Given growing concerns about climate tipping points and their risks, it is important to investigate the capability of identifying robust precursor signals for the associated transitions. In general, the variance and short-lag autocorrelations of the fluctuations increase in a stochastically forced system approaching a critical or bifurcation-induced transition, making them theoretically suitable indicators to warn of such transitions. Paleoclimate records provide useful test beds if such a warning of a forthcoming transition could work in practice. The Dansgaard–Oeschger (DO) events are characterized by millennial-scale abrupt climate changes during the glacial period, manifesting most clearly as abrupt temperature shifts in the North Atlantic region. Some previous studies have found such statistical precursor signals for the DO warming transitions. On the other hand, statistical precursor signals for the abrupt DO cooling transitions have not been identified. Analyzing Greenland ice core records, we find robust and statistically significant precursor signals of DO cooling transitions in most of the interstadials longer than roughly 1500 years but not in the shorter interstadials. The origin of the statistical precursor signals is mainly related to so-called rebound events, humps in the temperature observed at the end of interstadial, some decades to centuries prior to the actual transition. We discuss several dynamical mechanisms that give rise to such rebound events and statistical precursor signals.

A tipping point is a critical threshold beyond which a system reorganizes, often abruptly and/or irreversibly

Tipping point behavior is mathematically classified into three different types

The theory of critical slowing down (CSD) provides a framework to anticipate critical (or bifurcation-induced) transitions

Greenland records from the NGRIP ice core:

Dansgaard–Oeschger (DO) events are millennial-scale abrupt climate transitions during glacial intervals

The DO events are considered the archetype of climate tipping behavior

Recent studies have inferred that the AMOC is currently at its weakest in at least a millennium

In this study we report SPS for DO cooling transitions recorded in

The remainder of this paper is organized as follows. In Sect. 2, the data and method are described. In Sect. 3, we identify robust and statistically significant SPSs for several DO cooling transitions following interstadials with sufficient data length and show that rebound events prior to cooling transition are the source of observed SPSs. In Sect. 4 we discuss the results by using conceptual models. A summary is given in Sect. 5.

We explore CSD-based precursor signals for DO cooling transitions recorded in

Analysis of CSD-based precursor signals of abrupt DO cooling transitions, for the first six interstadials of NGRIP

Same as Fig. 2 but for the next six interstadials, from 74 ka to 12 kyr b2k.

We follow the classification of interstadials and stadials and associated timings of DO warming and cooling transitions by

The start (warming) and end (cooling) of each DO event are identified in 20-year resolution based on both

Based on the theory of critical slowing down (CSD), we posit that the stability of a dynamical system perturbed by noise is gradually lost as the system approaches a bifurcation point

Prior to calculating CSD indicators, we estimate the local stable state by using a local regression method called the

The statistical significance of precursor signals of critical transitions, in terms of positive trends of CSD indicators, is assessed by hypothesis testing

As CSD indicators we consider the variance and lag-1 autocorrelation, calculated in rolling windows across each interstadial. The 12 interstadials longer than 1000 years are magnified in Figs. 2 and 3 (top rows, blue) for the NGRIP

The variance is plotted in the third row of Figs. 2 and 3. Positive trends in the variance are observed for 9 out of 12 interstadials; the individual trends are statistically significant in 6 out of 12 cases (

Detection of precursor signals of DO cooling transitions for different interglacials, different proxy variables, different ice cores, and different CSD indicators.

We check the robustness of our results against changing smoothing span

The robustness analysis is performed for all the long interstadials of the six records and the two CSD indicators (Fig. 4d). Among the 12 interstadials, we find at least one robust SPS for eight interstadials (GI-25, GI-23.1, GI-21.1, GI-20, GI-19.2, GI-14, GI-12, and GI-8) and multiple robust SPSs for six (GI-25, GI-23.1, GI-21.1, GI-14, GI-12, and GI-8). If the data series is a stationary stochastic process, the probability of spuriously observing a robust SPS is estimated to be

We examine how much the rebound events affect the detection of CSD-based SPS. For this purpose CSD indicators are again calculated excluding the rebound events and their preceding cold spells (see Sect. 2.1). While eight interstadials (GI-25, GI-23.1, GI-21.1, GI-20, GI-16, GI-14, GI-12, and GI-8) exhibit robust SPS with the rebound events included, only four interstadials (GI-23.1, GI-14, GI-12, and GI-8) exhibit robust SPS without the rebound events (Fig. S23). The rebound events should hence be considered important, sometimes indispensable, sources for SPS of DO coolings.

We also examine the dependence of the results on the time resolution of the data. Here we use a high-resolution (5 cm depth)

Four potential dynamical mechanisms for the DO cooling transitions.

Rate dependence of CSD indicators for the fold bifurcation in the Stommel model. Its parameter

We detected robust precursor signals of DO cooling transitions for most interstadials longer than roughly 1500 years but not for shorter interstadials. The results suggest that long interstadials, the existence of rebound events, and the presence of SPS for the DO cooling transitions are all related (except for GI-19.2, which has no noticeable rebound event). These aspects may be related to generic properties of nonlinear dynamical systems. On the basis of conceptual mathematical models, we discuss four possible dynamical mechanisms leading to the precursor signals of DO cooling transitions. In three of four mechanisms, oscillations such as the rebound events can systematically arise prior to the abrupt cooling transitions. These modeling results justify the inclusion of the rebound events in the search for precursor signals presented above. Unless otherwise mentioned, details on model parameters as well as the hysteresis experiments conducted are given in Appendix B.

Here we consider the case that the system is excitable. For example, for

Mechanisms (2), (3), and (4) can generate behavior resembling the rebound events, leading to increases in the classic CSD indicators. In the toy models, rebound event-like behavior is generated when the trajectory passes by an equilibrium point with marginal stability (i.e., the equilibrium has neither strong stability leading to a permanent state nor strong instability leading to short interstadials) (Fig. 5b–g). In this case, the duration of the modeled interstadial is relatively long in relation to the marginal stability. By contrast, the absence of equilibrium or the presence of a strongly unstable equilibrium near the fold point of the critical manifold leads to brief interstadials without a rebound event and consequently a lack of SPS. This provides a possible explanation of why the rebound events and the robust SPS are simultaneously observed for long interstadials but not for short interstadials.

Another possible explanation for the lack of SPS for short interstadials is the following. The common assumption underlying CSD theory is that the parameter change is much slower than the system's relaxation time, and the latter is much slower than the correlation time of the noise

In this study we have explored statistical precursor signals (SPSs) and significant increases in critical slowing down (CSD) indicators (variance and lag-1 autocorrelation), for Dansgaard–Oeschger (DO) cooling transitions following interstadials, using six Greenland ice core records. Among the 12 interstadials longer than 1000 years, we find at least one robust SPS for eight interstadials longer than roughly 1500 years (GI-25, GI-23.1, GI-21.1, GI-20, GI-19.2, GI-14, GI-12, and GI-8) and multiple robust SPSs for six of them (GI-25, GI-23.1, GI-21.1, GI-14, GI-12, and GI-8) (Fig. 4d). Robust SPSs are, however, not observed for interstadials shorter than roughly 1500 years. One might link the increase in the proxy variance with the tendency of larger climatic fluctuations in colder climates

We have proposed four different dynamical mechanisms to explain the observed SPSs: (1) strict CSD due to the approaching of a fold bifurcation; (2) CSD in a wider sense, in stochastic slow–fast oscillations; (3) noise-induced oscillations prior to Hopf bifurcations; or (4) the signal of mixed-mode oscillations. In the last three mechanisms, oscillations such as the rebound events can systematically arise prior to the abrupt cooling transitions. These precursor oscillations are due to marginally (un)stable equilibria on the critical manifolds that cause a long-lived quasi-stable state (like long interstadials). This can explain why rebound events and SPSs are simultaneously observed only for long interstadials and are not observed for short ones. While the SPSs for bifurcation-induced tipping events (mechanisms 1 and 3) are established, detailed properties of SPSs for the stochastic slow–fast oscillations of the excitable system (mechanism 2) and for the mixed-mode oscillations (mechanism 4) remain to be elucidated.

We should mention the assumptions made in this study as well as alternative scenarios for the DO cooling transitions. First, the four dynamical mechanisms assume slow changes in parameters or slow variables which cause bifurcations in the fast subsystem. On the other hand, the rate-induced tipping mechanism has also been invoked for a possible AMOC collapse, where the rate of change of the external forcing (e.g., freshwater flux or atmospheric CO

We have shown that past abrupt DO cooling transitions in the North Atlantic region can be anticipated based on classic CSD indicators if they are preceded by long interstadials. However, it is found to be difficult to anticipate DO cooling events, at least from the 20-year-resolution ice core Greenland records, if they occur after a short interstadial. If the DO coolings transitions are actually associated with AMOC weakening (see the “Introduction”), our results may have an implication for the predicted weakening of the AMOC and its possible collapse in the future: the prediction with CSD indicators could be more difficult if the forcing changes fast. There is, however, a caveat to this implication because the past DO cooling transitions are different from the presently inferred AMOC changes. The time resolution (mainly

The statistical significance of precursor signals of critical transitions, in terms of positive trends of CSD indicators, is assessed by hypothesis testing

Here we describe specific settings for four conceptual models representing different candidate mechanisms for the DO cooling transitions. Unless otherwise mentioned, the stochastic differential equations below are solved with the Euler–Maruyama method with a step size of

The bistability of the AMOC strength

The FitzHugh–Nagumo-type (FHN-type) system is a prototypical model of slow–fast oscillators

To demonstrate the Hopf bifurcation mechanism in Fig. 5d and e, the same stochastic FHN-type model is used with

The mixed-mode oscillation model is obtained if the FHN-type model is extended to have multiple interacting slow variables. For example,

The Greenland ice core records used in this study can be obtained from

The supplement related to this article is available online at:

TM conceived the study and conducted the analyses with contributions from NB. Both authors discussed and interpreted the results. TM wrote the manuscript with contributions from NB.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors thank Keno Riechers and Maya Ben-Yami for their helpful comments.

The authors acknowledge funding by the Volkswagen Foundation. This is TiPES contribution #243; The TiPES (“Tipping Points in the Earth System”) project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement no. 820970. Niklas Boers acknowledges further funding by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 956170, as well as from the Federal Ministry of Education and Research under grant no. 01LS2001A.

This paper was edited by Bjørg Risebrobakken and reviewed by three anonymous referees.