The simulations were run with the fully-coupled atmosphere–ocean–vegetation–ice sheet–solid earth model MPI-ESM/mPISM/VILMA . It uses the coarse resolution MPI-ESM version 1.2 . The ESM consists of the ECHAM6.3 spectral atmospheric model at a T31 horizontal resolution (approximately 3.75°) and 31 vertical levels, the JSBACH3.2 land surface vegetation model and the MPIOM1.6 primitive equation ocean model with a nominal resolution of 3°. The ESM is coupled to the ISM mPISM based on PISM version 0.7.3. mPISM has a horizontal resolution of 10 km in the Northern Hemisphere and Antarctica. At the atmospheric boundary, the ESM and PISM are coupled through an energy balance model (EBM), which calculates a yearly SMB on 24 height levels based on hourly atmospheric data. This data is then interpolated onto the ice sheet surface (for details see ). At the ocean interface, salinity and temperature averages from 203 to 523 m are used to extrapolate the ocean conditions underneath the ice shelves and to estimate basal melt . Changes in the solid Earth and the change in relative sea level due to a redistribution of ice and water are computed with the global model VILMA . The model also includes an Eulerian iceberg model as well as interactive topography, bathymetry, land–sea mask and adaption of river routing directions . Thus, the model system accounts for the melt–elevation, melt–albedo and GIA feedback, as well as interaction between the ice sheets with the dynamic atmosphere, the ocean, the land surface and vegetation and icebergs. For more details on the model system refer to , who used a similar model system. Our model version differs from in that we included additional features, such as an updated parameter tuning and an expanded ocean model grid that extends the GR30 grid of across the land–sea boundaries of Greenland and Antarctica. An expanded grid is required to accurately simulate ocean dynamics beneath newly exposed ocean grid cells as the ice sheets retreat. mPISM and VILMA are asynchronously coupled to the ESM. This means, that following 10 years of simulated climate with MPI-ESM, mPISM and VILMA run with the repeated 10 year ESM forcing for 100 years. The simulated ice sheet geometry is then used as input into MPI-ESM. This allows for the simulation of long timescales that are necessary to capture the slow response times of ice sheets . At the end of the 100 year-long ISM cycle, the averaged meltwater and iceberg fluxes are transferred to the ESM. The asynchronous coupling could have an influence on the timing of transitions in the ice sheet due to the long response time scales of the ocean. Focusing on the equilibrated steady states, the asynchronous coupling method should have no significant impact on the results.

Aiming to study the existence of potential multiple steady states of the GrIS, we performed ten simulations that started from different GrIS volumes (0 %, 17 %, 21 %, 33 %, 43 %, 50 %, 67 %, 70 %, 83 % and 100 % of the PI value). These were run until equilibrium under prescribed constant PI greenhouse gas concentrations (CO2 of 282.59 ppm, CH4 of 711.09 ppb and N2O of 270.28 ppb; ) and orbital parameters . Figure a–j in the Appendix displays the initial ice thickness map and the initial volume of each simulation.

The simulation started at 100 % is an equilibrium run with a steady-state PI GrIS and PI AIS that was initiated from an equilibrated asynchronous fully-coupled spin-up simulation. Note that the PI state of both ice sheets is similar to PD. This simulation is referred to as g100, where “g” refers to the GrIS and “100” indicates its initial volume, and serves as reference PI simulation. The GrIS volume in g100 is only 1.7 % larger than the volume obtained from the IceBridge BedMachine of the NASA National Snow and Ice Data Center (NSIDC) Distributed Active Archive Center . In the simulation with 0 % initial GrIS volume (g0), we initially removed the GrIS in g100 and let the underlying surface bedrock adjust isostatically. We continued the run until the GrIS reached a new equilibrium under PI CO2 that is significantly smaller than its PI volume. The remaining initial GrIS volumes were obtained by branching off simulations at different volumes from a run in which this small GrIS in g0 continued to regrow under progressively decreasing CO2 concentrations. We continued these simulations under PI CO2 concentrations until reaching a new equilibrium or converging to the steady state of another equilibrium simulation. In all simulations the bedrock adjusted interactively to the changing ice sheets. Note that the bi-hemispheric model set-up allows for changes in the AIS. Hence, responding to the changes of the GrIS, the AIS also changes significantly. The initial ice distributions of the AIS are illustrated in Fig.  in the Appendix.

To explore the interactions between the GrIS and the AIS, and the impact of the AIS on the steady states of the GrIS, we branched off simulations from the steady-state simulations initialized at 0 % and 43 % (g0 and g43, respectively) in year 15 050 (Appendix Fig. k and l) and prescribed a constant PI AIS volume. These simulations are referred to as g0_pi_a and g43_pi_a, where pi_a stands for constant PI AIS. An additional sensitivity experiment performed to study feedbacks with the ocean will be introduced throughout the analysis. If not stated differently, we analyze the means over the final 1000 years of each simulation. The final steady-state simulations and the sensitivity experiments are summarized in Table .

With the aim of disentangling the feedbacks that constrain each steady state, we performed additional sensitivity experiments. These separate the impact of the different factors, including surface elevation, GIA, glacier mask and albedo, precipitation, and the ocean, on the stability of the steady states. To obtain the contribution of the surface elevation, we mapped the last 100 years of our 3D SMB data of each state onto the surface topography of the neighboring state. This yields which impact the total elevation effect has on the steady states. This effect is composed of the elevation change due to the ice-thickness change and the counteractive isostatic adjustment of the underlying bedrock. To separate the effects of the ice-thickness difference and GIA, we compared the elevation changes due to the ice thickness and GIA difference to the total elevation changes. We then linearly scaled their contributions to the SMB effect of the total surface elevation effect. The effect of the glacier mask and surface albedo has been isolated by running simulations of each steady state with MPI-ESM for 1000 years, using the glacier mask of the neighboring state, while keeping the ice sheets constant. We derived the SMB fields from these simulations using the EBM. To analyze the impact of the precipitation, we retained all conditions of each individual state, but scaled the precipitation fields to the ones of the neighboring state of the last 100 years (e.g., climate of one state but with precipitation of the next larger state to force the EBM), to yield the same climatology as the neighboring state. We then ran the EBM with the modified 100 years of atmospheric data to derive the precipitation effect on the SMB. Lastly, the ocean contribution on the stability of each state is filtered out by running MPI-ESM for 200 years started from the end of each steady state and nudging the sea-surface temperature (SST) and sea-surface salinity (SSS) towards the averaged last 200 years of the neighboring states, while keeping the ice sheets constant. In all cases, the resulting last 100 years of SMB is compared to the original SMB of each steady state. These experiments indicate how each feedback process restricts the mass gain or loss of the GrIS states.

Initialized from different initial ice sheet volumes (Fig. a–j), we obtain four steady states of the GrIS (Fig. ): a large, a medium, a small and a very small state (L_G, M_G, S_G and XS_G, respectively). L_G corresponds to the PI state with a maximum ice thickness of about 3200 m in eastern Greenland (Fig. a). In L_G, the GrIS holds an ice volume of about 3.0 × 10¹⁵ m3, corresponding to 7.3 m of sea-level equivalent (SLE). In XS_G, the GrIS is split into two separate parts that are confined to the eastern and southern mountain ranges, reaching elevations of up to 2600 and 1600 m, respectively (Fig. d). The eastern part grows to a maximum ice thickness of 2300 m, while the southern part has a maximum ice thickness of about 2000 m. The GrIS in XS_G retains 20 % of its PI volume, equivalent to about 0.6 × 10¹⁵ m3 or 1.4 m SLE of ice. In the next larger state S_G, the southern and eastern parts of the ice sheet are connected by a narrow stretch of ice and form one single small ice sheet (Fig. c). In S_G, the GrIS has a maximum thickness of about 2400 m. Its final volume amounts to about 30 %, which corresponds to 0.8 × 10¹⁵ m3 or 2.1 m SLE. In the last state M_G, a medium-sized ice sheet is present with a final volume of about 50 %, equivalent to 1.4 × 10¹⁵ m3 or 3.5 m SLE (Fig. b). Compared to S_G, the ice sheet in M_G extends further northwest into central Greenland. Its maximum ice sheet thickness is about 2600 m. Additionally, small ice caps cover parts of northern Greenland.

The large GrIS in L_G is stable due to the expansively glaciated terrain, with peaks of over 3000 m and its highly reflective surface (Figs. a and ). These cause a locally cold climate with an average annual temperature of about -17.8 °C and a minimum temperature in winter of on average -26.7 °C (Fig. a). The high orography blocks synoptic storm systems approaching Greenland from the west . Deflected on a more southerly trajectory, the storms move along the southern tip of Greenland, where they can cause precipitation. Additionally, the high topography along Greenland's southeastern coast plays a crucial role in generating orographic precipitation on the windward side of the mountains, driven by moist easterly onshore winds (Fig. a; ). Hence, accumulation is highest in southern Greenland, where the regions of maximum precipitation and high orography are located (Figs. i and a). As surface ablation is confined to the low-lying areas along the coast (Fig. e), the SMB is positive over the majority of the ice sheet (Fig. i).

Started from a completely disintegrated state (Appendix Fig. a), the GrIS regrows in the regions in the south and east of Greenland (XS_G) due to their favorable climatic conditions, including temperature and precipitation. The east of Greenland receives more precipitation than with a large GrIS as the atmospheric flow is less deflected by the lower orography of the smaller GrIS (Fig. l). As a consequence, moist air masses and storm tracks penetrate deeper into Greenland (previously shown by ), where they eventually precipitate on the windward side of the mountain range to the east of Greenland. This is reflected in a more homogeneous distribution of precipitation, with lower precipitation in the south and west of Greenland, but higher precipitation in the northeast in XS_G. The weaker Greenland Anticyclone (previously shown by ), in response to the reduced mechanical blocking and the higher near-surface air temperatures in the predominantly low-elevation areas of Greenland (Fig. d and h), further reduces the advection of moist air masses to southern Greenland. Hence, surface accumulation is lower in Greenland's south and west in XS_G than in L_G (Fig. d). Nevertheless, the southern and eastern regions continue to receive the highest annual precipitation, due to orographic effects and the proximity to the core of the storm track located south of Greenland. Only in the mountains are temperatures cold enough to preserve the snow throughout the year (Fig. h and l), which eventually favors the nucleation of a new ice sheet.

The differences in the climate of XS_G compared to S_G also inhibit ice sheet expansion into the center and north of Greenland. In the center and the north, a strong lapse-rate effect due to the lower surface elevation of up to 2300 m (XS_G compared to L_G) inhibits ice sheet formation by raising temperatures. Assuming a lapse rate as used in our EBM, we find a temperature increase of up to 10.6 °C in the areas of the largest change in surface elevation. Thus, the lapse-rate effect contributes most strongly to the annual-mean temperature change of up to 11.7 °C in XS_G as compared to L_G. This lapse-rate effect also explains why the highest temperature anomaly occurs in central and northern Greenland (Fig. d). Although the warming effect of the lower surface elevation due to the lower ice thickness of the smaller GrIS is slightly counteracted by GIA effects, which raises the bedrock surface by up to 600 m over central Greenland (Fig. d), cooling through GIA is not sufficient to allow for an expansion of the GrIS in XS_G. Another contribution arises from the smaller glacier mask and the absence of a snow cover in summer, which changes surface parameters to those of a non-glaciated surface. The latter enables the dynamic growth of grass and shrubs in ice-free areas. These surface changes reduce the summer albedo by about 0.6, leading to a strongly positive melt–albedo feedback. They also allow surface temperatures to exceed the melting point in XS_G. Owing to this surface-property effect, the warming relative to a large GrIS is stronger in summer (up to +17.0 °C) than in winter (up to +11 °C). Hence, no perennial snow cover can accumulate north of the northwestern margin of the GrIS in XS_G, although precipitation is larger with a smaller GrIS in these areas (Fig. i–l).

Another warming contribution in XS_G arises from the near-surface winds that approach Greenland on its southeast coast (Fig. a–d). Traversing Greenland, winds are forced to ascend over the very small GrIS and create a slight Föhn effect on the leeward side. This is expressed in warmer temperatures, a strong reduction in precipitation and down-slope winds on the leeward side as compared to the windward side of the ice sheet. This Föhn effect contributes to preventing an ice sheet expansion into the central, western and northern areas of Greenland as well as the connection of the eastern and the southern parts of the very small GrIS.

The smaller GrIS in XS_G is also stabilized by differences in the atmospheric circulation. In response to the reduced blocking effect of a smaller GrIS, the quasi-static wave at 500 hPa over Greenland is slightly shifted eastward and weaker (Fig. h). This shift reinforces the meridional flow pattern over Greenland and its surroundings, similar to findings of . Consequently, the southerly wind component over Greenland intensifies, advecting more warm air masses towards Greenland. This enhances the 2 m air-temperature rise over Greenland and contributes to preventing the expansion of the very small GrIS in XS_G. Over the adjacent Nordic Seas, the wind direction is increasingly northerly, amplifying the influx of cold polar air, as visible by colder 2 m air-temperatures over the Nordic Seas and Scandinavia (Fig. d and h). Additionally, the northerly winds drive sea ice further south and favor sea-ice expansion in the Nordic Seas particularly in winter (Fig. d). The larger sea-ice cover in XS_G reduces heat loss from the ocean to the atmosphere, enhancing the cooling of the overlying atmosphere. This also leads to colder upper ocean temperatures until a depth of approximately 150 m and warmer temperatures at deeper levels. A weaker AMOC strength at 30° N in XS_G (14.6 Sv) compared to L_G (17.3 Sv) further reduces the heat that is transported northwards, contributing to the colder upper ocean temperatures in the Nordic Seas. As the colder air is advected onto the GrIS by the southeasterly near-surface winds (Fig. d), this cold ocean anomaly likely contributes to preserving the southern part of the very small ice sheet in XS_G. Analyzing the variability of the AMOC and the SMB in XS_G, we find a linear relationship with an increase in SMB by 40.7 mm water equivalent (w.e.) per 1 Sv decrease of the AMOC strength.

To explore the importance of the ocean cooling in the Nordic Seas for the stability of the GrIS in XS_G, we conducted a sensitivity experiment. In this experiment, we used the same setup as in g0, where the GrIS stabilizes at XS_G, but with SST and SSS nudged towards the climatology of g100, which corresponds to the PI state. Hereafter, this experiment is referred to as g0_pi_oce (Table ). Hence, this experiment only considers interaction of the ice sheets with the atmosphere and sea ice, while suppressing feedback with the ocean. In the absence of the ocean cooling in the Nordic Seas (g0_pi_oce), a new ice sheet develops only in the east of Greenland, while no regrowth occurs in the south of Greenland. Hence, the cooling of the Nordic Seas, caused by an absent or much smaller GrIS, is a necessary prerequisite for the development of an ice sheet in Greenland's south and feedbacks with the ocean maintain the southern GrIS.

Above an initial volume of 21 %–33 % of its PI volume (0.6–1.0 × 10¹⁵ m3), the GrIS transitions into the next larger state, S_G. The GrIS in S_G as well as M_G is stabilized by similar climate conditions and feedbacks, as described for XS_G (Figs.  and ). However, being strongly controlled by orography and ice sheet area, the atmospheric circulation signals are weaker when the GrIS is larger.

In S_G, the ice ridge between the eastern and the southern GrIS originates from a colder climate. The ice ridge increases the surface elevation and albedo, and also keeps surface properties in a glaciated state, leading to colder temperatures than in XS_G (i.e., elevation and albedo feedback; Fig. c and g). Due to orographic effects, accumulation is higher on the windward side and atop of the ice ridge compared to XS_G (Fig. c). The lower ablation and higher accumulation stabilize the connection between the eastern and the southern part of the GrIS. On the leeward side of the ice ridge, however, a Föhn effect inhibits an ice sheet expansion towards the northwest in S_G. S_G only has a small range of stability. Above a threshold of 33 %–43 % of its PI volume (1.0–1.3 × 10¹⁵ m3), the GrIS becomes unstable and transitions into M_G, due to the effects of the higher surface elevation, the larger glacier mask, a weakening of the southerly winds, weaker Föhn winds, a less strongly redistributed precipitation and more ice flowing into the central areas.

M_G can be attained from a relatively large range of initial values (Fig. ). However, it is less stable than the other states. This is evident in the simulation g43, which is a stable M_G state for more than 80 000 years before it abruptly transitions into the next smaller state S_G. The state transition coincides with a slightly stronger AMOC, whose increase is within the bounds of natural variability e.g., . In the first 1000 ice sheet years of the transition, the AMOC is stronger by on average 1.3 Sv compared to the preceding millennia. The stronger AMOC transports more heat northward into the North Atlantic Ocean, where it leads to a warm anomaly in the Irminger Sea and the Nordic Seas. This warm anomaly is advected onto the GrIS by southeasterly winds off the southeast coast of Greenland (Fig. b). Over the GrIS, the warmer air triggers melting of the northwestern part of the ice sheet. Together with the climate–ice sheet feedbacks described, the medium GrIS in g43 enters self-sustained melting and transitions into S_G. In contrast, the medium GrIS in the simulation initiated with 70 % of the GrIS volume (g70) remains stable throughout our simulated time period. As both simulations have a similar and stable ice volume and ice distribution for about 80 000 years, we consider g43 before the transition also as M_G state. However, subjected to small disturbances, the M_G initiated from 43 % of GrIS volume eventually destabilizes and transitions into a more stable equilibrium state. Hence, we consider the final part of g43 as S_G state.

Similar to the smaller states XS_G and S_G, an expansion of the GrIS in M_G is impeded by a strong lapse-rate effect, the absence of glacial surface conditions in the northwest, and to a lesser degree by a slightly different atmospheric circulation and a redistribution of precipitation, which cause a negative SMB in central and northwest Greenland (Figs. b, f, j and j). Resting primarily on low-elevation and flat bedrock, the northwestern part of the GrIS is exposed to warmer temperatures and lacks stabilizing pinning points over high-elevation terrain (Fig. b), controlling ice sheet growth in this region. Hence, even started from a significantly larger GrIS volume of 70 % with the ice edge further in the northwest (Fig. h), the GrIS returns to M_G (Fig. ). Only above a threshold of 70 %–83 % of its PI volume (2.1–2.5 × 10¹⁵ m3), does an ice cover in the northwest become stable, as it connects with the northern part of the GrIS and transitions into L_G (Fig. ). Hence, there is no stable state with an ice cover in the flat and low-lying northwest that is not connected to the northern part of the GrIS. This explains the absence of a stable state between 50 % and 100 % of its PI volume.

But how important is each of these described feedbacks and processes for the stability of the GrIS states? Figure  summarizes the contribution of each feedback, based on the figures in Appendix . The figure indicates, for instance, that the GrIS in S_G is stable mostly due to the impact of the surface elevation feedback. As explained in Sect. , the surface elevation effect is composed of the effect of the ice thickness and the counteractive GIA. In S_G, the surface elevation is significantly lower over large parts of Greenland than in M_G, due to the smaller ice sheet and associated lower ice thickness. Although uplift through GIA counteracts the effect of the lower ice thickness, the net impact of these two feedbacks is destabilizing, as the ice thickness-induced elevation change is larger than the isostatic uplifting-induced elevation change. The combined glacier mask and surface albedo feedback further impedes an ice sheet expansion along the margins of the GrIS in S_G, mainly by raising the surface ablation. On the other hand, the higher surface elevation and larger glacier mask than in XS_G stabilize the GrIS in S_G. Feedback with the precipitation opposes the elevation and glacier mask feedback mainly due a redistribution in the accumulation, similar to Fig. j–l. While the precipitation of S_G favors ice-sheet expansion towards the north of Greenland, thus stabilizing the small GrIS, the redistribution reduces the SMB in the south of Greenland. For the transition from a small to a very small GrIS (S_G→XS_G), the redistribution of precipitation has a counteracting effect and destabilizes the GrIS in S_G. The ocean feedback has the weakest impact on the stability of the small GrIS in S_G. Warmer ocean conditions in the Baffin Bay and Irminger Sea compared to colder conditions in the Nordic Seas in S_G than in M_G (Fig. j) favor ice sheet growth in the west, but inhibit it in the east. Spatially averaged, feedback with the ocean thus contributes to impeding the shift of the small GrIS into the medium state. However, the ocean changes slightly contribute to preventing a retreat of the GrIS in S_G, due to a cooling-related lower surface ablation in S_G than in XS_G (Fig. j). Although the ocean feedback has a smaller or even minor contribution compared to the elevation feedback, even these small contributions can be decisive for critical transitions of the ice sheet, as explained in the preceding analysis, and should therefore be included in analyses of the stability of the GrIS.

The contributions to the stability of the other states are mostly comparable. An exception is the positive precipitation contribution for the transition of a medium to a large GrIS (M_G→L_G) as compared to the negative precipitation contribution for the transitions of a very small and a small GrIS to the next larger state (XS_G→S_G and S_G→M_G). It is attributed to the precipitation-induced higher surface ablation with a medium than with a large GrIS, as higher summer rainfall and lower summer snowfall increase the energy available for surface melt. Thus precipitation has a stabilizing effect on the M_G state, limiting further GrIS regrowth.

By including interactive ice sheets in both hemispheres, the model setup allows for changes in both the GrIS and the AIS volumes. Here we discuss how changes in the GrIS's geometry impact the AIS and vice versa.