The model used in this study is the three-dimensional thermomechanical ice-sheet model Yelmo coupled with the regional energy-moisture balance climate model REMBO at the ice-sheet surface. This model setup is similar to that of but with a different ice-sheet model, and has also been used in the study of . The model domain covers the entirety of Greenland at a horizontal resolution of 16 km.

A full description of the ice-sheet model is not repeated here. Instead we focus this section on an illustration of the processes present at the base of the ice that are most important for the results of this article and which rely on several parameterizations. The Stokes stress balance equations that solve for the dynamic evolution of the ice sheet are parametrized as a combination of shear flow when the base of the ice is frozen to the bedrock (such that basal velocity is zero) and regions where there is sliding at the base (either for floating ice or ice streams) . Thus, ice streams can be identified in model simulations by regions of grounded ice with nonzero basal velocity.

The basal frictional stress τb that resists basal sliding is modeled using a regularized Coulomb friction law , 1τb=-cb(|ub||ub|+u0)qub|ub|, where ub is the two-dimensional basal velocity vector. The threshold speed u0 = 100 ma-1 allows the basal stress to saturate at high velocities, where it becomes independent of the basal velocity . Its value, along with the empirical parameter q=0.2, is derived from laboratory experiments . The bed friction coefficient cb is a field that depends linearly on the effective pressure N at the base of the ice, 2cb=λN.

The factor λ represents the till strength of the bedrock and depends on the elevation above or below sea level. This factor leads to larger sliding velocities at lower elevations, as the till in these areas is primarily composed of sediments that are softer and more easily deformable. A similar elevation-dependent till strength is also seen in , , and .

The effective pressure N^ differs from the overburden pressure P0 depending on the basal water saturation fraction s∈[0,1], 3N^=N0(δP0N0)s10e0Cc(1-s), following the parameterization of . The parameter δ=0.02 is an empirical scaling factor of the overburden pressure. This means that the effective pressure at the base is equal to 2 % of the overburden pressure when the till is saturated (s=1), and any excess water is considered to be drained from the system . The parameter N0 = 1000 Pa is a reference pressure at reference void ratio e0=0.69, and Cc=0.12 is the coefficient of compressibility of the till. Finally, since the overburden pressure is an upper limit for the effective pressure, the minimum of the two is taken, 4N=min⁡{P0,N^}.

The basal water layer thickness Hw has a maximum value of Hw,max = 2 m, and the saturation fraction is defined as s=Hw/Hw,max. In turn, Hw changes with the basal mass balance b˙g and is removed via a constant drainage rate Cd, 5∂Hw∂t=-ρiρwb˙g-Cd, where ρi and ρw are the densities of ice and water, respectively, and 6b˙g=-1ρiL(Qb+k∂T∂z|b+Qr).

The function b˙g therefore converts the heat flux at the base to a melt rate via the latent heat of ice fusion, L. This heat flux consists of the heat generated by friction due to sliding of the ice along the base, Qb, as well as the heat conducted into the column of ice above given by the coefficient of conduction of ice k and the temperature gradient of the ice at the base, ∂T∂z|b. In addition to this, there is a geothermal heat flux boundary condition Qgeo imposed 2 km below the surface, and the heat diffuses vertically through the bedrock. The term Qr is then the flow of heat into the ice at the interface of bedrock and ice. The isostatic adjustment of the bedrock under the ice sheet uses the elastic lithosphere-relaxing asthenosphere (ELRA) model with a relaxation timescale of 3000 years.

At the ice-sheet surface, Yelmo is coupled to REMBO . REMBO is a two-dimensional model with vertically integrated equations for energy and moisture balance. These variables are diffused throughout the model domain to represent processes at the dominant synoptic scale. The strength of this diffusion decreases with increasing latitude to represent decreased atmospheric activity at higher latitudes. REMBO also accounts for orography, whereby precipitation rates increase as the horizontal surface gradient increases. The temperature and precipitation fields of REMBO are calculated at 100 km resolution, and a bilinear interpolation is used to input these as boundary fields to the surface of Yelmo.

Ice sheets lose mass through dynamic processes such as ice streaming and calving of floating ice, but also through a negative surface mass balance (SMB). The SMB is the difference between the mass gained by precipitation in the form of snowfall and lost due to ablation, either due to melting, sublimation, or wind erosion. In the model, the SMB is determined by the air temperature at the ice-sheet surface and precipitation rates calculated by REMBO. The SMB can be separated into a positive contribution from precipitation and a negative contribution from surface melt Ms. The latter is calculated using an insolation-temperature melt method (ITM), in which insolation S and albedo αs are explicitly taken into account : 7Ms=ΔtρwLm[τa(1-αs)S+c+λT], where Δt is the length of a day in seconds, ρw is the density of water, and Lm is the latent heat of melting. The atmospheric transmissivity τa=0.46+0.00006zs is a function of the surface elevation zs, increasing absorption of solar radiation with altitude and thereby acting as a negative feedback for large ice-sheet thicknesses. The surface albedo is calculated as 8αs=min⁡(αg+ddcrit(αs,max⁡-αg),αs,max⁡), where αs,max⁡ is the maximum albedo of snow and αg is the albedo of the ground, and ddcrit is the ratio of snow thickness d to a maximum value dcrit. Finally, c and λ are free parameters that fit surface temperature T to present-day melt rates of the GrIS .

To investigate r-tipping of the GrIS, an equilibrated initial state is required. This allows us to disentangle r-tipping from any effects of inertia associated with transient systems approaching equilibrium. The present-day GrIS simulation is initialized using ice-sheet thickness and bedrock topography from version 5 of the BedMachine mapping of Greenland and the ERA-40 climatology as boundary conditions for REMBO, and allowed to relax to an equilibrium state over a 200 ka simulation. This equilibrated initial state is shown in Fig. . A comparison of the surface velocities of this initial state to those of the present-day GrIS is seen in Appendix Fig. . The states of the GrIS at 300 and 400 ka are also saved as initial states B and C, respectively, representing a small perturbation to initial state A. A comparison of the surface elevations and surface velocities of the three initial states is found in Fig. . The use of an ensemble of multiple initial states is not common for ice-sheet simulations, but is implemented to test the robustness of the results and a possible dependence on initial conditions.

The model is externally forced in subsequent experiments by applying a constant summer temperature anomaly to the surface temperature and an annual mean ocean temperature at the boundary of the model domain of REMBO, as in and . This regional summer surface temperature anomaly is approximately equivalent to a global mean temperature increase. Since r-tipping occurs for a forcing temperature less than the bifurcation point value, the latter must first be estimated. To this end, an adaptive quasi-equilibrium forcing (AQEF) function is used. The AQEF is a transient simulation with a variable forcing rate, and is implemented as such: The GrIS is considered to be quasi-equilibrated when the rate of mass loss of the last 100 years of model time is less than 2 Gta-1. While the ice sheet is not in quasi-equilibrium, the forcing is kept constant to allow it to reach quasi-equilibrium. Once quasi-equilibrated, the forcing is increased in an adaptive manner: the longer it takes the model to achieve quasi-equilibrium, the less the forcing is increased. The maximum rate at which the forcing can be increased is 10⁻⁵ Ka-1, and the minimum rate is 0. In this way, the forcing parameter does not increase while the collapse of the ice sheet is occurring, preventing rate-dependent hysteresis .

Once the bifurcation point has been estimated, it is used to determine a range of parameters for the subsequent ramping experiments that are used to assess the possibility of r-tipping. In these ramping experiments, the forcing is increased at a linear rate to some value less than the bifurcation point. Thereafter, the forcing is kept constant and the ice sheet is allowed to equilibrate over 400 ka. An ensemble of simulations is generated by applying different rates of increase and maximal forcing values, allowing the effect of the rate and magnitude of warming to be evaluated. These ramping experiments are also performed starting at each of the three initial states to investigate the dependence on initial condition.

Figure  shows the ice volume as a function of the forcing parameter, the regional summer temperature anomaly, for the three initial states. There is a clear tipping behavior for a forcing of +1.28 K, and it is essentially independent of the choice of initial state. This tipping resembles what is expected for a saddle-node bifurcation.

The retreat of the GrIS during the tipping event is seen in Fig. . Spatially, the retreat begins in the north. The spatial tipping pattern is equivalent to that of . The pattern of ice-sheet loss is also similar to that of (their Fig. 3b) albeit without eventual partial regrowth of the ice sheet after collapse (their Fig. 3c).

The estimated bifurcation point is used to inform the parameter range for the ramping experiments. Since the bifurcation point is located at +1.28 K, the maximal forcing of the ramping experiments are chosen as +1.00, 1.05, 1.10, 1.15, 1.20, and 1.25 K. A value of +1.30 K past the bifurcation point is also included to confirm that tipping does indeed always occur when forced past +1.28 K. The forcing increases linearly at a rate of 10⁻¹, 10⁻², 10⁻³, 10⁻⁴, or 10⁻⁵ Ka-1, up to one of the seven maximal forcing values noted previously. The time series of the forcing for each experiment are shown in Fig. .

Each of the ramping experiments is initialized from one of three states A, B and C, and the entire ensemble consists of 7 × 5 × 3 = 105 members, which are all shown in Fig. . The appearance of tipping in Fig. c to f suggests rate-induced tipping, as tipping is observed for simulations forced to values lower than the b-tipping value of +1.28 K. Tipping also occurs for all simulations forced past the bifurcation point, that is, those forced to +1.30 K. What is absent, however, is some critical rate below which tipping does not occur, implying that r-tipping of the GrIS can occur even for very slow rates of forcing (Fig. l and n).

One prominent feature in this ensemble of simulations is the non-monotonicity in the time before the tipping occurs, hereafter referred to as the “tipping time”, with rate of forcing, magnitude of forcing, and initial condition. For the same initial conditions, longer tipping times are often seen for faster forcing rates. For instance, for initial condition C at a forcing level of +1.25 K, the trajectory forced at a rate of 10⁻² Ka-1 tips around 300 ka, where the trajectory forced at slower rate of 10⁻³ Ka-1 tips earlier, around 100 ka.

In addition, tipping times for a maximum forcing of 1.30 K can be longer than those for 1.25 K. For example, for initial condition B and the slowest rate of 10⁻⁵ Ka-1, the tipping occurs around 250 ka for a forcing of +1.25 K and around 400 ka for +1.30 K. Finally, even for the same magnitude of forcing and rate of forcing, the tipping times are not consistent over initial conditions. For example, the simulations forced to +1.20 K at a rate of 10⁻³ Ka-1 tip at 300, 350 and 250 ka for initial conditions A, B and C, respectively. Overall the tipping behavior seems to be sensitively dependent on the initial state of the system and the rate of forcing.

While investigating the cause of these random tipping times, irregular oscillations in the ice-sheet volume before tipping were observed. These oscillations can be isolated to a single region of the ice sheet: the northern drainage basin, where the ice-sheet also begins its retreat during a tipping event (Fig. a–d). It is hypothesized that the random tipping times are linked to these irregular oscillations. An example of the oscillations for each maximal forcing is shown in Fig. a. For all these simulations, there is an initial loss of mass during the first 50–80 ka of simulation time. Thereafter, for a maximal forcing of +1.00 K, there is a relatively small amount of variability around a steady ice volume of about 3.09 × 10⁶ km3. For maximal forcing values between +1.05 and +1.15 K, the mean ice volume is lower, between 2.9 and 3.0 × 10⁶ km3. The ice sheet experiences volume fluctuations with magnitudes on the order of 0.05 × 10⁶ km3 and periods between 8 and 30 ka. Finally, the simulations for maximal forcing of +1.20 to +1.30 K in Fig.  retreat to a much smaller ice volume, representing a collapsed GrIS as seen in Fig. c.

Figure b shows these oscillations for a fixed forcing magnitude of +1.15 K and varying rates of forcing, showing that the variability is not uniquely determined by the forcing magnitude, but is still similar in amplitude and period. Tipping for one of the rates (10⁻³ Ka-1) is also observed, albeit not for the fastest rate, as can be expected for r-tipping.

Two simulations at a forcing of +1.00 and 1.05 K are compared, as this represents the onset of the large-amplitude variability seen in panel (a) of Fig. . Further examination of the oscillations show they are a result of two ice streams that alternate between periods of stagnant and rapid basal sliding. These are the Humboldt and Petermann glaciers, which lie in close proximity to each other on the northern ice-sheet edge (Fig. ). These glaciers are both regions of fast-flowing ice and behave as ice streams in the model simulations.

The first difference to be seen is the extent of the ice sheet in this region. For a forcing of +1.00 K, the ice-sheet margin is such that the Humboldt ice stream (HIS) is marine-terminating, with the Petermann ice stream (PIS) covering the Petermann fjord. The ice sheet in this case is termed “unretreated”. In the simulation with a forcing of +1.05 K, the ice sheet extent is much reduced. From the mean ice-thickness profiles, the ice margin is almost 100 km further inland (Fig. a, c, d and f). The HIS is no longer connected to the ocean, although the PIS terminates at the Petermann fjord. We refer to this as the “retreated” configuration, with the two separate ice streams designated as the retreated HIS and retreated PIS.

The temporal behavior of the unretreated and retreated configurations is compared by taking the spatial mean in two grid boxes, one containing the PIS/retreated PIS and the other the HIS/retreated HIS, as seen in Fig. a and j. For the unretreated case, the basal velocities in the two different ice streams behave quite differently. The PIS is in a state of steady flow of around 100 ma-1, facilitated by a constant basal water layer thickness. The HIS, in contrast, alternates between near zero basal movement and sliding velocities of 100 ma-1. The periods of stagnation correspond with an increase of basal friction caused by a reduction in the water content of the till due to freezing or drainage. While the basal velocities are minimal, the ice thickness increases, establishing a “build up” phase. The subsequent loss of mass due to rapid ice streaming is a “surge” phase. The period of these oscillations varies between approximately 5 and 9 ka. This steady cycle of mass gain during the build-up phase and loss during the surge results in an oscillation in mean ice thickness between 100 and 200 m in amplitude. The PIS also shows a smaller alternation in ice thickness with a similar temporal pattern while maintaining constant ice-stream flow, suggesting the thickness variations are influenced by the oscillations of the nearby HIS.

In the retreated configuration, the oscillations of ice thickness and basal velocity have larger amplitude and longer period, as mentioned previously. While the exact location of the retreated PIS and retreated HIS differs slightly in all the simulations showing oscillations, their existence is robust among the ensemble. In contrast to the unretreated PIS, the retreated PIS is no longer in a steady-flow state and instead displays the same build-up/surge variability as the unretreated and retreated HIS.

Two general patterns emerge for the oscillations of the retreated configuration. The first is one of a long, asymmetric build-up and surge event. An example is seen around 225 to 260 ka in Fig. c, e, g, and i. While the basal velocity in the retreated PIS switches abruptly between the build-up and surge phases, the basal velocity in the retreated HIS increases gradually. This accelerates mass loss until a point where the basal water layer thickness rapidly decreases in both ice streams when the ice thickness reaches its minimum, resulting directly from the thermomechanical coupling. The second pattern is that of short oscillations with a period of around 8 ka. These are seen in the last 60 ka of the time series in Fig. c, e, g, and i. The basal velocity in the retreated PIS abruptly switches between maximal and minimal, with corresponding drops in basal water-layer thickness when the velocity is minimal. In the retreated HIS, the till remains saturated with water. However, the basal velocity, and thus the flow, is not steady. It increases during the surge and decreases during the build-up of the nearby retreated PIS. This indicates an influence of the retreated PIS on the retreated HIS.

Thus the distinguishing factor between the two patterns is that during the short oscillations, the retreated PIS is in a build-up/surge mode, whereas the retreated HIS has a constant till saturation. This causes the retreated HIS to be in a steady-flow state, while still experiencing small oscillations due to the proximity to the retreated PIS. On the other hand, both the retreated HIS and the retreated PIS are in the build-up/surge mode during the longer-period events. A third mode is seen in some simulations that is shown in Fig. , and is identified by a long period of minimal ice volume. This mode corresponds to a steady flow state of intermediate mean basal velocities of around 100 ma-1 in both the retreated HIS and the retreated PIS.

The differences between the behavior of the PIS and the HIS in both configurations can be explained by slight variations in their topography and forcing fields, as seen in Fig. . The constant geothermal heat flux in the PIS is slightly higher than in the HIS, with a spatial average of 48.75 and 48.25 mWm-2, respectively. During the periods where oscillations occur, the basal mass balance in the PIS varies much more than in the HIS, primarily due to the larger frictional heating (Qb in Eq. ) which is a direct result of larger basal velocities in the PIS while it is streaming (Fig. g). The bed elevation of the HIS is lower than the PIS, leading to lower average basal friction and thereby making it more prone to the steady-streaming state. Finally, the HIS receives more precipitation than the PIS on average, about 0.28 m.w.e.a-1 compared to 0.26 m.w.e.a-1, respectively. This might allow for the mass lost by the HIS during streaming to be better balanced by the accumulation, as opposed to the PIS, where lower accumulation means the ice thickness can only regrow once the streaming has quiesced.

Glacial isostatic adjustment (GIA) has been demonstrated to cause oscillations in ice sheet volume , and the relaxation time of 3000 years is relevant on the timescales of variability observed in the HIS and the PIS. Figure  shows the magnitude of variation in the altitudes of the bedrock and ice-sheet surface as well as the ice thickness. The maximum bedrock altitude in both the HIS and the PIS is close to 200 m, which is reached at the end of a long surge event (around 260 ka in Fig. ). However, this value is only approximately 40 m greater than the minimum bedrock altitude at the beginning of the surge (around 230 ka in Fig. ), meaning the bedrock is not the primary contributor to the difference in the surface altitude during the period of variability. In contrast, the ice thickness changes by about 600 m for the HIS and 500 m for the PIS during this time. So, while there is some effect of the GIA, the magnitude of the uplift of the bedrock is never so large as to affect the build-up/surge variability of the ice sheet. If the regrowth of the ice sheet after a surge were triggered by the GIA bringing the bedrock to an altitude that allows for a positive SMB, we may expect to see a period of minimal ice thickness during which the bedrock is uplifting, and at a certain altitude the ice thickness to start increasing again. Instead, we see the GIA simply lagging the ice thickness. The ice regrowth instead corresponds to a minimal basal velocity, i.e. an end of the surging. So, while the GIA does have some influence on the ice surface elevation, it does not set the pace of the oscillations.

To assess whether the lack of predictability of the tipping is due to ice-stream oscillations or due to other factors, we change the value of a parameter in the basal friction law to eliminate the tendency of ice streams to oscillate and investigate the consequences. As described previously, the ice-stream oscillations are surges in the basal velocity of the ice stream due to the thermomechanical coupling at the base of the ice sheet. To remove the oscillations but maintain the ice stream, the basal velocities in the region of interest need to be lower but non-zero. For lower basal velocities, the mass lost due to streaming is closer to the accumulation rate, bringing the ice stream to a steady flow state . To achieve this, the value of δ in Eq. () is increased. This raises the minimal effective pressure and thereby increases the basal frictional stress.

Increasing the value of δ and thereby the basal frictional stress has the effect of making the entire ice sheet less dynamically active, reducing the amount of mass loss due to ice streaming and calving. This means that the temperature forcing required to induce the collapse of the ice sheet is greater for larger values of δ.

Simulations with the original value of δ = 0.02 as well as increasing values of δ to 0.04 and 0.10 are shown in Fig. . As δ is increased, we note an increase in the oscillatory period until the oscillations disappear completely. As this factor affects ice streams across the entire ice sheet, the bifurcation point for each of these parameter values is different. Specifically, the bifurcation point increases as δ increases, as the ice sheet is losing less mass due to lower ice stream velocities.

For δ = 0.02, all (no) simulations with the strongest (weakest) forcing tip, but for intermediate maximum forcing values the tipping times do not decrease monotonically with the forcing rates. For a large enough value of δ = 0.1, the ice-stream oscillations disappear and the tipping time for a maximal forcing of +2.10 K occurs at approximately the same time for each rate. That is, the dependency on the forcing rates disappears and the tipping is much more predictable. This suggests that the ice-stream oscillations introduce a delay in the tipping.

Accelerating mass loss is typically seen in marine-terminating outlet glaciers due to their sensitivity to oceanic forcing , suggesting rate-induced effects may be more prevalent in these areas due to the fast timescale of the marine ice sheet instability . However, future loss due to negative SMB forced by increasing atmospheric temperatures could outweigh that of ice-sheet dynamics due to ocean forcing . In addition, atmospheric forcing may be more critical in the north of Greenland compared to the south , which is reflected in the model simulations. The ice extent shown in Fig.  indicates that mass loss does not start in regions with many marine-terminating outlet glaciers, specifically the southeast . In fact, many remain after the tipping has completed. This indicates that the possibility of r-tipping of the GrIS primarily by oceanic forcing may not be relevant.

The remainder of the ice sheet interacts predominantly with the atmosphere, and the mass loss occurs either through surface melt or dynamically through ice streams which may not necessarily be marine-terminating, meaning the activation of ice streams does not depend on oceanic basal melting. The answer to whether the large-scale mass loss of the GrIS can be influenced by the sensitivity of these fast-moving ice streams to the rate of atmospheric warming is not entirely clear on the basis of the results of this study. There is no apparent critical forcing rate for which r-tipping occurs, at least not for a rate faster than 1 K of warming over 100 ka. The simulations indicate that the tipping behavior is affected by the rate of forcing but in a non-monotonic manner. However, the long tipping times are not explained by non-monotonic r-tipping , as we would expect any tipping events to occur at roughly the same time at a given forcing magnitude. While it may be valid to claim that r-tipping does indeed occur since the final state of the GrIS depends on the rate of forcing in some way, it is important to approach this point with nuance.

Ice streams not only represent a mechanism of rapid mass loss in ice sheets, but have also been shown to be a source of internal periodic variability in models. Periodic behavior of ice masses can be seen in glaciers that are confined to some topographical valley and/or have a basal slope, for example . Their reduced spatial extent also means their periodic behavior is on a much shorter time scale of decades to centuries and thus directly observable. While the same cannot be said of oscillating ice streams with periods of millennia, present-day ice streams have been demonstrated to be accelerating or decelerating , due to thermomechanical coupling at the base of the ice sheet, see also . Studies of oscillatory behavior in ice sheets include parameterized models and comprehensive ice-sheet models with both idealized geometries and realistic topographies . Additionally, some studies include a coupling to additional components of the climate system . Similarities between the oscillations seen in this paper and those observed in paleoclimate modeling studies are discussed in Appendix C.

Large-scale oscillations in the GrIS ice volume have been reported in the modeling studies of and , but the characteristics of those oscillations are an order of magnitude higher than what is seen in the present work, having periods over 100 ka and amplitudes between 0.4 and 2.1 × 10⁶ km3. The values of mantle viscosity and atmospheric lapse rate used in Yelmo and REMBO are 1 × 10²¹ Pas and 6.5 Kkm-1, respectively, which places it in the oscillatory regime of under moderate temperature forcing. As all three models use an ELRA model of glacial isostasy, the primary difference between the observed behavior in those studies compared to the present work is the regional atmospheric model. In , the SMB is calculated using a positive degree day (PDD) method that depends on the surface temperature which is calculated locally as a function of the altitude. In , the SMB is calculated offline using the Community Earth System model, and then adjusted via a lapse rate during the runtime of the ice sheet model to reflect changes in ice sheet topography. However, there is no back-coupling of the ice sheet to the atmosphere. In contrast, REMBO allows for diffusion of energy and moisture, which decreases the local strength of the melt-elevation feedback, thereby also reducing the sensitivity to the GIA feedback.

Common to ice-sheet modeling is the use of a single initial state for a given combination of parameters rather than an ensemble. The use of an ensemble of states for investigating the evolution of an ice-sheet model is not too common, appearing in the studies of and to be able to account for the variability of the atmosphere and ocean when forcing the ice sheet. This is distinct to the results of this study, where the model is not coupled to an external climate model beyond a simple, deterministic diffusive energy and moisture balance atmosphere and thus the variability is internal to the ice sheet itself. That is, in those studies there is an external chaotic forcing that necessitates the use of an ensemble, whereas our model setup experiences constant external forcing after the temperature ramping period has ended. Despite the constant external forcing, we see simulations at a given forcing magnitude exhibit significant deviations from each other, indicating a sensitive dependence on the initial state which is a hallmark of an internal chaotic mode of variability.

While it is outside of the scope of the present study to formally prove that this mode of variability is indeed chaotic, there are some features that can be neatly interpreted as manifestations of chaos, particularly the unpredictable tipping times which are discussed in Sect. 4.3. Whether the chaos is a genuine physical phenomenon or simply a result of parameterization is under contention and will require further investigation. As this irregular variability of surging ice streams been reported in many other ice-sheet modeling studies, its existence is not entirely unfounded . What is unique to this study is the period and amplitude of the oscillations in the GrIS, as well as its switching between two different modes due to a coupling of nearby ice streams. The study of examines this variability using a simplified model, establishing that this chaos can arise in a system with no spatial extent.

An important caveat of the results of this paper is that oscillations in the model are highly sensitive to the basal friction parametrization. The physical mechanisms at the base of the ice that allow for ice streaming are not perfectly understood, and therefore the behavior of ice streams in models can depend heavily on the specific parameterization . In situations where ice streams are marine-terminating, buttressing of ice shelves can serve to regulate the pace of mass loss, and thus decrease their sensitivity to the basal friction . Due to the lack of floating ice shelves that might provide buttressing in the HIS and PIS, it is to be understood that the oscillations are highly sensitive to the basal parameterization. Indeed, this is also reflected in the results of changing the parameter value of δ in Sect. 3.5, which is further discussed in Sect. 4.4. Further limitations are discussed in Sect. 4.5.

Due to the large mass of the ice sheet and the long timescales associated with surface mass balance processes, there is considerable inertia when transitioning from ice-covered to ice-free, especially when the system is kept very close to the tipping point. This causes the system to spend a long time in the ice-covered state after the bifurcation point has been passed, before eventually collapsing. If the ice sheet has an internal mode of chaotic variability, then the time spent before the collapse is sensitively dependent on the configuration of the ice sheet when the bifurcation point is crossed, and these transitions are called chaotic transients . The long and unpredictable tipping times seen in this study are proposed to be due to such chaotic transients, and they can obscure the detection of r-tipping.

The interplay between chaotic transients and r-tipping has been previously reported in a study by for the AMOC. In their study, trajectories forced at different rates are brought close to the basin boundary between the stable “AMOC on” state and a chaotic saddle that separates it from the stable “AMOC off” state. This basin boundary is fractal, such that different rates of forcing may land the system differently on the saddle, where it experiences a chaotic transient resulting in transitions to the AMOC off state that are non-monotonic in rate. Alternatively, chaotic transients can arise due to a “ghost attractor” that appears for forcing values just past a bifurcation point, where the drift away from the ice-covered state, which is now no longer a stable fixed point, is slow . The oscillatory behavior of the build-up/surge variability can generate unstable periodic orbits that “collide” with the chaotic attractor of the ice-covered GrIS at the bifurcation point, generating the ghost attractor. Thus the chaotic transients in this study may either arise when crossing a chaotic saddle before the bifurcation point due to r-tipping, or due to being caught in a ghost attractor after passing a bifurcation point (i.e. b-tipping).

To answer the question of whether the GrIS experiences r-tipping in the ramping experiments, we must examine the chaotic transients. The system-specific critical forcing value was estimated as +1.28 K, and multiple simulations were observed to experience tipping for a forcing value less than this but with a fast rate of change of the forcing, implying r-tipping. However, the adaptive quasi-equilibrium forcing itself may exhibit a chaotic transient, which would result in overestimation of the bifurcation point. Comparing the simulations ramped to +1.25 K (below the assumed critical value) and +1.30 K (above the assumed critical value), we note that they are similar in oscillatory amplitude and period before tipping. If the oscillations are due to crossing a chaotic saddle before the assumed bifurcation point of +1.28 K, and therefore r-tipping, then they must be qualitatively different from the behavior of the system after the bifurcation point is passed. Since the variability of the simulations at the two forcing values are qualitatively similar in period and amplitude, we conclude that they are generated by similar non-attracting sets . This implies that the critical value is not between +1.25 and 1.30 K as estimated by the AQEF. This reasoning can be extended to all parameter values that result in tipping (i.e. +1.10 to +1.30 K). Thus, either the bifurcation point is between +1.05 and 1.10 K and all transients are due to a ghost attractor, or the bifurcation point is larger than +1.30 K and all of the transients are due to an r-tipping through a chaotic saddle.

The behavior of the chaotic transients in the present study more closely resembles those due to a ghost attractor after a bifurcation than due to r-tipping through a chaotic saddle. First, scaling laws indicate that the lifetime of the chaotic saddle, and thereby the tipping time, should increase as the maximal forcing approaches the bifurcation point . In contrast, the mean lifetime of the chaotic transients, 〈τ〉, scales with the magnitude of the parameter p past the bifurcation value pb , 9〈τ〉∼(p-pb)-γ, where γ is system-specific parameter value. This scaling of the tipping times mirrors the pattern seen in Fig. . Secondly, if r-tipping was present, we may expect to see tipping not occur below some critical rate, which is absent in the rates examined in this study. While it is not possible to evaluate the limit of the rate going to zero in finite simulation time, the rates investigated cover a wide range of physically realizable values. As the lowest rate of 10⁻⁵ Ka-1 acts over the same time span as changes in orbital forcing, rates slower than this are not relevant.

Removing the oscillations by increasing δ makes the tipping more predictable. At a value of δ = 0.1, the tipping occurs at approximately the same time for both the fast and slow rates of forcing increase. This is in contrast to the simulations with δ = 0.02, where oscillations are present and the tipping at a given maximal forcing can vary by tens to hundreds of thousands of years depending on the rate and in a non-monotonic manner. This corroborates the hypothesis that the oscillations cause the delay in tipping. For an intermediate value of δ = 0.04, tipping to an ice-free state only occurs for the largest forcing magnitude of +1.85 K. Interestingly, two simulations at lower forcing magnitudes see a tipping to an intermediate ice-sheet state. Such a state has been seen before in studies such as and , but is not explored further in the present paper.

Increasing δ also reveals some interesting interplay between the parameterization that allows for the oscillations and the tipping. As increasing δ decreases the ice-stream velocity, the ice sheet loses less mass dynamically and thus the forcing magnitude required for tipping increases. However, the tipping is now no longer delayed by the oscillations. Since the lifetime of the transients can be over 100 ka, it may be the case that the tipping occurs later for a lower forcing value. As a result, the oscillations simultaneously serve to lower the value of the bifurcation point as well as to increase the time before tipping occurs.

The caveat of the results of this paper is that they are built on numerical considerations that may be highly sensitive to modeling choices. The coarse resolution of the ice-sheet model results in many small-scale processes, such as those of smaller ice streams or the topographic roughness of outlet stream such as the Petermann fjord, which may introduce additional oceanic forcing that can influence the retreat of the ice extent . This is particularly important for the PIS, which extends through a fjord and thus its retreat is sensitive to model resolution. However, the retreated HIS and the retreated PIS observed in the model does not extend to the ocean, meaning that the chaotic variability observed is due to the ice dynamics and basal hydrology rather than due to connectivity to the ocean and the numerical instabilities associated therewith. Additionally, the retreated HIS and PIS lie on a bed topography that has much lower relief than that of the fjords at the extent of the unretreated PIS , indicating it may not be as sensitive to model resolution . That being said, vital processes such as calving in fjords cannot be properly represented on coarser resolutions, so whether the PIS would retreat in a way that allows for sustained oscillations is questionable. More directly, it has been demonstrated that the mechanism of the ice stream oscillation itself is resolution-dependent .

The geothermal heat flux is a boundary condition that is highly uncertain but has a critical effect on the behavior of these surges, as it sets the pace of basal melting and meltwater production that is critical to the transition of an ice stream from the build-up to the surge mode . For example, report much larger geothermal heat fluxes in the region containing the PIS and HIS, which would tend the ice streams towards a steady-streaming mode, leading to a more predictable collapse of the GrIS by avoiding chaotic transients entirely.

Uncertainties in precipitation fields are yet another limitation, as accumulation rates are vital to maintaining the build-up/surge pattern. If accumulation is too large, the thinning of ice stream due to the surge might not decrease the ice thickness enough to cause re-freezing at the base, maintaining a steady-streaming mode . A poor estimation of the precipitation rates in the northern region of Greenland can result in the build-up surge variability never being realized.

The lack of coupling to an oceanic model decreases the impact of marine-terminating glaciers on the collapse of the ice sheet, which are an important but poorly constrained source of ice sheet mass loss . While there is oceanic forcing applied in the ramping experiments, it is spatially constant, which may underestimate the warming in the southeast and overestimate it in the north . A proper implementation of oceanic forcing could induce an ice-sheet collapse that begins in the south, invalidating the presence of the oscillations that delay the tipping. This mechanism may also re-introduce the possibility of r-tipping of the GrIS due to oceanic forcing. There is also no bi-directional coupling between the ice sheet and the ocean, which may serve as a negative feedback that can prevent the oscillations. This could results in dynamic mass loss of the GrIS that outpaces the period of the oscillations, rendering their effect on the time before tipping invalid.

Another important clarification is that the chaotic transients observed only appear for a very small range of forcing magnitudes past the bifurcation point. This is due to the strong power-law scaling of the chaotic transient lifetime (Eq. ), meaning that many other modeling studies that investigate the tipping of the GrIS may only consider temperature forcing values outside of this narrow band. Ultimately this means the period, amplitude, or even appearance of these surges that generate the chaotic variability are highly dependent on the model configuration and sensitive to parameterization. That is, any misrepresentation of the dynamics due to the numerical modeling considerations might change how often and at what thresholds these surges occur. However, their appearance alone is worthy of investigation and is the purpose of the present study.