We use the ice-sheet model Yelmo coupled with the regional energy-moisture balance atmospheric model REMBO . Yelmo is a three-dimensional thermomechanical ice-sheet model that has been previously used to simulate the evolution of the Greenland , the North American and the Antarctic ice sheets . To compute the ice velocities, Yelmo uses the depth-integrated-viscosity approximation (DIVA) that considers longitudinal, lateral, and vertical shear stresses . In DIVA, the effective viscosity and the horizontal velocity gradients are replaced by their depth-integrated values in the effective strain rate equation. This allows for a more efficient two-dimensional solution, which yields close results to the computationally much more expensive Stokes problem . A more detailed description of Yelmo is found in .

The basal friction is calculated using the regularized Coulomb friction law as follows: 1τb=cb|ub||ub|+u0qub|ub|, where ub is the basal velocity, q the friction law exponent, u0 the regularization term for the Coulomb law set at 100 myr-1, and cb the basal yield stress, defined as: 2cb=tan⁡ϕN. Here, N is the effective pressure, calculated following . The till friction angle, ϕ, is a linear function of bedrock elevation, with a lower limit of 1° when the bedrock is 700 m below sea level and an upper limit of 40° when the bedrock is 700 m above sea level. As in , we impose a scaling factor of 0.1 on cb in the northeastern basin. This accounts for the softer properties of the bed and allows for a better representation of the North East Greenland Ice Stream (NEGIS).

To represent glacial isostatic adjustment (GIA), Yelmo is coupled to the regional isostasy model FastIsostasy . We employ here an elastic lithosphere and a viscous mantle (ELVA) with a spatially homogeneous upper-mantle viscosity of 1×10-21Pas in the first 88 km and 2×10-21Pas in the following 600 km, and gravitational interactions between the ice sheet, the solid Earth and the ocean.

The surface mass balance (SMB) is calculated by the regional energy-moisture balance (REMBO) model, which follows an ITM scheme, described below in Sect. . The geothermal heat flow is taken from and is imposed as a boundary condition at a depth of 2 km into the bedrock. Submarine melting at the base of ice shelves is parametrized by a linear equation dependent on ocean temperature (Sect. ). Calving follows the Von Mises stress criterion and it is defined as a function of the effective stress at the ice front . While other calving criteria exist that represent the calving phenomenon with greater complexity, the computational cost of applying these methods at the full ice-sheet scale makes it preferable to employ simpler criteria such as Von Mises. However, this law produces satisfactory results consistent with previous studies of the GrIS using Yelmo and has demonstrated good performance in reproducing observed calving rates across different glaciers . In addition, since the presence of ice is considered implausible in the deep ocean, calving is applied where the bedrock depth exceeds 1000 m.

REMBO is a 2D regional energy-moisture balance climate model that simulates daily precipitation and temperature fields for Greenland, as well as the surface mass balance (SMB) through a one-layer snowpack model. The temperature and humidity over the ocean around Greenland are imposed as boundary conditions, for which the climatological mean from years 1958–2001 of ERA-40 reanalysis is used to represent the present-day conditions. The atmosphere over Greenland is then simulated through the vertically integrated energy balance and moisture balance equations. Additionally, REMBO takes into account the orography of Greenland and the planetary albedo, including its seasonal changes, thus reproducing the associated albedo and elevation feedback, the latter with a lapse rate of 6.5 Kkm-1 like in. It does not simulate the atmospheric broader general circulation and has its own biases, particularly in precipitation. Despite relatively low resolution of REMBO (100 km), the surface boundary conditions – and therefore the melt-elevation and melt-albedo feedbacks – are computed at the higher ice-sheet model resolution of 16 km. As a result, the model is able to represent the key atmosphere-ice feedbacks relevant to this study at a relatively high resolution. It has been previously used to simulate both the past and the future evolution of the GrIS .

The SMB is defined as the difference between the accumulation and the runoff. REMBO follows the insolation-temperature melt (ITM) method, which was originally developed by and . The surface melt rate Ms follows the linear equation: 3Ms=1ρwLm[c+λT+τa(1-αs)S], where ρw is the water density; Lm is the latent heat of ice melting; c and λ are empirical parameters set to -55Wm-2 and -10Wm-2K-1; τa is the transmissivity of the atmosphere; αs the surface albedo; and S the insolation at the top of the atmosphere. Therefore, the dependence of melting on both absorbed insolation and temperature is taken into account. The surface albedo is parametrized as a function of the snow thickness and the type of ground surface .

REMBO is bidirectionally coupled to the Yelmo ice-sheet model. The SMB and surface temperature obtained by REMBO force the ice-sheet evolution in Yelmo and the evolution of the topography calculated in Yelmo is an input for REMBO .

To force REMBO, we impose monthly temperature anomalies at the boundaries. Here, as in , we assume that annual temperatures follow a sinusoidal cycle where ΔTDJF=2ΔTJJA, so that the annual temperature anomaly is ΔTann=(ΔTDJF+ΔTJJA)/2=1.5ΔTJJA. Since temperature during the summer months is more important for melting, we use ΔTJJA as input.

While relative humidity r is held constant, specific humidity depends on temperature according to 4Q=Qsat(T)⋅r, where Qsat(T) is the saturation specific humidity, which follows the Clausius–Clapeyron relation. The choice to maintain constant relative humidity rather than specific humidity in REMBO is based on the stronger temperature dependence of the latter. Total precipitation is then calculated as a function of specific humidity, thus accounting for temperature variations: 5P=(1+k|∇zs|)Qτ, where ∇zs is the surface elevation gradient, τ is the water turnover time in the atmosphere, and k is an empirical parameter.

REMBO overestimates precipitation in the southwestern and northern regions . To address this systematic bias, we apply a spatially varying bias correction following and based on PMAR, the 1961–1990 mean monthly precipitation from . Thus, the corrected precipitation is: 6Pcorr=P⋅δP,whereδP=PMARPREMBO,ref where P is the precipitation without correction, and PREMBO,ref is the precipitation simulated by REMBO for the present day without correction. Gaussian smoothing is first applied to both PMAR and PREMBO,ref so that we only correct large-scale features. After this, the correction factor δP is computed internally within REMBO at its native resolution of 100 km. To maintain a consistent field with the boundary conditions at each moment and to avoid introducing a bias from present-day conditions, δP is restricted to values between 0.6 and 1.4 (see Fig. ). This correction serves to improve the agreement of the seasonality and overall magnitude of precipitation, particularly in the north.

The ocean forcing of the GrIS is treated in very different manners throughout modelling studies. Some fully neglect it, taking into account only atmospheric forcings while others parameterize sub-shelf melting as a function of the ice-shelf draft i.e., the ice-thickness below the ocean surface;. Still, others do not use an explicit forcing but constrain their model with relative sea level . Finally, some include parameterizations of basal melting as a function of ocean temperatures e.g.. Results show that ice-ocean interactions play a key role in the past GrIS evolution, notably at times and in areas where the ocean-ice contact zone is largest, such as the LGM, when the GrIS margins reached the continental shelf break. Even if present marine-terminating glaciers cover only a small part of Greenland in the present day, their advance and retreat significantly affects the ice geometry and dynamics in the continental zone, and this effect is amplified as the ice-ocean contact zone expands .

As in , we formulate the basal melting rate at the grounding line Bgl using an anomaly approach: 7Bgl(t)=κΔTocn(t)+Bref, where κ=15 m i.e.yr-1 °C⁻¹ is the heat flux between the ice and the ocean at the ice-ocean interface; ΔTocn(t) is the anomaly between the ocean temperature at time t and at present day; and Bref represents the basal melting rate at the present day. Here, Bref is set at 50m i.e.yr-1 following observations of basal melting at the grounding line . All parameters in Eq. () are assumed to be spatially uniform along the marine boundaries of the GrIS. Using a single value is a coarse approximation; however, in the absence of a complete sub-shelf melt map for Greenland, constructing a reliable 2D spatial field would be subjected to large uncertainties. Therefore, for the idealized experiments conducted in this study, we consider this approximation to be appropriate.

On the other hand, the sub-shelf basal melting rate Bsh is lower than that at the grounding line: 8Bsh=γBgl, where γ=0.1, in accordance with observations of Greenland glaciers .

ΔTocn is related to the annual atmospheric temperature anomaly ΔTann as follows: 9ΔTocn=focnΔTann.

The scaling factor between oceanic and annual atmospheric temperature focn is an empirical parameter that is highly uncertain. Here we calculate the mean ratio between oceanic and atmospheric temperatures over the ocean surrounding Greenland using PMIP3 model outputs CCSM4, CNRM-CM5, FGOALS-g2, IPSL-CM5A-LR, MIROC-ESM, MPI-ESM-P, MRI-CGCM3, and removing the outlier values. This yields focn,LGM=0.18±0.02 for the LGM and focn,Hol=0.5±0.2 for the mid-Holocene. These values show therefore considerable spread. We chose focn=0.22 as it falls within this range and is closer to LGM values, which is appropriate since the ocean forcing has greater influence in the temperature range we investigate. Equation () can be rewritten as a function of ΔTJJA: 10ΔTocn=0.22ΔTann=0.33ΔTJJA. Therefore, considering the values from Eq. (), basal melting in ice shelves (Bgl≥0) is activated at ΔTJJA=-10.1 K.

Here, in contrast to but following subsequent studies , Bgl is limited to positive values, preventing refreezing at the base of ice shelves. While local variations in basal melting and refreezing can occur in reality, given our simplification of applying spatially homogeneous melting, we also prevent refreezing at the base of ice shelves. This is consistent with recent studies in the Antarctic ocean showing that basal melting can still be active under glacial conditions .

To study the stability of the GrIS, warming (retreat) and cooling (regrowth) simulations (hereafter branches) were performed, starting at different initial states, respectively: an LGM-like state and a virtually ice-free state (Fig. ). To obtain these initial states, we performed two spin-up experiments (one for each branch), initialized with present-day topography and ice thickness , and ran them for 60 kyr with a constant forcing. The ice-sheet model has ten vertical layers in sigma-coordinates and a horizontal resolution of 16 km, the same as in other recent studies examining the stability of the GrIS . While a 16 km resolution may misrepresent certain processes, it enables long-term simulations that otherwise would be not possible due to their high computational cost (see Appendix Sect. ). At this horizontal resolution, the interaction with the ocean is not resolved in the narrowest fjords. For this reason, as in , the bedrock elevation is reduced by 1–1.5σ, where σ represents its standard deviation, in the areas in contact with the ocean and higher uncertainty in the value of the bedrock elevation.

The LGM-like state (Fig. a) was obtained by imposing a regional summer air temperature anomaly of ΔTJJA=-12 K and an ocean temperature anomaly of ΔTocn=-4 K, according to Eq. (). It is important to note that this LGM-like state is not meant to be a reconstruction of the LGM, but an idealized configuration designed to analyze the effect of temperature on ice-sheet stability in isolation from other major paleoclimatic forcings that are relevant to achieve a realistic LGM state, such as changes in sea level, insolation, or atmospheric CO₂. Nevertheless, even though the boundary conditions do not fully represent those of the LGM, we obtain a configuration that closely resembles simulations with full LGM conditions (see Appendix ). We obtain an ice sheet that is considerably larger than the present-day one. It has a simulated extension of 3.07 million km², a total volume of 5.15 million km³ and a volume above flotation of 4.6 million km³, corresponding to an anomaly of 3.66 m of sea level equivalent (SLE). The latter value is within the range reported in literature, ranging between 2.6 and 4.7 m SLE . Low atmospheric and ocean temperatures allow for the expansion of the GrIS up to the continental shelf break, with floating ice shelves appearing in the southeast and in Baffin Bay, fed by multiple ice streams. Large agreement is found between the simulated and the reconstructed margins for the LGM . An exception is the northeastern area, where our simulation reaches the continental shelf break while the reconstruction by does not. Whether the GrIS reached the shelf break in that region or not is still under debate but a more recent study based on high-resolution hydro-acoustic data suggests that was the case , thus matching our simulations. What is almost certain is that the GrIS expanded beyond the limits suggested by , whose reconstruction for the LGM appears to be a lower limit as compared to the former studies.

The virtually ice-free state was obtained through the simulation of a warmer state (Fig. b), with a regional summer air temperature anomaly of +4 K and an ocean temperature anomaly of ca. +1.3 K. Under these conditions, the ice sheet is drastically reduced both in volume and extent (Fig. b). Ice persists only in the high-mountain areas: at the southern tip of Greenland and in the easternmost region, around the Watkins range. The mass balance of the remnant ice caps is almost zero in their interior (all incoming accumulation is melted away) and negative at their margins, especially in areas in contact with the ocean (not shown).

Finally, we performed a third spin-up for present-day conditions, for which we find a close agreement between our simulation and the observations (see Fig.  in the Appendix).

To study the wider stability of the GrIS, we perform a series of simulations, outlined in Fig. . First, quasi-equilibrium simulations were performed with different rates of forcing (between 5×10-3 and 1×10-5Kyr-1) in a range of regional summer air temperature anomalies from -12 to +4 K relative to the average value between 1958–2001, which is used for the boundary conditions of the model as in and . For the warming experiments, a positive constant temperature anomaly rate is applied to the LGM-like state, until a temperature anomaly of +4 K, relative to the average value, is reached. In parallel, for the cooling experiments a negative constant temperature anomaly rate is applied, starting from the virtually ice-free state until a temperature anomaly of −12 K is reached. Therefore, to achieve the full temperature anomaly range of 16 K from the initial state, the durations of the simulations vary between 3200 years and 1.6 million years, depending on the forcing rate.

In addition, equilibrium states are derived through two types of simulations: (1) applying an instantaneous summer air-temperature anomaly to the initial state during 60 kyr across different temperature levels ranging from -12 to +4 K, in 1 K increments; and (2) similar to , branching off from the 1×10-4Kyr-1 quasi-equilibrium simulation, then maintaining a constant temperature at different forcing levels (ranging from -12 to +4 K for the atmospheric temperature, with 1 K increments and 0.2 K increments near the tipping points) for 80, 120 or 400 kyr, depending on the time required for the simulations to stabilize.

By decreasing the forcing rates in the quasi-equilibrium simulation, we bring the transient simulations closer to the equilibrium states. Ideally, this would be achieved with an infinitely slow forcing rate, which is however impossible in practical terms. Nonetheless, the combination of the transient quasi-equilibrium simulations with the equilibrium states allows us to perform an exhaustive analysis that characterizes the stability phase space over the chosen range of study – where the tipping point values are taken from the branching-off experiments, as these are the most reliable due to their gradual convergence toward equilibrium.

Finally, in order to differentiate the atmospheric and the oceanic effects, we perform two more quasi-equilibrium experiments with a forcing rate of 3×10-5Kyr-1 under the following conditions:

Only OCN: ΔTJJA remains constant at -12K during the warming branch and +4K during the cooling branch, corresponding to initial state values.

Through the different equilibrium and quasi-equilibrium experiments of the GrIS, we obtain two different branches (warming and cooling, or retreat and regrowth) that constitute the bifurcation diagram (Fig. a). As the forcing rates decrease (and therefore the forcing periods increase) the transient quasi-equilibrium response converges to the equilibrium states. This is a consequence of the large inertia of the ice sheet: only long forcing periods allow for the ice-sheet response to the forcing to fully develop and to approach the equilibrium states. In the slowest simulation (1×10-5Kyr-1), which takes 1.6 million years, the volumes coincide with the equilibrium values for the entire interval, except in the vicinity of the tipping points, where the ice sheet takes even longer to stabilize.

We identify two bifurcation points in the warming branch. As noted earlier, their values are best determined from the branching-off equilibrium simulation, which provides the most reliable representation of the equilibrium states. The first one, between -10 and -9 K, begins with the loss of most of the GrIS ice shelves, followed by the retreat of the northeast margin (the state of the ice sheet before and after the transition is shown in Fig. b and c and is analyzed in greater depth in Sect. ). Once a temperature anomaly of -9 K is reached, a gradual decrease in volume begins, showing an almost linear trend with a slope close to zero relative to the forcing. This behavior persists until a temperature anomaly of 1.2 K is reached (Fig. d), at which point the positive melt-albedo and melt-elevation (atmospheric) feedbacks are triggered (Sect. ). This second tipping point is identified in this study between +1.2 and +1.8 K.

Figure  shows a zoomed-in view around this tipping point. Near the bifurcation point – particularly at +1.4 and +1.6 K – in both equilibrium branches of the branching-off experiments, the ice sheet oscillates between two states (shown in more detail in Fig. ). These oscillations show periods ranging from 70 to 120 kyr. At +1.8 K, the GrIS reaches a stable virtually ice-free state with a volume of 1.58 m of SLE (Fig. e). In this transition, we move from an ice sheet that almost completely covers the surface of Greenland to a virtually ice-free state, with some residual ice only at the highest altitudes, mainly in the Watkins Range and the southern tip.

In the regrowth (cooling) branch, the GrIS starts at its virtually ice-free state (Fig. b), with ΔTJJA=+4 K, and it is cooled. The ice sheet remains nearly ice-free until 1.6 K, where oscillations resume, as in the warming branch. The ice sheet remains in an intermediate state (Fig. f) until 0 K, with no ice in the north due to the high temperatures and the low precipitation in the area.

In this cooling branch, quasi-equilibrium simulations are farther away from equilibrium than in the warming branch for the same rates, especially during the critical transition (Fig. a, quasi-equilibrium simulations with rates of forcing between 10-5–10-3Kyr-1 and the temperature range 0–+1.8 K). Starting from the ice-free state, the recovery of the ice sheet requires lower temperatures and more time. This is due to the fact that melting or calving can occur almost instantaneously with high rates, while the processes of accumulation and regrowth are slower and can require thousands of years. Between 0 and -9 K the ice sheet does not show large variations, but just a linear regrowth with small sensitivity to temperature changes. Finally, from -9 K onwards, the sensitivity is increased again: the ice shelves are recovered, and the northeastern margin expands until it reaches the continental shelf break. By -12 K, it attains a volume above flotation of 4.45 million km³, almost converging to the initial state of the warming branch.

Hysteresis persists throughout almost the entire temperature range (between -10 and +1.4 K), with the largest volume difference for temperature anomalies between 0 and +1.4 K. It is important to highlight how, for the branching-off equilibrium states, the hysteresis is narrower and the regrowth in the cooling branch allows intermediate equilibrium states like that shown in Fig. f. At ΔTJJA=+1.2 K multistability becomes evident, with two different ice-sheet configurations (Fig. d and f) and respective volumes of 2.99 and 1.82 million km³. These results confirm the multistability induced by atmospheric feedbacks related to elevation and albedo identified in previous studies , as well as the irreversible nature of this transition on long timescales: if the present-day GrIS collapses, even if temperatures return to present-day values, there would still be a 1.4 m SLE difference.

Within the range from -9 to 0 K, the differences between branches especially affect the ice thickness in the margins. For example, for a summer air temperature anomaly of -5 K (Fig. a–c), the equilibrium ice sheet on the warming branch extends well beyond the margin of the equilibrium state in the cooling branch for the same temperature and has a 12 % larger volume. This nonlinearity originates from different mechanisms. To explain these, we analyze the ice-sheet evolution at selected transects (1 and 2; Fig. d, e). These are located in areas with the greatest differences between the grounding lines, following the ice-flow streamlines. Along these transects, the Disko Island (Qeqertarsuaq in Greenlandic) and a bathymetric peak in the northeast act as pinning points for the warming branch equilibrium states analogous to those found for the Antarctic ice sheet in . These slow down the outflow and allow the ice sheet to have a higher volume.

On the other hand, in the cooling branch the ice sheet shows a more irregular margin than in the warming branch. As it grows from the interior, it encounters the coastline (which is highly irregular) as a boundary, where ocean-induced melting occurs, preventing further growth. In contrast, in the warming branch, since it originates from an ice sheet in equilibrium and of greater extent (the LGM-like state), the continuity of the ice allows basal melting caused by the ocean to melt the margin more uniformly, resulting in a more compact and regular ice sheet shape. This leads to a larger perimeter in the cooling branch: the cooling branch state has a margin of ca. 40 000 km, whereas the warming branch state's margin is ca. 27 000 km long. As a result, even though some areas in the cooling branch have not yet reached the coastline, the total margin in contact with the ocean is greater (ca. 10 000 km vs. ca. 7000 km in the warming branch). This generates a higher basal melting and calving, making the regrowth difficult and preventing the ice sheet from reaching the size of the warming branch at the same temperature.

In order to analyze the mechanisms leading to ice loss and recovery along the stability diagram, we show the surface, basal and lateral mass fluxes in Fig. . This analysis can only be performed with the quasi-equilibrium simulations, so we selected the simulation with the lowest forcing rate (1×10-5Kyr-1), as it is the closest to equilibrium. As we saw in the previous section, for each quasi-equilibrium simulation, the system is farthest from equilibrium near the tipping points (shaded gray). This is reflected in the total mass balance (Fig. ), which remains stable over the entire bifurcation diagram except in the proximity of the tipping points, where it becomes negative for the warming branch and positive for the cooling branch.

For the warming branch, the SMB increases between -12 and -4 K as higher temperatures allow for more precipitation. However, from -4 K onwards, ablation increases and the SMB gradually decreases. Finally, at ca. +1.5 K SMB drops abruptly and becomes negative, showing that atmospheric feedbacks have been triggered. The basal mass balance (BMB) is negligible between -12 and -10.1 K due to the fact that low ocean temperatures do not allow for sub-shelf melting, and to the small value of the melting under grounded ice. When temperatures rise enough to produce sub-shelf melting, an ice-shelf reduction, a margin retreat, and a decrease in the ice-ocean contact zones is induced (see Appendix ). This eventually causes a reduction in calving. BMB continues the increase with ocean warming until around -5 K. After that point, sub-shelf melting begins to decrease as the contact with the ocean strongly diminishes, becoming almost negligible by +2 K. Calving is highest in the temperature ranges where ice shelves are present (-12 to -10.1 K). It decreases, as the BMB does, when the contact with the ocean decreases.

Therefore, before the first transition (at ca. -9.5 K for this quasi-equilibrium simulation), SMB is increasing, so only the sub-shelf basal melting – which intensifies with rising ocean temperatures – can drive the response, directly triggering ice-mass loss and accelerating ice dynamics. In contrast, during the second transition (at ca. +2 K), the influence of the ocean diminishes, as BMB decreases in amplitude and therefore cannot account for the observed mass loss. Instead, SMB decreases sharply, implying a positive feedback mechanism where surface forcing becomes the dominant driver.

In the cooling branch, surface processes are responsible for the ice-sheet growth, with accumulation increasing and ablation decreasing as temperature decreases. Basal melting and calving also increase as both the ice sheet and its contact zones with the ocean increase. Calving, in particular, strongly increases at around -9 K, when ice shelves recover and temperatures are too low to allow for basal melt. The transitions in this regrowth branch are more gradual than in the retreat branch, since the ice growth processes are slower.

To further investigate this issue, in Fig.  we show the results of the simulations with atmosphere and ocean separately. Regarding the warming branch, the OCN-only simulation follows the default run almost exactly until ΔTocn=-1 K, when the margin retreats to the coast and reaches a constant state regardless of the increase in ocean temperature. There is no second tipping point in this simulation, which confirms that the latter was due to the atmospheric feedbacks. Conversely, the ATM-only simulation shows minimal changes until ΔTJJA=-3 K (ΔTocn=-1 K). At around ΔTJJA=-1 K the ice sheet loses the northeast sector, then retreats to the present-day margin and subsequently reaches a virtually ice-free state.

Regarding the cooling branch, in the OCN-only experiment the ice sheet remains in its virtually ice-free state as expected, given that although oceanic temperatures decrease, it is impossible for the ice sheet to grow without a decrease in atmospheric temperatures. In contrast, the ATM-only experiment follows the control run almost exactly until ΔTJJA=-7 K, when in the control run the ice sheet recovers the marine sectors and in the ATM-only experiment the warm ocean does not allow the grounding line to advance.

These results confirm the findings shown in Fig. : ocean warming drives the abrupt transition at the first bifurcation point, while atmospheric warming triggers positive surface feedbacks, specifically melt-elevation and melt-albedo feedbacks . In the absence of ocean forcing, an abrupt loss in the northeastern region still occurs, but substantially higher atmospheric temperatures are needed to initiate such a margin retreat.

We next analyze the regional changes in ice-sheet volume by dividing the GrIS into three main regions: north, west, and southeast (Fig. ). This division is based on grouping basins with similar topographical, atmospheric, and oceanic characteristics. The northern region is formed by the basins at high latitudes around the Arctic Ocean, including also the North East Greenland Ice Stream (NEGIS) area basins 1 and 2 in; the western region comprises basins discharging ice into Baffin Bay basins 6, 7 and 8 in; and the southeastern region includes basins with mountain chains, where the ice persists for higher temperatures basins 3, 4 and 5 in.

In Fig. , we can see that while the first tipping point has a regional character, affecting only the northern zone (the only one with dM/dt>10Gtyr-1), the second one is a global tipping point that influences the entire ice sheet. Additionally, when comparing the mass fluxes in the three regions in the quasi-equilibrium simulation with the lowest forcing rate as above (Fig. b–d), we find that in the northern area there is very little accumulation throughout the phase space. In contrast, the southeast area has a higher elevation and the SMB is significantly higher, with positive values even after exceeding the second tipping point.

At about -10 K, the ocean warming triggers sub-shelf melting, leading to a gradual volume decline (see Appendix ). Volume losses are concentrated in the western and southeastern regions, where the ice at the margin is floating and exhibits a higher sensitivity to oceanic forcing at lower temperatures. While these zones experience nearly linear losses up to approximately +2 K, in the northern region – where the ice is grounded and its thickness is higher – the volume remains nearly constant until the temperature anomaly reaches ca. -9.5 K (in this quasi-equilibrium simulation), when the abrupt ice loss occurs. The increased basal melting generates an imbalance that causes grounding-line retreat. Before the tipping point (Fig. ), the ice sheet remains grounded on a flat bedrock, but when retreat begins, the ice-sheet margin retreats across a mostly retrograde slope. There, the grounding line becomes prone to marine ice-sheet instability (MISI) as described by and studied in different areas of Greenland . This triggers enhanced ice flux toward the margin, progressive thinning, transition from grounded to floating conditions, and eventually slightly increased calving, as shown by the spike at -9.5 K in Fig. c (left shaded area). The major retreat is not compensated by precipitation, as accumulation is minimal at these latitudes. This is evident in Fig. c (north), where the SMB is lower than in the southeast and west (Fig. b and d), preventing the ice-sheet recovery in this region.

At the second tipping point, the quasi-equilibrium simulation shows that losses also occur slightly earlier in the northern and western regions, where the decline of the SMB is more notable than in the southeast (Fig. b to d). However, when the ice sheet reaches equilibrium for ΔTJJA=+1.8 K, the feedbacks that have triggered the ice losses result in the collapse of the entire ice sheet. Therefore, once ice loss begins in these regions, if the forcing persists in time, the complete loss of the GrIS becomes inevitable. Finally, as we saw in Fig. b, at 4 K the ice only remains in the southeast zone.