as in Fig. f1), or it may be absent, resembling Fig. h1 (EWN) and g1 (EW). In cases with the smallest final M, the north-western lobe (“W”) vanishes, and ice remains only on the eastern mountains (“E”). For example, in the experiment ending in Fig. e1 (marked with “e” in Fig. b), the southern and north-western domes detach and vanish within 3 kyr. Subsequently contraction continues on all sides, but there is a slow and small regrowth after the minimum mass is reached.

The transient and final states of all experiments lie close to a common power-law relationship between ice sheet mass M and area A with M∝A1.31 (Fig. f), similar to the exponent of 1.36–1.38 derived for glaciers from observations and theory and references therein. Final states with the same configuration have a characteristic deviation from the common relationship; e.g. EWN states have greater M.

Because of the local lapse-rate feedback, the mass loss sometimes accelerates by a few tenths of a millimetre per year SLE, while one of the outlying portions becomes separate or is eliminated, in a few cases after some millennia of relatively slow change. This is a similar phenomenon to the saddle collapse during the separation of the Laurentide and Cordilleran ice sheets during the last deglaciation but an order of magnitude smaller. For example, in the experiment ending in Fig. f1 (the dotted green line, marked “f”, in Fig. b), the rate of ice loss accelerates after 10 kyr, at the start of the retreat of the northern margin, which is completed by 15 kyr.

In Sect.  we described why there might be a threshold ΔSAT beyond which the ice sheet would be eliminated by the small ice cap instability, whereas with smaller ΔSAT it would have mass and area little reduced from its present-day state. In Sect. , instead of such a well-defined threshold, we found a range of steady-state ice sheet mass and area, generally smaller for larger ΔSAT. The ice sheet endures, albeit in a much reduced state, even for ΔSAT giving large negative initial SMB.

For studying the evolution of the ice sheet as its area A contracts, it is helpful to consider the specific SMB s=S/A, where obviously S and s have the same sign. We can write Δs=ΔsT+ΔsL+ΔsC, where ΔsT and ΔsL are the changes in specific SMB due to climate change and the local lapse-rate feedback, as in Sect. . When the warmer climate is initially imposed, ΔsT<0, and the perturbation is amplified by ΔsL<0 due to thinning of the ice sheet.

The term ΔsC represents the effects of change in the climate experienced by the ice sheet, arising both because the climate changes in all areas and because the ice sheet changes the areas it occupies. An important example of the latter is the retreat of the ice sheet margin (or a glacier tongue, in general) to higher altitude in a warmer climate because this reduces the ablation while preserving the accumulation. In this and other cases, the climate effects can give ΔsC>0. Thus they can counteract the local lapse-rate feedback ΔsL<0, prevent a runaway feedback loop and eventually reverse the sign of Δs so that a steady state is reached with SMB and discharge in balance again, even if with greatly reduced area.

In cases where specific SMB is initially positive, it becomes more positive (Fig. e) because the areas from which the ice sheet retreats are predominantly those of relatively larger ablation or smaller snowfall. Consequently the area-integral SMB (the product of increasing specific SMB and decreasing area) changes relatively little (fairly vertical trajectories in Fig. b). For instance, under MIROC5 RCP4.5 climate with medium albedo, the initial SMB, snowfall and ablation are 150, 735 and 585 Gt yr-1 (Table ). The final SMB is the same as the initial because ablation and snowfall both decrease by 115 Gt yr-1 (Fig. f2, f4), a larger fractional decline in ablation (20 %) than in snowfall (15 %). The steady state is achieved by the withdrawal of the margin from the coast in some sectors, reducing discharge sufficiently (by 209 Gt yr-1 or 60 %) to balance the smaller SMB.

In cases where specific and area-integral SMB are initially negative, they become positive (Fig. b, e). This happens because snowfall decreases less than ablation. For instance, under HadGEM2 abrupt4xCO2 climate with low albedo, the initial SMB, snowfall and ablation are -756, 797 and 1554 Gt yr-1 (Fig. b2, b4). The final state is a small eastern ice cap (like Fig. e1 but smaller) with SMB, snowfall and ablation of 9, 31 and 21 Gt yr-1; snowfall is 3.9 % and ablation 1.4 % of the initial value. The ice cap receives greater precipitation and snowfall than the same region did initially (compare Fig. b4, e4) and has more cloud and less surface downwelling shortwave radiation (Fig. b3, e3) because of the effect of topography on atmospheric circulation and climate.

In the initial state of all the experiments of the second set, the ice sheet has a smaller mass than present, and it grows to reach a new steady state; there are none in which it continues to lose mass (Fig. c). However, the mass of the regrown steady state depends on the initial state and the albedo.

With high albedo, the ice sheet regrows, in about 50 kyr, from the minimal state to a steady state with the extent of the present day's (EWNS configuration; Fig. c). Since this starting state is a limiting case, we assume that the ice sheet would reach the same final state from any initial state, implying that this is only a steady state for historical climate with high albedo. Therefore the loss of the ice sheet would be reversible, albeit on a long timescale, if the high albedo is realistic.

On the other hand, with the medium and low albedo, two distinct sets of steady states can be reached in the second set of experiments: one set with final mass of 7 m SLE or more, the other with final mass of 5–6 m SLE. Initial states are divided between these two sets of final states by a threshold of initial mass at 4.0 m SLE with the medium albedo and 3.9 m SLE with the low albedo.

Starting above the threshold, the ice sheet regrows to the EWNS configuration with medium albedo as with high albedo (Fig. c), but with low albedo there are two steady states. The larger is the EWNS configuration (7.3 m SLE; Fig. a1), while the smaller lacks the southern dome (7.0 m SLE, EWN configuration; Fig. h1). The southern dome has positive SMB in the historical climate (Fig. a2), but in the warm climates it is readily lost due to increased ablation, and in the historical climate without the dome there is negative SMB inhibiting readvance at the new southern margin (Fig. h2). The snowfall is however little changed in that region (Fig. a4, h4). The southern dome is the last part of the ice sheet to reappear with the high and medium albedos (solid green and blue lines in Fig. c after 30 kyr).

Starting below the threshold, the ice sheet attains steady states lacking the northern portion, which we refer to collectively as no-north states. The steady state with medium albedo has the EWS configuration (5.7 m SLE; like Fig. f1 in extent but thicker). With the low albedo, there are two steady states, having masses of 5.3 m SLE (EW configuration; Fig. g1) and 5.0 m SLE (E configuration; like Fig. e1 but much larger), which differ because the north-western dome is missing in the latter case. This dome is the last part to regrow with medium albedo.

Other authors have likewise found that the present state of the Greenland ice sheet is not the only steady state under historical climate . A minimal state with ice solely or mostly in the east is a common feature of all these studies and ours. In other respects the steady-state configurations are dissimilar. The medium state of most resembles our no-north states. Our results are more complex than others in showing five steady states. We suppose that this is because greater detail in the interaction of the ice sheet topography with atmospheric circulation and SMB can be simulated by FAMOUS–ice than by the simpler approaches of previous studies.

To summarize our second set of experiments, transient states which have passed below the threshold regrow to no-north steady states, while those still above the threshold regrow to EWNS or EWN steady states. Consistent with this, we note that all final states lying below the threshold in the first set of experiments are no-north states (Fig. b). States taken from below the threshold on trajectories of rapid decline show no tendency for the northern portion to regrow, even after tens of millennia under historical climate. Thus about 2 m of GMSLR will become irreversible once the Greenland ice sheet mass drops below the threshold if the medium or low albedo is realistic.

Under the same BCs, initial states which differ only slightly in topography (the minimum separation of our initial states in M is actually 0.2 m SLE) can lead to final states which differ substantially (by more than 1 m SLE) because ice sheet–climate feedbacks amplify the initially small difference in SMB. The probable reason is that ablation exceeds accumulation in the northern region without the ice sheet (shown by negative SMB at the northern margin in Fig. g2), partly because snowfall is reduced (Fig. a4, g4).

The low- and medium-albedo no-north steady states following regrowth are 1–2 m SLE above the threshold and yet grow no further, unlike states along trajectories of rapid decline having M in the same mass range, i.e. between the threshold mass (4 m SLE) and the no-north mass (5–6 m SLE). The implication is that, for states in this mass range, the outcome depends on the history. To test this, we have conducted further experiments (dashed lines in Fig. c) beginning from the two steady states in this range (large circles at the end of dashed lines in Fig. b), which were reached by slowly declining trajectories. These two are no-north (EWS) states. Initially the ice sheet mass grows, but, unlike when starting from rapidly declining transient states in this range, it soon becomes nearly constant at a slightly higher M than is reached from states below the threshold. The difference in M is due to a large southern dome, which was kept during the slow decline (along the dashed lines leading to the large circles in Fig. b) but had been lost already in states of the same mass in the warmer climate that produces the fast decline (the solid lines with red and green circles in Fig. b) and is not rebuilt in the historical climate. This result suggests that, for slow or quasi-static decline of the ice sheet, the no-north mass itself is the threshold of irreversibility.

Using the linear relationship between the initial rate of mass loss and the final steady-state mass in the first set of experiments (solid line in Fig. d), we can translate the threshold of irreversibility (M=3.9–4.0 m; horizontal red and green lines), which applies during trajectories of rapid decline, into a threshold on the rate of loss (vertical red and green lines). Under a warm climate which initially gives a more negative ΔSMB than the threshold rate, the ice sheet will eventually decline to a state which is smaller than the threshold mass. Roughly estimating a range from the scatter in the relationship, the results suggest that the threshold ΔSMB lies between -500 and -150 Gt yr-1 (Fig. d). Since recently observed ΔSMB e.g. is at the upper end of this range, i.e. a relatively small rate of decline, we recall from the previous paragraph that the relevant threshold may instead be the no-north steady-state mass of about 5.5 m SLE. In that case the linear fit indicates that the recently observed ΔSMB is close to the threshold rate which will eventually lead to partially irreversible loss of the ice sheet.

If the recently observed rate of mass loss of about 0.7 mm yr-1 SLE persisted, it would take 4900 years for the ice sheet mass to reach the threshold of irreversibility and about 2700 years to reach the no-north steady-state mass. At the highest rate of loss simulated in our experiments for the end of this century, of about 2 mm yr-1 SLE, it would take 1700 years to reach the threshold. Allowing for systematic uncertainty, the AR5 predicted even larger rates of mass loss due to SMB perturbation, of up to about 6 mm yr-1 SLE by the end of the century, at which rate the threshold would be reached in 600 years.

In order to avoid eventual irreversible ice loss, the climate must be returned to near pre-industrial before the threshold mass is reached. Reversing climate change requires extracting heat from the ocean as well as removing the radiative forcing. If that can be done at all, it could not be done instantaneously, and mitigating climate change in the short term will buy more time to save the ice sheet in the long term. Further simulations would be required to evaluate whether particular trajectories of future climate would avoid irreversible ice loss.