These authors contributed equally to this work.

During the last glacial interval, the Northern Hemisphere climate was punctuated by a series of abrupt changes between two characteristic climate regimes.
The existence of stadial (cold) and interstadial (milder) periods is typically attributed to a hypothesised bistability in the glacial North Atlantic climate system, allowing for rapid transitions from the stadial to the interstadial state – the so-called Dansgaard–Oeschger (DO) events – and more gradual yet still fairly abrupt reverse shifts.
The physical mechanisms driving these regime transitions remain debated.
DO events are characterised by substantial warming over Greenland and a reorganisation of the Northern Hemisphere atmospheric circulation, which are evident from concomitant shifts in the

Recently, evidence was reported for the destabilisation of climatic subsystems likely caused by continued anthropogenically driven climate change

The possibility of alternative stable states of the entire climate system or its subsystems (and transitions between these) has been discussed at least since the 1960s

In this context, our study investigates the Dansgaard–Oeschger (DO) events, a series of abrupt warming events over Greenland first evidenced in stable water isotope records from Greenland ice cores

An important branch of research has assessed the performance of low-dimensional conceptual models in explaining the DO variability in the Greenland ice core records

In the state space spanned by

This article is structured as follows.
We first present the paleoclimate proxies analysed in this study and explain how we pre-processed the data to make them suitable for estimating the two-dimensional drift (Sect.

The analysis presented here is based primarily on the joint

The ratio of stable water isotopes, expressed as

In Fig.

The analysis conducted in this work relies on the following assumptions and technical conditions:

The data-generating process is sufficiently time-homogeneous over the considered time period.

The process is Markovian at the sampled temporal resolution.

The data are equidistant in time.

The relevant region of the state space is sampled sufficiently densely by the available data.

With regard to (i), a low-frequency influence of the background climate on the proxy values and on the frequency of DO events is evident (see Fig.

Unit root test of the detrended data.
ADF refers to the augmented Dickey–Fuller test;
ADF-GLS refers to the augmented Dickey–Fuller–GLS test.
We reject the presence of a unit root in each of the time series at

Stationarity tests provide further confirmation that the detrended data are free of any slow underlying trends:
we have applied two separate tests to assess the stationarity of the detrended data on timescales beyond single DO cycles.
These tests are the augmented Dickey–Fuller test (ADF) and the augmented Dickey–Fuller–GLS test (ADF-GLS).
Both tests test the possibility of a unit root in the time series (null hypothesis).
The alternative hypothesis is that the time series does not have a unit root; i.e. it is stationary.
We can safely reject the presence of a unit root in each time series at

There is a trade-off between conditions (i) and (iv) concerning the choice of the data window. While an even shorter window would assure time homogeneity of the dynamics with higher confidence, the sampling of the state space would become insufficiently sparse. The above choice (59–27 kyr b2k) guarantees a sufficient number of recurrences of the pre-processed two-dimensional trajectory to the relevant state space regions to perform statistical analysis. To obtain time-equidistant records (iii), the data are binned into temporally equidistant increments of 5 years.

Autocorrelation

The question of Markovianity (ii) is the most difficult to answer unambiguously.
Here we draw on the following heuristic argument:
the autocorrelation functions of the increments of both proxies shown in Fig.

Finally, further preconditions for our endeavour are the fact that the NGRIP record exhibits an exceptionally high resolution (iv) compared to other paleoclimate archives and that the two time series share the same time axis.

In this work, we treat the combined

In practice, in order to carry out the estimation of the first-order KM coefficients as defined in Eq. (

Similar to selecting the number of bins in a histogram, when employing kernel density estimation with a Nadaraya–Watson estimator for the Kramers–Moyal coefficients

The selection of an appropriate bandwidth

All numerical analyses were performed with Python's

We first discuss the two drift components

Two-dimensional drift reconstruction.

The estimated dust drift

The dust nullclines' structure supports the possibility for abrupt transitions in two ways:
either random fluctuations move the system across the unstable branch (if present, depending on the value of the control parameter), or the control parameter, in this case

We now focus on the reconstructed drift

Figure

Redrawing of Fig.

Above we argue for the existence of a double-fold bifurcation in the dust variable.
In order to show that the coupling of the dust and

We use the two-dimensional Kramers–Moyal equation to investigate the deterministic drift of the combined dust and

In the following, we discuss how the results presented here relate to the findings of previous studies.
An important branch of research around DO events draws on low-dimensional conceptional modelling and, related to that, inverse modelling approaches with model equations being fitted to ice core data.
Many of these studies build on stochastic differential equations and in particular on Langevin-type equations.
Our study follows the same key paradigm, regarding the paleoclimate record as the realisation of a stochastic process.
However, as far as we know, it is the first study to assess the two-dimensional drift non-parametrically in the

For the period investigated here

Our results contradict the interpretation that

Clearly, the state space spanned by

We have analysed the records of

Importantly, our findings question the prevailing interpretation of the two regimes observed in the isolated

Similar investigations to ours should be applied to other pairs of Greenland proxies to investigate the corresponding two-dimensional drift. Finally, our study underlines the need for higher-resolution data, as the scarcity of data points is a limiting factor for the quality of non-parametric estimates of the KM coefficients.

The code to reproduce the analysis and all figures is available at

All ice core data were obtained from the website of the Niels Bohr Institute of the University of
Copenhagen

KR and LRG designed the study with contributions from all authors. KR and LRG conducted the numerical analysis. KR and LRG wrote the paper with contributions from all authors.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Leonardo Rydin Gorjão and Dirk Witthaut gratefully acknowledge support from the Helmholtz Association via the grant “Uncertainty Quantification – From Data to Reliable Knowledge” (UQ; grant agreement no. ZT-I-0029). This work was performed by Leonardo Rydin Gorjão as part of the Helmholtz School for Data Science in Life, Earth and Energy (HDS-LEE). Niklas Boers acknowledges funding from the Volkswagen Foundation. This is TiPES contribution no. 162; the “Tipping Points in the Earth System” (TiPES) project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement no. 820970.

This research has received funding from the European Union's Horizon 2020 research and innovation programme (TiPES; grant no. 820970), the Helmholtz-Gemeinschaft (grant nos. ZT-I-0029 and HIDDS-0004), and the Volkswagen Foundation.This work was supported by the Technical University of Munich (TUM) in the framework of the Open Access Publishing Program.

This paper was edited by Valerio Lucarini and reviewed by Peter Ditlevsen, Tamas Bodai, Christian Franzke, Mohammed Reza Rahimi Tabar, and one anonymous referee.