We consider the Stommel box model of the Atlantic thermohaline circulation with added noise to represent variations in the atmospheric forcing on short timescales. The model assumes the overturning flow ψ in between well-mixed polar and equatorial ocean basins to be proportional to the density difference 1ψ∝(ρp-ρe)=[αT(Te-Tp)-αS(Se-Sp)], where the density is given by the equation of state of seawater 2ρe,p=ρ01-αT(Te,p-T0)+αS(Se,p-S0), with some reference densities, temperatures and salinities represented by ρ0, T0 and S0, respectively. The two model variables represent the dimensionless temperature difference T=αT(Te-Tp) and salinity difference S=αS(Se-Sp) in between the boxes. This defines the dimensionless overturning circulation strength 3q=T-S. Temperature and salinity in the boxes relax towards an atmospheric temperature and salinity forcing Te,pa and Se,pa. The meridional difference of the forcing drives the circulation and is represented by the two parameters η1∝(Tea-Tpa) and η2∝(Sea-Spa). A third parameter represents the timescale ratio of the temperature and salinity relaxation η3=τTτS. The model is then defined by the stochastic differential equations 4dTt=η1-T-|T-S|Tdt+σTdWT,t5dSt=η2-η3S-|T-S|Sdt+σSdWS,t, with the Wiener processes WS,t and WT,t. Time is scaled with respect to the ocean timescale τT=200 years. For a more detailed derivation of the model, see . The deterministic system features a parameter regime with two stable equilibria, which are referred to as the circulation “on” and “off” states. For the on state we have T>S, where the temperature forcing gradient dominates the opposing salinity forcing gradient and drives the circulation. The off state (S>T) corresponds to a reversed circulation, which is weaker and dominated by the salinity forcing gradient. In Fig. a and b we show deterministic bifurcation diagrams of q with respect to the parameters η1 and η2. In both cases, the on state loses stability in a regular saddle-node bifurcation, whereas the off state destabilizes in a non-smooth saddle-node bifurcation. The latter is also known as a non-smooth fold and is due to the fact that the Stommel model is a non-smooth dynamical system owing to the absolute value in its equations (see Sect. S3 and Fig. S3 for more detail). The existence and extent of bi-stability depends on the parameter η3. A large timescale separation (slower salinity damping) leads to a large region of bi-stability, whereas as the salinity damping approaches the timescale of temperature damping, the bistability disappears (Fig. c). This is because a faster salinity damping disables the positive salt advection feedback, which gives rise to the bi-stability.

The ocean model is coupled to a sea ice component in the polar ocean box, which is the energy balance model described in and , modified by neglecting the seasonal cycle and effects of the sea ice thickness. The changing sea ice cover acts to insulate the polar ocean to varying degrees from the cold atmospheric temperature forcing Tpa, thus modulating the temperature forcing gradient η1∝(Tea-Tpa). A schematic of the coupled model, including model variables and important parameters, is given in Fig. . The deterministic sea ice component is defined by 6dIdt=Δtanh⁡Ih+R0Θ(I)-BI+L-F-1+R, with the Heaviside step function Θ(⋅). Time is scaled with respect to τI=1 year. The parameters and their values are described in Table . While I>0 corresponds to a positive sea ice cover, I<0 represents zero sea ice cover, and the variable is instead a measure of the enthalpy of the surface ocean . The control parameter R models influences on the sea ice concentration due to external factors, such as export or import of sea ice into the North Atlantic via changes in wind stress. While in the climate system R is driven by slower dynamic processes, such as changes in ice sheet topography, we treat it as a control parameter. We use parameter values from , which yield a sea ice component that is bi-stable with respect to R. As seen in Fig. , for a range of R there exists a stable state with a positive sea ice cover (red), as well as a state with zero sea ice cover I<0 (black). This range is bounded by two saddle-node bifurcations. The stable state with sea ice cover collapses at R=-0.282. We define the state at R=0 as the stadial state, yielding a fixed point with positive sea ice cover I0+≈1.156. A slight deviation from the parameters of is our larger value of h, which gives a more gradual albedo transition from an ice-free to an ice-covered state. Accordingly, the bifurcation diagram is more “S-shaped” instead of “Z-shaped” (see Sect. S1 and Fig. S1 in the Supplement for more details).

To model transitions from stadial to interstadial conditions in the coupled sea ice–ocean model, we consider the following mechanism. The glacial polar ocean is insulated by a high sea ice concentration from the atmospheric temperature forcing, preventing it from losing heat efficiently. As the sea ice concentration decreases, the polar ocean becomes more and more exposed to the cold atmosphere and loses heat. Thus, the sea ice variable modulates the parameter η1, which we now define as η1(I)=η10-κΘ(I)I, with the Heaviside function Θ(⋅) since I<0 corresponds to zero sea ice cover. Adding noise as a model of fast atmospheric perturbations (Wiener process WI,t), this yields the following coupled equations: dIt=Δtanh⁡Ih+[R0Θ(I)-B]I+L-F-1+Rdt7+σIdWI,t,8τTτIdTt=η10-κΘ(I)⋅I-T-|T-S|Tdt+σTdWT,t,9τTτIdSt=η2-η3S-|T-S|Sdt+σSdWS,t. The value of κ reflects the change in atmospheric temperature forcing when removing the sea ice cover. In this conceptual framework it can only be chosen heuristically. We can for instance assume η10=3.0 for an open ocean, and atmospheric temperature forcings in a glacial climate of 20 and -10 ∘C in the equatorial and polar box, respectively. Full sea ice cover would limit the polar temperature forcing to 0 ∘C, corresponding to η1=2.0. Even if the glacial polar atmosphere were above 0 ∘C, given that it was colder than the surface ocean, extensive sea ice cover would severely reduce heat loss to the atmosphere and thus effectively reduce η1. Here we choose a scenario where during the stadial the sea ice reduces the atmospheric forcing from η1=3.0 to η1=2.65. κ is then chosen such that η1=2.65 at the stadial fixed point I0+ and η1=3.0 for I<0, yielding κ=0.35/I0+. As a result, the ocean component is in the bi-stable regime for both full and zero sea ice cover. A transition from stadial to interstadial will then be captured by decreasing R from zero beyond the bifurcation point, which tips the sea ice component towards a state of I<0, while the ocean remains in the bi-stable regime.

Due to the unidirectional, linear coupling of the model, and our focus on a specific dynamical regime, we restrict our presentation of the coupled dynamics to the individual bifurcation diagrams of the sea ice component with R as control parameter and of the ocean component with η1(I) as effective control parameter. The full bifurcation structure of the coupled model with R as the only control parameter is presented in Sect. S2 and Fig. S2.

Here we investigate the tipping dynamics in the ocean component in the deterministic limit (σT=σS=0). As noted above, there is a non-smooth fold in the Stommel model as the off state loses stability, which leads to a resurgence of the AMOC. In the bifurcation diagrams of Fig.  it can be seen that, both in terms of T and S, the stable fixed point (red line) moves in the same direction as the saddle point when the non-smooth fold bifurcation is approached. Thus, in a sufficiently fast parameter shift towards the fold, the saddle point can outpace the system state, which is trying to follow the moving equilibrium. This is illustrated in Fig. , where instantaneous parameter shifts and the corresponding movements of the system state vector in the bifurcation diagrams are indicated. When the saddle point moves past the system state, the system will tip towards the alternative stable state, which is the on circulation in our case. Thus, tipping can occur even before the bifurcations points are reached, which is known as rate-induced tipping. While in the Stommel model this can happen for both η1 and η2 as control parameter, it occurs for a larger range of amplitudes and rates of the parameter shift when changing η1.

To be more rigorous one has to consider the movement of the basin boundary as the control parameter is changed. The basin boundary is the stable manifold of the saddle, and it separates the basins of attractions of the on and off states, i.e., the sets of initial conditions that converge to the respective attractors. In Fig.  we illustrate the movement of the fixed points and basin boundary as η1 is changed from 2.65 to 3.0. This corresponds to the scenario of a transition from stadial to interstadial sea ice cover in the coupled model, as described in Sect. . Figure b shows that the off fixed point before the parameter shift (open circle) lies inside the basin of attraction of the on fixed point after parameter shift (blue area). This is a sufficient condition for rate-induced tipping, which has been called basin instability , since for an instantaneous parameter shift, the system would tip to the other attractor. Similarly, as the system tries to follow the moving fixed point during a sufficiently fast parameter shift, it will fail to reach the off basin (orange area) at the end of the parameter shift and tip to the on fixed point. This happens for the blue trajectory, where the parameter is ramped up linearly within 300 years. In contrast, the purple trajectory shows that tipping does not occur for a ramping duration of 500 years. For this given amplitude of the parameter shift, there is a critical rate of parameter change in between these two values.

Figure  shows time series of q for simulations with different ramping durations. The realizations in Fig. a and b tip to the on attractor, while the realizations in Fig. c and d track the moving off equilibrium. The critical ramping duration is in between the 388.5 and 390 years employed in Fig. b and c. Comparing Fig. a to b, one observes a delay in the tipping in Fig. b of thousands of years. This occurs because, for close-to-critical rates, the system state passes by very closely to the saddle point, where it remains for a long time as the dynamics slow down before being ejected. The close approach of the saddle happens because the system state is attracted by the saddle's stable manifold, which is also the basin boundary. If one were to use the exact critical ramping duration, the system state would evolve precisely towards the saddle and remain there. Such trajectories are called maximum canards . This behavior is also seen for trajectories that eventually track the moving equilibrium, as in Fig. c. It is worth noting that the attraction by the stable manifold of the saddle continues after the parameter shift is already over, as shown in the inset in Fig. c.

The critical ramping duration depends on the amplitude Δη1 of the parameter shift (Fig. a). Rate-induced tipping becomes possible at a certain Δη1, where the basin instability condition is first satisfied. Increasing Δη1 then leads to a very rapid increase of the critical ramping duration Dc. Thereafter, Dc keeps increasing and actually diverges as the bifurcation is approached. This is due to the non-smoothness of the fold bifurcation, where the attractor and saddle meet in a cusp (see Sect. S3 and Fig. S3 for more detail). As a result, the attractor gets close to the basin boundary very quickly as the bifurcation is approached. This leads to a super-linear scaling of the shortest distance to the basin boundary. In Fig. b, we compare this to the square-root scaling of the smooth fold bifurcation in the sea ice component. In the non-smooth case, as the bifurcation is approached the basin boundary gets arbitrarily close to the attractor. Then, even very small and slow parameter increases lead to tipping. Thus, the non-smooth fold leads to what could be called a soft tipping point. In practice, there is no hard critical threshold of the parameter, but for any parameter shift at finite rate, the tipping will occur at some point prior to the bifurcation. The precise location of the tipping point depends on the trajectory of the parameter shift.

We now consider additive noise in the ocean component, which models variations in atmospheric forcing on short timescales. In addition to the soft tipping just described, the stochastic perturbations further blur the critical threshold leading to tipping. For a given amplitude of the parameter shift, there is no longer a critical rate but a range of rates where the probability of tipping goes from 0 to 1. Figure a shows how this range of rates expands for increasing noise level. Note that since the system features unbounded noise, we consider finite time-tipping probabilities during a simulation time of 5000 years. Eventually, there will always occur a noise-induced transition to the on attractor, especially from the off attractor at η1=3.0 for higher noise levels.

By introducing noise, tipping becomes a mixture of rate-induced and noise-induced transitions, since the unbounded noise allows the system to cross the basin boundaries of the deterministic system in any circumstances. Still, for low noise levels the behavior strongly resembles the deterministic case. As discussed earlier, for a ramping speed relatively close to the critical rate, the tipping involves an escape from the saddle. This behavior is robust for low noise levels, where the stochastic fluctuations cannot overcome the attraction of the stable manifold of the saddle. Thus, the system approaches the saddle before being ejected from its vicinity.

As the noise level is increased, there are noise-induced early tippings as well as significantly delayed tippings. In order to quantify when a tipping is “early” or “late”, we need to define the moment when the system actually tips. For the deterministic system, a sensible choice would be the time when the moving, quasi-stationary basin boundary is crossed, since this is the first moment that the system would tip in case the parameter shift would be stopped suddenly. However, for the noisy system this does not guarantee tipping, since the system may cross back to the other basin at any time. As a heuristic definition of tipping, we can instead detect the departure from the vicinity of the saddle in terms of the overturning q, as the tipping is associated with a monotonic increase of q (see Fig. ). Thus, as tipping we define the first crossing of q=0.1, which is a slightly larger value than at the saddle to allow for some fluctuations around it. In phase space this defines a straight line.

Figure  shows the crossing of this threshold, as well as the basins at the time when the basin boundary is first crossed, for three different realizations with a ramping duration of 300 years and σT=σS=0.2. The time of tipping varies significantly and depends primarily on the proximity of the approach to the saddle and the subsequent time spent in its vicinity. Whereas Fig. b shows a realization with tipping close to the deterministic scenario, the realization in Fig. a leaves the stable manifold early and does not approach the saddle closely. The realization in Fig. c approaches the saddle very closely and remains there for a long period of time.

The tipping time distribution and its dependence on the noise level is shown in Fig. b. In our case of a ramping duration slightly below the critical value of the deterministic system, there are three regimes of noise levels. For low noise (σ=0.01, σ=0.02, σ=0.04 and σ=0.06 in Fig. b) the trajectories are very similar to the deterministic case, and it is very unlikely that the noise pushes the system closer to the saddle. Thus, the tipping time is distributed closely around the deterministic value. For intermediate noise (σ=0.1 and σ=0.2 in Fig. b), some early noise-assisted tippings are possible, as seen by the shift of the distribution mode towards earlier times. For other realizations there is a good chance that the noise pushes the system closer to the saddle, where it can stay for a long time (thousands of years) as the dynamics slow down before escaping. This leads to the development of a long tail in the tipping time distribution. For larger noise (σ=0.4 in Fig. b), even earlier noise-assisted tippings are seen, as well as some delayed tippings. However, the latter do not occur as frequently as for intermediate noise since the average residence time at the saddle is also shortened.

We now consider the coupled model and investigate how a stadial–interstadial transition can arise as a cascading tipping of the two components. The cascade is initiated by a change in the control parameter R leading to a decrease and eventual tipping of the sea ice to I<0. Subsequently, the modulation of the parameter η1(I) due to the decrease of I can be expected to induce a rate-induced resurgence of the AMOC. On the one hand, this is because the short sea ice timescale leads to very fast dynamics as the sea ice tips. On the other hand, even if the sea ice does not change quickly, when the amplitude of the change in η1(I) becomes larger, there will be rate-induced tipping anyway due to the soft tipping point in the Stommel model described earlier. We thus choose the robust scenario where the coupling κ is such that the ocean component remains in the bi-stable regime with respect to η1(I), and thus a rate-induced AMOC resurgence is the only pathway to tipping. As described in Sect. , this can be exemplified by a change in η1(I) from η1=2.65 (at the stadial sea ice fixed point for R=0) to η1=3.0 for a collapsed sea ice cover I<0. Simulations with these parameters are qualitatively representative for a wider range of coupling strengths and rates of changing R.

Figure  shows trajectories for a cascading stadial–interstadial transition in the deterministic limit when R is ramped down from R=0 to R=-0.3 over 340 years. The transition can be divided into several stages. First, the sea ice slowly decreases as R is decreased and the ocean component tries to track the moving equilibrium (green segment of trajectories in Fig. ). At point A, 325 years after the start of ramping, the sea ice passes the bifurcation point and rapidly tips to I<0 (purple segment in Fig. ). This leads to a quick movement of η1(I) towards η1=3.0, which is reached at point B, 350 years after the start of ramping (Fig. b). As a result, the ocean state crosses the moving basin boundary (gray surface in Fig. c) from above and is thus determined to undergo rate-induced tipping to the on attractor (solid black curve). However, before tipping the ocean state is attracted by the stable manifold (i.e., the basin boundary) of the saddle (yellow segment). Finally, at point C (700 years after the start of ramping) the ocean component escapes the vicinity of the saddle and tips towards the on state (blue segment).

There is a critical timescale below which such a cascading transition with a rate-induced tipping is possible. This is a combination of the rate of change in the control parameter R and the speed of the tipping of the sea ice, which is held fixed here. As additive noise is included in the model, the boundary of tipping in terms of the ramping time of the control parameter is again blurred. Figure a shows the tipping probabilities for different noise levels as a function of the ramping time of R. The result is very similar to the ocean-only case, except that because of the fast tipping in the sea ice, the average ramping times leading to tipping are slightly higher. The picture looks different as we increase the noise level in the sea ice component, as seen in Fig. b. Here, the ramping times that yield significant tipping probabilities simply increase with the noise level without a large simultaneous decrease of the tipping probability for lower ramping durations. This is because noise-induced transitions to I<0 occur before the bifurcation of I is crossed. Since these transitions happen on the fast sea ice timescale, a rate-induced tipping of the ocean model becomes possible even when R is changed very slowly. As in the ocean-only case, the tipping cascade involves a saddle escape, which can lead to significant tipping delays as noise forcing of intermediate strength is included. Next, we will discuss this in more detail and relate it to potential pre-cursor signals leading up to such transitions.

Due to their irreversible nature, it is important to foresee impending tipping points using generic early warning signals that do not require detailed knowledge of the system dynamics. These are typically obtained from time series by estimating a statistical indicator in a sliding window with appropriate detrending (see Sect. S4). For bifurcation tipping, a system often exhibits critical slowing down, which can be measured by increasing variance and autocorrelation. In Fig.  we show these indicators estimated in a sliding window for the cascading transition in Fig. . As expected there is an increase in variance and autocorrelation of I leading up to the bifurcation (Fig. c and d). Because of the speed of the parameter shift necessary to induce the cascade, the increases in the indicators do not fully exceed the variability prior to the parameter shift but could still provide early warning with a reasonable skill. Due to the coupling one might expect a signature of the sea ice critical slowing down in the ocean component. This is not seen here (Fig. e and f), since increasing fluctuations due to the sea ice are small compared to the variability in the ocean component for the chosen σ. If no noise is added to the ocean variables, critical slowing down can be detected in T or S. This might be an example of scenarios proposed in and , where it is hypothesized that a bifurcation in the sea ice system is detectable as increased variance in the high frequencies of ice core data prior to DO events. Similarly, the fluctuations in I of increasing amplitude and temporal correlation may influence the ocean subsystem in a more consistent way as the bifurcation is approached. This should increase the cross-correlation, especially on longer timescales that can be measured with detrended cross-correlation analysis (DCCA). This has been proposed as early warning indicator for cascading transitions . The method is similar to detrended fluctuation analysis, but instead of scaling in the variance, it measures scaling in the covariance of two signals with increasing timescales (for details, see or ). We can detect a slight increase on average in the DCCA exponent of I and T (Fig. g) for the transition in Fig. . However, the increase found in individual time series is not statistically significant, owing to the large variance of the DCCA estimator.

During the rate-induced transition of the ocean component there is an increase in the ensemble variance, as can be seen by the shadings in Fig. b. This increase, as well as a corresponding increase in ensemble autocorrelation, has been proposed before as an early warning signal for rate-induced tipping . However, we show here that this results from the large spread in the amount of time spent by individual realizations at the saddle before tipping to the other attractor (see Fig. ). In contrast, the fluctuations in individual realizations, as used for operational early warning, do not show an increase in variance and autocorrelation. This can be seen in Fig. e and f, where no increases in sliding window variance and autocorrelation accompany the increase in ensemble variance. For the estimation of variance and autocorrelation in a sliding window, a detrending of the time series is necessary, such that remaining trends in the residuals are not larger than the fluctuations themselves. For our detrending method using cubic functions, the severity of detrending, and thus the ability to remove sharp changes in the signal trend, depends only on the sliding window size (see Sect. S4 for more details). In order to remove the trend due to the parameter shift regarded here, a window size of no more than 200 years is required (Fig. S4).

Detrending inevitably removes some of the original fluctuations. To show that the lack of increased fluctuations in the detrended time series is not a consequence of too severe detrending, we extract segments of time series simulated from the Stommel-only model, where the system is in the vicinity of the saddle and there are no sharp trends. The fluctuations around the saddle are then compared to time series segments where the system fluctuates around the initial attractor. We define the vicinity of the saddle by the time periods in the simulations where q=0.06 is first crossed until q=0.1 is first crossed (Fig. ). We regard the time series segments of an ensemble of realizations where the system stays in this vicinity for at least a certain duration. After detrending the segments by cubic functions, we calculate variance and autocorrelation. This yields empirical distributions of these quantities, describing the fluctuations in the system shortly before tipping. For each realization, we also choose a segment of the same duration taken just before the parameter shift starts, yielding distributions of variance and autocorrelation at the initial attractor. Figure  shows that variance and autocorrelation at the saddle are not increased but actually slightly decreased compared to the initial attractor. This is best seen for longer segments (Fig. c and d), since there the uncertainty in the estimators is smaller. One can also see that in this case the average variance and autocorrelation are larger compared to Fig. a and b because the detrending in longer windows removes less variability on longer timescales.

It thus does not appear that critical slowing down indicators apply to rate-induced tipping. Instead, we exploit that the system is attracted towards the saddle where the dynamics are different to those at the initial attractor. If this difference can be detected before the system tips, a small perturbation in the right direction or a reversal of the parameter shift could push the system back in the desired basin of attraction. Saddles, which have at least one unstable direction in phase space, can be distinguished from attractors by a change from a negative to a positive real part of the largest eigenvalue of the Jacobian. Estimating the Jacobian from the time series in a sliding window could thus be a generic tool to detect the saddle escape involved in rate-induced tipping, and we describe a method to do this in the Appendix . With this method the elements of the Jacobian during rate-induced tipping of the Stommel model can be inferred and allow for the distinction of the dynamics around the different fixed points (Fig. S5). However, there are quantitative biases in the estimates of individual elements, and as a result the estimates of the real part of the largest eigenvalue in the vicinity of the saddle are not consistently positive. These biases could be a result of the detrending, of a too high noise level or because the unstable dynamics are “suppressed” since we consider time series segments taken before the escape from the saddle.

As a more reliable indicator we propose the actual elements of the Jacobian, since they are inferred in a qualitatively robust way (Fig. S5). This lowers the estimator variance compared to the eigenvalues, which are composed of the estimates of all elements. The off-diagonal elements record changes in sign of the feedbacks in between the system variables. Such changes in feedback are common as a system moves towards a saddle. We combine the off-diagonal elements to a scalar early warning indicator J, as defined in Eq. (). Figure a–f shows that J can distinguish the dynamics around the attractor (black) and the saddle (red) before tipping. The panels correspond to different minimum lengths of the time windows used to estimate J. The figure also shows probabilities p of observing a value of J estimated around the attractor that is larger than a value of J in the vicinity of the saddle. This measures the performance of J as an early warning signal. For longer time windows, the distributions become better separated since the uncertainty of the estimator is reduced. Still, even for relatively short windows the indicator correctly identifies the departure from the attractor for most realizations.

An operational early warning signal can be constructed by estimating J in a sliding window, and raising an alert as soon as a threshold Jc is exceeded. Choosing a location of Jc relative to the tails of the distributions in Fig. a–f is a trade-off in between maximizing the rate of true positives and minimizing the rate of false positives (alerts). The performance of the alert as a binary classifier can be summarized in receiver operating characteristic (ROC) curves. The curve of a perfect classifier collapses to the point (0,1). Figure g shows that for realizations that spend a longer time at the saddle, the indicator J comes close to a perfect classifier, detecting the saddle approach with very low false positive and very high true positive rates. Figure  shows J estimated from time series in a sliding window, along with critical slowing down indicators. J begins to rise sharply roughly 200 years after the ramping started and decreases slightly as most realizations leave the saddle towards the on attractor. In contrast to the ensemble variance (orange), the variance and autocorrelation in the sliding window show no signal, apart from a small artifact around the parameter shift, which is a remnant of imperfect detrending in the 200-year windows.