Risk of tipping the overturning circulation due to increasing rates of ice melt

Significance

Ongoing greenhouse gas emissions put elements of the Earth system at risk for crossing critical thresholds (tipping points), leading to abrupt irreversible climate change. Measures for reducing emissions should keep Earth in the safe operating space away from tipping points. Here we show that increasing rates of change of ice melt can induce a collapse of the Atlantic Meridional Overturning Circulation in a global ocean model, while no critical threshold in ice melt is crossed and slower increases to the same level of ice melt do not induce tipping. Moreover, the chaotic dynamics of the climate make such a collapse hard to predict. This shows that the safe operating space of the Earth system might be smaller than previously thought.

Abstract

Central elements of the climate system are at risk for crossing critical thresholds (so-called tipping points) due to future greenhouse gas emissions, leading to an abrupt transition to a qualitatively different climate with potentially catastrophic consequences. Tipping points are often associated with bifurcations, where a previously stable system state loses stability when a system parameter is increased above a well-defined critical value. However, in some cases such transitions can occur even before a parameter threshold is crossed, given that the parameter change is fast enough. It is not known whether this is the case in high-dimensional, complex systems like a state-of-the-art climate model or the real climate system. Using a global ocean model subject to freshwater forcing, we show that a collapse of the Atlantic Meridional Overturning Circulation can indeed be induced even by small-amplitude changes in the forcing, if the rate of change is fast enough. Identifying the location of critical thresholds in climate subsystems by slowly changing system parameters has been a core focus in assessing risks of abrupt climate change. This study suggests that such thresholds might not be relevant in practice, if parameter changes are not slow. Furthermore, we show that due to the chaotic dynamics of complex systems there is no well-defined critical rate of parameter change, which severely limits the predictability of the qualitative long-term behavior. The results show that the safe operating space of elements of the Earth system with respect to future emissions might be smaller than previously thought.

Catastrophic and unexpected shifts in nature and society have been ubiquitous throughout history. In complex, nonlinear systems they can arise when a critical threshold in the boundary conditions is crossed. This is referred to as a tipping point and is of specific concern in the context of anthropogenic climate change. Several elements of the climate system have been identified to be at risk for crossing a critical threshold with increasing greenhouse gas concentrations, including Arctic sea ice, the Amazon rain forest, Boreal permafrost, and the Atlantic Meridional Overturning Circulation (AMOC) (1, 2). Tipping points are most often associated with a bifurcation or attractor crises, i.e., a loss of stability of a stable system state (a so-called attractor). A catastrophic shift occurs as a system parameter is changed beyond the bifurcation point and the system state evolves to another attractor. Consequently, there have been substantial efforts to assess whether elements of the Earth system indeed possess such critical thresholds (3–5) and whether one can determine the proximity of the current state to the threshold (6, 7). Furthermore, the existence of generic precursors or early-warning signals preceding tipping points has been explored (8–10).

While these are important issues to address, it is now known that catastrophic transitions to undesired attractors can be induced when the parameter is changed at a rate exceeding a certain critical value, even though the parameter does not cross a critical threshold (bifurcation point) (11–13). This is known as rate-induced tipping and in the context of real-world systems like the climate, such transitions further limit the range of parameters that span the safe operating space. Here, we focus on a potential collapse of the AMOC with increasing freshwater input into the North Atlantic. This is a concern since there is observational evidence for accelerating meltwater runoff from Greenland (14–16), as well as a slowing down of the AMOC (17). As illustrated in Fig. 1A, if the climate system displays rate-induced tipping, the increasing rates of change of freshwater runoff could push it into a regime (gray region) that has been thought safe otherwise. Previous evidence for rate dependency of an AMOC collapse comes from conceptual ocean box models (18, 19) as well as two-dimensional ocean and climate models with varying freshwater (20) or greenhouse gas forcing (21).

Fig. 1.

Tipping of the ocean circulation. (A) Observational evidence for accelerating Greenland meltwater runoff (Materials and Methods) in comparison to conceptual boundaries of the safe operating space of the ocean circulation. (B) Maximum value of the meridional stream function, zonally averaged over the Atlantic basin, in a continuous model simulation. The forcing is increased to $F_{max}=0.31$ in small increments within 300 y each (black curve) and subsequently decreased to zero again (orange curve). (C, Inset) Time series of the forcing parameter $F$ in a parameter shift experiment with ramping duration $T=160$ y. (C, full plot) Corresponding values of the AMOC maximum (time is color coded). The black circles are the values shown in B. (D) Same as C but for $T=140$ y.

Here we show that rate-induced transitions are indeed a concern for the climate system, by demonstrating explicitly the existence of a rate-induced collapse of the AMOC in a three-dimensional model of the global thermohaline circulation with time-dependent freshwater forcing. In addition, by performing large ensemble simulations, we show that there are fundamental difficulties in defining and determining safe rates of parameter changes in the presence of chaotic dynamics, as is the case for the climate system. This makes the boundary separating safe from unsafe climate conditions fuzzy (striped area in Fig. 1A).

Results

Hysteresis of the Ocean Circulation.

The ocean model employed here has an equilibrium with a stable and vigorous AMOC and a maximum transport of roughly 10 Sv (1 Sv $\equiv 10^6{\text{m}}^3\cdot {\text{s}}^{-1}$) under present-day conditions. It features small ($<$1 Sv) quasi-periodic multidecadal oscillations, which may be related to known multidecadal variability in the Atlantic (22). We perturb this ocean state by gradually increasing a freshwater anomaly $F$ uniformly over the North Atlantic convection regions (Materials and Methods). As we increase $F$ slowly to $F_c\approx 0.23$ Sv, the AMOC strength decreases gradually (Fig. 1B). Until this critical value $F_c$, the structure of the overturning remains largely intact and similar to present day. As $F$ is increased further, the circulation jumps abruptly to a very weak state. The northern cell of the overturning circulation is collapsed completely, and a reverse circulation cell expands from the south (SI Appendix, Fig. S5). Apart from a 2-practical salinity unit freshening of the surface waters, this AMOC collapse causes a cooling of up to 3 °C in the North Atlantic region due to reduced northward heat transport (SI Appendix, Fig. S6). As the freshwater anomaly is slowly decreased again there is an abrupt jump from a weak to a strong AMOC (orange curve in Fig. 1B). However, this occurs at different $F$ than the shutdown. Thus, the model shows hysteresis with respect to changing freshwater forcing. Hence, there is a window of parameter values, where two stable configurations of the AMOC exist. At the edges of this window, one of these two states loses stability. This indicates an underlying structure similar to a fold–fold bifurcation, which has been shown to exist in a range of ocean models, from very simple (23), to higher dimensional and comprehensive (24). The existence of bistability and tipping points of the AMOC in the present-day climate is still an active matter of discussion, and in ocean and climate models this depends on boundary conditions (25). The ocean model employed here uses simplified boundary conditions regarding surface heat, water, and momentum exchange. By neglecting wind stress forcing, we isolate the thermohaline component of the ocean circulation. A sensitivity analysis with respect to different boundary conditions is given in Materials and Methods and SI Appendix. While we cannot answer the question of bistability in the real climate, our aim in the following is to demonstrate behavior that could be potentially observed in addition to more traditional notions of tipping points.

Rate-Induced Tipping.

Given a parameter regime supporting multiple equilibria, we test whether rate-induced tipping can occur. An ensemble of simulations is branched off from the hysteresis experiment at $F_1$ in the bistable region, whereafter the parameter is ramped linearly to $F_2<F_c$ at a rate $r=\left(F_2-F_1\right)\cdot T^{-1}$ and then held constant to allow the model to equilibrate (Fig. 1C). In the following we refer to the ramping duration $T$ instead of the rate. Fig. 1 C and D shows the AMOC maximum as a function of $F$, for two parameter shift experiments with different $T$. When choosing $T=160$ y, the AMOC evolves toward the state that is also obtained in the slower hysteresis experiment. It thus tracks the moving present-day attractor that it was initialized on. In contrast, for $T=140$ y the model evolves toward the state of collapsed AMOC that is obtained in the hysteresis branch with decreasing parameter. From the quasi-stationary hysteresis diagram (Fig. 1B), such a rate-induced transition prior to crossing the tipping point would not be expected. Whether this can occur depends on the movement of the manifold separating two attractors (the so-called basin boundary), relative to the moving, desired attractor during a parameter shift. In Fig. 2 we illustrate this mechanism for rate-induced tipping with a conceptual model. This is a minimal dynamical system of two state variables $x$ and $y$, which closely resembles the behavior in the ocean model (compare Fig. 2 C and D). In this model it is seen that the condition for rate-induced tipping to occur is if the basin boundary (red manifold) moves such that it crosses states on the initial, desired attractor (dashed black line). See Fig. 2 legend for more details and Ashwin et al. (26) for a rigorous discussion of rate-induced tipping.

Fig. 2.

Dynamical mechanism for rate-induced transitions in a conceptual model. For the forcing parameter $0<\beta <{\beta }_c$ there is bistability of a present-day AMOC limit cycle (gray surface) and a collapsed AMOC (black line), separated by the unstable edge state (red surface). While the AMOC collapses in a bifurcation at ${\beta }_c$, a rate-induced collapse is possible when shifting from ${\beta }_1$ to ${\beta }_2$, depending on the model parameter $\gamma$ (Materials and Methods). (A) For $\gamma =0$ the model tracks the limit cycle for any rate of parameter shift. The orange and blue lines are trajectories for a fast and a slow rate, respectively. (B) For $\gamma =3$, there exist states on the limit cycle at ${\beta }_1$ (purple arch) that lie across the basin boundary at ${\beta }_2$. These tip to the other attractor in an instant parameter shift. Indeed, the orange trajectory crosses the basin boundary and tips to the collapsed AMOC, while the blue trajectory tracks the limit cycle. The green trajectory is obtained for a critical rate and evolves to the edge state. (C) Time series corresponding to the trajectories in $x$-$y$-$\beta$ space shown in B. (D) Analogous time series of parameter shift simulations with the ocean model using three different ramping durations.

Critical Rates of Change in Forcing.

Whereas the previous result shows that knowledge of and staying away from a tipping point of the AMOC do not always prevent a catastrophic transition, one can instead hope to constrain a critical rate of parameter change that separates the scenarios of tipping to the undesired and tracking the desired attractor. While Fig. 1 C and D suggests a corresponding critical $T$ in between 140 and 160 y, we show instead that there is no sharply defined critical rate. Fig. 3A shows an ensemble of simulations with a range of different rates. The states are color coded such that blue corresponds to a present-day, vigorous AMOC, and red corresponds to a collapsed AMOC. These simulations result in three different qualitative behaviors, denoted as tipping outcomes. While most realizations either tip to the collapsed attractor or track the moving present-day attractor, a few realizations, indicated by yellow bars, do not settle to either of the two attractors during the simulation time. We believe that they arise when the system state gets close to the stable manifold of a saddle, also known as edge state (27), and gets attracted. Such trajectories are known as maximum canards and occur for parameter shifts at critical rates (12), as demonstrated for the conceptual model in Fig. 2. We find that these realizations can spend 10,000 y or more in the vicinity of the edge state. However, they will eventually decay to one of the two attractors (SI Appendix, Fig. S4).

Fig. 3.

Dependence of tipping on rate and initial conditions. (A) (Top) Ensemble of model simulations where the freshwater forcing parameter is increased linearly to $F_2=0.193$ over different durations $T$ starting after a 600-y spin-up at constant forcing $F_1=0.219$. Shown are color-coded time series of the AMOC maxima of the ensemble members. (Bottom) The outcome of the parameter shift is given as a bar plot. (B and C) Dependence on initial conditions for a fixed ramping duration of $T=70$ y. (B) (Bottom) The time series of a spin-up realization, given as 2-y averages of the AMOC maximum. The ensemble simulations branched off at the corresponding time points are shown above as time series of the AMOC maximum. The black line indicates when the parameter shift starts in the individual simulations. (C) As in B, but for a refined ensemble based on a 2-monthly sampling of a 10-y segment of the spin-up.

From Fig. 3A it is clear that there is no well-defined critical rate separating tipping from tracking realizations. For $T>150$ y all realizations track. For 50 y $<T<$ 150 y there is a mixed pattern with some realizations tipping, some tracking, and others visiting the edge state. While for $T<50$ y most realizations tip, this is still not guaranteed, since we find a realization for $T=10$ y that evolves toward the edge state. Nevertheless, the probability of tipping increases with the rate, comparable to systems with added noise (28). Importantly, a nonsharp and even nonmonotonic rate dependence can also arise in deterministic chaotic systems, as has been recently shown in a low-dimensional dynamical system (29). Furthermore, for attractors that are not fixed points, there can be both tipping and tracking initial conditions on the attractor for a given rate, which has been referred to as partial tipping (30). Thus, critical rates vary across initial conditions.

Sensitive Dependence on Initial Conditions and Predictability.

Next, the mechanisms that give rise to the pattern observed in Fig. 3A are explored by testing the sensitivity of the tipping outcome to different initial conditions. To this end, we create an ensemble of initial conditions sampled along the attractor at $F_1$, which are ramped to $F_2$ at fixed $T=70$ y (Materials and Methods). This represents an intermediate regime of $T$ where the system neither surely tips nor tracks. Fig. 3B shows the initial conditions along the attractor, as well as color-coded time series and a bar plot indicating the tipping outcome. Depending on the initial conditions, tipping and tracking realizations, as well as realizations that evolve to the edge state, exist. Thus, the model displays partial tipping. The pattern of tipping and tracking initial conditions on the attractor is nontrivial and has no apparent periodicity. Nearby initial conditions do not seem to have a similar outcome. This is likely a result of the chaotic dynamics, where initially nearby trajectories diverge exponentially during the ramping time. At the time when the parameter shift is over they may end up on either side of the basin boundary for $F_2$. In Fig. 3C we show an ensemble that is even more closely spaced on the attractor. Again, adjacent initial conditions often have different tipping outcomes and the obtained pattern is consistent with a random, uncorrelated sequence (Materials and Methods). Shortly after the parameter shift for every ensemble member has started (year 15 in Fig. 3C), the ensemble standard deviation of the AMOC maximum is 0.0073 Sv. Seventy years later it is 0.1139 Sv, and in year 150, shortly before the first realizations tip, the trajectories are largely uncorrelated and the standard deviation is 0.3838 Sv. Even though the decorrelation time is larger than $T$, this divergence of trajectories until the end of the parameter shift will influence the tipping outcome of initially nearby trajectories (see SI Appendix, Fig. S3 for more details).

Fractal Basin Boundaries.

An additional cause for the sensitive dependence on initial conditions could be a fractal or riddled geometry of the basin boundary (27, 31). To isolate this effect we consider an instantaneous parameter shift, so that the initial states and the model states at the end of the parameter shift are identical. Fig. 4 shows three ensemble simulations with initial conditions that are spaced successively closer along the attractor. Even for this instantaneous parameter shift not all initial conditions tip, which shows that the basin boundary at $F_2$ actually intersects with the initial attractor at $F_1$. Fig. 4 A and B shows refined ensembles of initial conditions, with a temporal spacing on the attractor of 1 wk and 2 mo, respectively, which span over a transition from tracking to tipping initial conditions in the coarser resolution ensemble. Even though the attractor in this region must cross the basin boundary, there is no clear transition from tracking to tipping. Instead, the patterns are consistent with a random, uncorrelated sequence (Materials and Methods), which strongly indicates that the basin boundary is fractal. Further evidence comes from the chaotic transient dynamics of the trajectories approaching the edge state (SI Appendix, Fig. S4), suggesting it is a chaotic saddle whose stable manifold is associated with a fractal basin boundary (32). This would imply that even for trajectories that end up close to another in the vicinity of the boundary at the end of the parameter shift, it is not possible to know whether the model will tip or track.

Fig. 4.

Tipping for an instantaneous parameter shift. Different initial conditions are generated by branching off a realization from the spin-up simulation every week, 2 mo, and 2 y for the ensembles in A, B, and C, respectively. The outcome of the instantaneous parameter shift from $F_1=0.193$ to $F_2=0.219$ branched off at the respective times is given by the bar plots. The time series above show the average of the AMOC maximum in between successive branch-off times.

Conclusions

Our results show that in addition to its location, the rate at which a tipping point of the AMOC can be safely approached needs to be constrained for proper risk assessment, which diminishes the safe operating space (Fig. 1A). While our modeling study may not be accurate enough to provide quantitative constraints, we show that the boundaries of this safe operating space are in fact not sharp. Due to chaotic dynamics and fractal basin boundaries, the outcome of a parameter shift depends sensitively on the initial conditions, and even for fixed initial conditions there is no sharp boundary that defines a safe rate of change. Such a behavior can be expected to exist across large parts of the climate model hierarchy, where chaotic dynamics are the norm. To minimize risk, one could suggest avoiding rates of change that are higher than the minimum rate for which tipping is observed in future modeling studies similar to ours. However, there might always exist initial conditions with a lower minimum rate. Thus, it seems necessary to adopt a probabilistic framework for risk assessment, which requires large ensembles of model simulations.

To summarize, our results suggest that the existence of alternative and undesired stable states may be already a risk even if the tipping point is relatively far away. Thus, when assessing risks of tipping in scenarios of future greenhouse gas emissions, it is important to consider safe rates of change in order not to enter the regime of unpredictable rate-induced tipping. Our findings point to fundamental limitations in climate predictability and corroborate the need to keep the boundary conditions of vulnerable elements of the climate system as stable as possible.

Materials and Methods

Veros Ocean Model.

The simulations are performed with the primitive equation finite-difference ocean model Veros (33) in a global present-day configuration. The model represents mesoscale turbulence with the Redi (34) and the Gent and McWilliams (35) parameterization for isopycnal and thickness diffusion with a diffusivity of $\mathrm{1,000}\hspace{0.17em}{\text{m}}^2/\text{s}$. Diapycnal mixing is parameterized after Gaspar et al. (36) with a background diffusivity of $10^{-5}\hspace{0.17em}{\text{m}}^2/\text{s}$. To enable long model simulations and a large number of ensemble realizations, a horizontal resolution of 90 longitudinal and 40 latitudinal grid cells is chosen, along with 40 vertical layers, which range in thickness from 332 m at the bottom to 12 m at the surface. The latitudinal grid resolution increases from 5.${\mathrm{3}}^{○}$ at the poles to 2.${\mathrm{1}}^{○}$ at the equator (37). The bathymetry is based on the ETOPO1 global relief model (38), smoothed with a Gaussian filter to match the grid resolution. The model domain ends at ${\mathrm{80}}^{○}$N and thus does not feature an Arctic connection of the ocean basins. The ocean is forced by heat and salinity exchange with the atmosphere, which act through upper boundary conditions. These are specified by various present-day climatological fields, based on the ERA-40 reanalysis (39), which have an explicit time dependence that is omitted in the following notation. In the present simulations, we isolate the thermohaline component of the circulation by neglecting wind stress forcing of the ocean. We find that in our model, this leads to a reduction of the overturning circulation by 30% compared to simulations that employ ERA-40 wind stress forcing (SI Appendix, Fig. S2). Thus, wind-driven upwelling is not the dominant driver. Furthermore, the model including ERA-40 wind stress forcing displays abrupt transitions of the AMOC qualitatively similar to the model without wind stress forcing (SI Appendix, Fig. S2C). Thus, we conjecture that the underlying positive feedback is preserved and the results presented here are applicable to model systems that include a wind-driven circulation. The boundary condition for heat exchange is expressed through a first-order Taylor expansion of the heat flux given the anomaly of the modeled surface temperature with respect to a fixed surface temperature field (40):

$$Q\left(T_i\right)=Q\left(T_i^{obs}\right)+{\left.\frac{\partial Q}{\partial T}\right|}_{T_i^{obs}}\left(T_i^{obs}-T_i\right).$$

For this, a prescribed surface temperature field $T_i^{obs}$ as well as fields of net heat flux $Q\left(T_i^{obs}\right)$ and the derivative of the heat flux with respect to changes in surface temperature ${\left.\frac{\partial Q}{\partial T}\right|}_{T_i^{obs}}$ are used. The latter field is derived following ref. 41 using the ERA-40 reanalysis data and has a global yearly average of $\frac{\partial Q}{\partial T}=30.26\hspace{0.17em}\text{W}\cdot {\text{K}}^{-1}\cdot {\text{m}}^{-2}$. The salinity boundary condition is given by a relaxation of the surface salinity to an observed field $S^{obs}$. At each grid point $i$ this relaxation results in a salinity flux, the strength of which is governed by the deviation from the value of the forcing field, as well as a relaxation time scale ${\tau }_S$ and the upper ocean layer thickness $h$:

$${\varphi }_i=h{\tau }_S^{-1}\left(S_i^{obs}-S_i\right).$$

Such a relaxation boundary condition is commonly used in ocean-only models to obtain a stable ocean circulation. In reality, the salinity flux ${\varphi }_i$ is governed by the local balance of evaporation, precipitation, and inflow of freshwater due to river discharge. A relaxation boundary condition implies a strong feedback between this balance and the surface salinity. However, since local evaporation and precipitation are largely independent of the local ocean salinity, it is not clear to what extent such a feedback is physically justified. Importantly, strong relaxation disables the positive salt advection feedback, which is thought to be responsible for the existence of multiple stable regimes of the ocean circulation, and can lead to tipping points of the AMOC (25). This feedback is also present in coupled models, where the processes that are represented by the boundary conditions discussed here are resolved. A salinity flux forcing field independent of the modeled surface salinity might be more realistic. However, as found in previous studies (42) as well as for the present model, in this case the thermohaline circulation becomes highly variable and displays intermittent episodes of vigorous and severely reduced AMOC. While this is an interesting instance of self-sustained, large-scale chaotic variability, the present study focuses on a dynamical regime where more well-defined, stable circulation states exist. This is obtained by using a relaxation boundary condition with suitable ${\tau }_S$. The time scale ${\tau }_S$ effectively controls the strength of the salt advection feedback. This is demonstrated in SI Appendix, Fig. S1. As a consequence, the main results of the paper are obtained with ${\tau }_S=360$ d.

Hysteresis Experiment.

We initialize the model with a 4,000-y spin-up simulation from present-day initial conditions (43) with present-day forcing. Then, a salinity flux anomaly $\tilde{\varphi }$ is applied at the surface in the grid cells between ${\mathrm{296}}^{○}$W to ${\mathrm{0}}^{○}$W and ${\mathrm{50}}^{○}$N to ${\mathrm{75}}^{○}$N. This corresponds to an area of roughly $A=1.5\hspace{0.17em}\text{mio.}\hspace{0.17em}{\text{km}}^2$. Due to the salinity relaxation boundary conditions, adding a constant freshwater flux (negative salinity flux) is equivalent to simply changing the values of the salinity forcing field $S^{obs}$ in the according region. In this paper, we give the values of the equivalent total freshwater forcing $F=\tilde{\varphi }A{S}_{ref}^{-1}$ in Sv, with the reference salinity $S_{ref}=3.5\hspace{0.17em}\text{g}\cdot {\text{kg}}^{-1}$. This freshwater anomaly is ramped up in small increments, where one increment consists of a 200-y linear increase of the freshwater anomaly, followed by a 100-y relaxation period at constant anomaly. At a value of $F=0.31$ Sv, the anomaly is ramped down to 0 again using the same increments. The total simulation time of this transient hysteresis experiment was 22,900 y.

Ensemble Simulations for Different Forcing Rates.

We branch off linearly ramped parameter shift experiments from the hysteresis experiment where the forcing parameter has been increased slowly to $F_1=0.193$, which is at the left edge of the bistable parameter regime. Here, the AMOC is still vigorous. To ensure further relaxation to the attractor, we continue the simulation for 600 y at constant forcing. Thereafter, an ensemble of model runs is branched off where we linearly ramp up the forcing parameter at different rates to $F_2=0.219$, which is still below the tipping point. After the parameter shift is completed, the simulations are continued for at least 400 y to allow equilibration. Several realizations have been continued for 3,000 y to verify that this equilibration time is sufficient.

Ensemble Simulations for Different Initial Conditions.

We create ensembles of different initial conditions by sampling along the attractor at $F_1=0.193$. To this end, we continue the 600-y spin-up run and branch-off simulations after given time intervals, at which point the parameter shift to $F_2=0.219$ starts. For the results shown in Fig. 3 B and C, we used time intervals of 2 y and 2 mo, respectively. The same is true for Fig. 4 C and B, whereas for Fig. 4A we used intervals of 1 wk. The branch-off and simulation times given in Figs. 3 and 4 start after 9,100 y of simulation, which corresponds to 4,000 y of spin-up at $F=0$, 4,500 y of ramping to $F_1$, and 600 y of spin-up at constant $F_1$. Compared to using a snapshot or pullback attractor framework (44), our method of sampling the attractor in time greatly reduces the computational cost.

Testing Randomness of Ensemble Tipping Outcomes.

We test whether the ternary sequences of tipping outcomes obtained by sampling the attractor in time, as shown in Figs. 3 B and C and 4, are consistent with a random, uncorrelated sequence of three possible outcomes $\mathcal{X}=\left\{\mathrm{0,1,2}\right\}$. We do this with a one-tailed Monte Carlo hypothesis test using the conditional entropy of adjacent elements $X_i$ and $X_{i-1}$ in the sequence as a test statistic:

$$H\left(X_i|X_{i-1}\right)=-\sum _{x\in \mathcal{X}}\sum _{y\in \mathcal{X}}p\left(X_i=x,X_{i-1}=y\right)\log \frac{p\left(X_i=x,X_{i-1}=y\right)}{p\left(X_i=x\right)}.$$

This measures whether the elements $X_i$ in the sequence are dependent on their preceding elements $X_{i-1}$. In our context, it thus measures to which degree close-by initial conditions display a similar tipping outcome. If adjacent elements are independent, then $H\left(X_i|X_{i-1}\right)=H\left(X_i\right)$. Otherwise, $H\left(X_i|X_{i-1}\right)<H\left(X_i\right)$. For each tipping sequence given by the bar plots in Figs. 3 B and C and 4, we generate ensembles of 1 million synthetic, ternary independent and identically distributed sequences of the same length as the data. The probabilities $p\left(X=\left\{\mathrm{0,1,2}\right\}\right)$ are estimated from the data as the fraction of realizations that are tipping, are tracking, or go to the edge state, respectively. For each synthetic sequence we calculate the conditional entropy. We reject the null hypothesis of independence of adjacent elements in the sequence at 95% confidence if 95% of the synthetic test statistics are larger than the data test statistic, corresponding to a P value of $P=0.05$. If this is the case, it indicates that the tipping outcomes of adjacent initial conditions on the attractor are likely to be similar. In contrast, for the data of Figs. 3C and 4 A and B we obtain $P=0.604$, $P=0.704$, and $P=0.713$, respectively.

Conceptual Model for Rate-Induced Collapse of the AMOC.

To illustrate the rate-induced mechanism responsible for the AMOC collapse in our ocean model simulations, we introduce a conceptual dynamical systems model of two variables,

$$\begin{array}{ll}\hfill \frac{dx}{dt}& =\left(-r^4+2r^2-\beta \right)x-\omega \tilde{y}\hfill \\ \hfill \frac{d\tilde{y}}{dt}& =\left(-r^4+2r^2-\beta \right)\tilde{y}+\omega x,\hfill \end{array}$$

with $r^2=x^2+{\tilde{y}}^2$ and $\tilde{y}=y-\gamma \beta$. Here, $\beta$ is a control parameter analogous to the freshwater forcing parameter, and $\omega$ a frequency. The parameter $\gamma$ specifies a dependence of $y$ on $\beta$ and thus a tilt of the invariant manifolds in the $y-\beta$ plane, which allows for rate-induced tipping to occur. The bifurcation structure is governed by the equation $2r^2-r^4-\beta =0$. The solutions are $r_{\pm }^2=\pm \sqrt{1-\beta }$, where $r_{+}$ is real for $\beta <1$ with a corresponding stable limit cycle (attractor) with radius $r_{+}$ and frequency $\omega$. We identify this attractor with the vigorous, present-day AMOC. Additionally, there is a fixed point $r=0$, which we identify with the collapsed AMOC state. It is unstable for $\beta <0$. At $\beta =0$ an unstable limit cycle $r_{-}$ emerges in a Hopf bifurcation while $r=0$ becomes stable. $r_{-}$ corresponds to the edge state that separates the two attractors (basin boundary). At ${\beta }_c=1$ the stable and the unstable limit cycles merge and disappear in a saddle node bifurcation. In Fig. 2 we show the quasi-stationary attractors and basin boundary as a function of $\beta$ to illustrate the heuristics of rate-induced tipping. A rigorous analysis of a similar system using pullback attractors can be found in Alkhayuon and Ashwin (30).

Greenland Freshwater Runoff Data.

In Fig. 1A we show freshwater runoff data from the Greenland ice sheet. The colored trajectory is based on an ice core reconstruction spanning the years 1675 to 2013 in annual resolution (14), which has been low-pass filtered with a 40-y Gaussian kernel. The rate of change of these smoothed runoff data is estimated by the linear trend in a 15-y moving window. We furthermore show a data point that is derived from the Gravity Recovery and Climate Experiment (GRACE) dataset of Greenland mass balance during the years 2003 to 2016 (15, 45). The monthly data for Greenland mass anomaly is yearly averaged and the difference of these yearly averages estimates the runoff per year. We estimate the rate of change by the increasing linear trend of this runoff over the entire time period, which is plotted against the mean runoff.

Data Availability

The data shown in Fig. 1A are available from Trusel et al. (14) and at http://products.esa-icesheets-cci.org/products/details/greenland_gravimetric_mass_balance_rl06_dtuspace_v2_0.zip/. The entire Veros source code is available under a GPL license on GitHub (https://github.com/dionhaefner/veros). Code for analyzing the data, the Veros setup files used to change the boundary conditions and forcings, and the raw data used to produce the figures are available at https://github.com/johannes-lohmann/rtip_veros (DOI: 10.5281/zenodo.4428228). The raw model simulation data that underlie the findings of this study are deposited in the Electronic Research Data Archive repository of the University of Copenhagen and can be accessed at https://sid.erda.dk/sharelink/He6H3pL0Xp.