Dansgaard–Oeschger (DO) events constitute the most pronounced mode of centennial to millennial climate variability of the last glacial period. Since their discovery, many decades of research have been devoted to understand the origin and nature of these rapid climate shifts. In recent years, a number of studies have appeared that report emergence of DO-type variability in fully coupled general circulation models via different mechanisms. These mechanisms result in the occurrence of DO events at varying degrees of regularity, ranging from periodic to random. When examining the full sequence of DO events as captured in the North Greenland Ice Core Project (NGRIP) ice core record, one can observe high irregularity in the timing of individual events at any stage within the last glacial period. In addition to the prevailing irregularity, certain properties of the DO event sequence, such as the average event frequency or the relative distribution of cold versus warm periods, appear to be changing throughout the glacial. By using statistical hypothesis tests on simple event models, we investigate whether the observed event sequence may have been generated by stationary random processes or rather was strongly modulated by external factors. We find that the sequence of DO warming events is consistent with a stationary random process, whereas dividing the event sequence into warming and cooling events leads to inconsistency with two independent event processes. As we include external forcing, we find a particularly good fit to the observed DO sequence in a model where the average residence time in warm periods are controlled by global ice volume and cold periods by boreal summer insolation.

During the last glacial period, lasting from approximately 120
to 12 kya BP (thousands of years before present), a large number of abrupt
large-scale climate changes have been recorded in Greenland ice cores
and other Northern Hemisphere climate proxies. These so-called
Dansgaard–Oeschger (DO) events

NGRIP oxygen isotope ice core record in
20-year binned resolution and associated Dansgaard–Oeschger
warming and cooling events. The numbers above the time series indicate the
warming transitions into the respective Greenland interstadials. The
nomenclature is adopted from

Since the discovery of these unexpected climate events with no known cause,
questions of this kind have been addressed. Whereas high-resolution coupled
climate models under glacial conditions typically lack DO-type variability,
models of intermediate complexity and simpler conceptual models have been
proposed to explain qualitative features of the sequence of last glacial
climate changes. Starting from the discovery of an approximate 1500-year
spectral signature in the GISP2 ice core record

In this work, we want to expand on this idea by testing whether the observed sequence of events is indeed consistent with one or more random, stationary processes, or whether the changes over time of the properties of the observed event sequence require modulating parameters of the governing process over time. To this end, we consider the whole glacial period, as opposed to previous efforts focusing on a rather regular period in the middle of the glacial. We investigate two different levels in detail of description by first only regarding the sequence of warming events and second the combined sequence of alternating transitions in between cold and warm conditions. We proceed by testing two null hypotheses: (1) the sequence of DO warming events is a realization of a Poisson process with fixed rate parameter; (2) the sequence of stadials and interstadials is a realization of two independent Poisson processes with fixed rate parameters giving rise to transitions in between stadials and interstadials. In order to test the hypotheses, we consider the evolution of the number of warming events in a moving window of 20 kyr. This quantity measures how variable the average event frequency is over time, a property which we denote as irregularity, and in the DO sequence it deviates strongly from a constant occurrence frequency of events over time. We test whether samples from the abovementioned stationary processes show similar irregularity.

In addition to the evolution of the frequency of warming events we look at the evolution of the abundance of the stadial over the interstadial condition, which changes significantly over time in the DO sequence. This additional non-stationary structure in the data is the basis for another hypothesis test we perform. Finally, we test how the models' support with respect to the data is improved as we force the rate parameters with a combination of a global climate proxy and orbital variations of insolation to incorporate changing background climate conditions. The main findings of this study are as follows: (1) a Poisson process with fixed rate parameter, modeling warming transitions only, is consistent with the time variations in the NGRIP DO warming event sequence; (2) a model composed of two independent stationary Poisson processes governing transitions in between stadials and interstadials is not consistent with the time variations in the observed DO event sequence; (3) forcing the aforementioned models with a combination of a global ice volume proxy and a summer insolation curve leads to good statistical agreement with the observed sequence. Specifically, we find good agreement for a model with two individual processes, where the average transition rate from interstadial to stadial is controlled by global ice volume forcing, obtained from independent ocean core isotope records, and the average transition rate from stadial to interstadial is controlled by boreal summer insolation.

The paper is structured in the following way. In Sect. 2 we introduce in more detail the data used in this study, the summary statistics used to investigate irregularity in the event series, the models used to explain the data and the hypothesis test procedure. In Sect. 3 we present the results of the hypothesis tests on the different models. We discuss and interpret the results in Sect. 4.

Our study of the sequence of DO events is based on the refined dating
represented by the GICC05 timescale

Time-varying irregularity indicators
calculated from the NGRIP DO sequence, and climate forcings.

Given sequence and timing of transitions in between stadials and
interstadials, we construct time-varying indicators of irregularity in the
sequence of event timings, which are shown in Fig.

While no significant correlation between duration of individual stadials and
preceding or subsequent interstadial is observed (Pearson's

We now describe the models which are used to evaluate our hypotheses on the
data using the test statistics described above. The first model used in our
study models the process generating the sequence of warming events as a
Poisson process with fixed rate parameter

As a second model, labeled “two-process model” hereafter, we propose two
individual processes for generating warming transitions from stadials to
interstadials and cooling transitions from interstadials back to stadials.
Each is represented by a Poisson process with a fixed rate

The average interstadial and stadial durations of the data seem to behave differently over the course of the glacial, as captured by our second test statistic. This motivates us to study whether this behavior is likely to be encountered by chance assuming randomness and independence of both warming and cooling transitions.

As comparison to our hypothesis of stationary random processes, we consider
the same models with time-varying rate parameters, which are given by a
linear combination of two external climate factors:

The hypothesis tests are performed in the following way. For a given model we
simulate a large number of realizations, which are collections of subsequent
events with the same total duration as the record (104 kyr). For each
realization we calculate the time-varying indicator of interest and the
corresponding scalar test statistic. We then use the distribution of test
statistics for a one-sided hypothesis test. The test simply counts how many
test statistics in the ensemble are as large as or larger than the test
statistic obtained from the data. Divided by the sample size, this yields a

The results of the hypothesis test on the stationary one- and two-process
models are shown in Fig.

Empirical distributions from Monte Carlo
simulation of the test statistics for the stationary

To better visualize the outcomes of the hypothesis tests, we show confidence
bands for the time-varying indicators from our Monte Carlo simulations in
Fig.

Point-wise 95 % confidence bands and
model mean (black line) for the time-varying indicators

In the following we present the hypothesis tests performed on the one- and
two-process models forced with insolation

The best-fit two-process model has warming transition rate

Time-varying transition rate parameters of
the best-fit one-process

The hypothesis tests for the fitted models are shown in
Fig.

Summary of model parameters, hypothesis test results and goodness of fit of the mean model time-varying indicators with respect to the data.

We additionally report how the goodness of fit of the forced models changes
when using only partial forcing and thus a reduced number of parameters. When
forcing the one-process model with both ice volume and insolation, we yield
an
RMSD of the model mean

We show the mean time-varying indicators for this model in
Fig.

Point-wise 95 % confidence bands
and model mean (black curve) for the time-varying indicators

Our first result considers only the warming events. While the distribution of waiting times in between warming events is well modeled by an exponential distribution (not shown here), we show that the number of events in a moving window of 20 kyr (and thus the mean waiting time) clearly changes over time, but no more than would be expected from a realization of a stationary Poisson process. Thus, if there is a unique process giving rise to the warming transitions, it need not be changing over time due to external factors. Although the description of DO events solely by the timing of the abrupt warming is very simplistic, we still think it is a useful result since the abrupt warming events are the most robust feature in ice cores and other proxy records and are commonly used to assess synchronicity and pacing of abrupt climate change in the last glacial.

The second result indicates, however, limits to the stationarity in the
sequence of events as we increase the detail of description. Assuming two
independent processes giving rise to transitions from stadials to
interstadials and vice versa, the null hypothesis of stationarity can be
rejected with both our statistics. Specifically, both the variations over
time of the number of warming events and the relative durations of stadials
and interstadials are too large to be consistent with our two-process model
using constant parameters. This model gives rise to a more regular sequence
of warming events, compared to the one-process model. This is because one DO
cycle is the sum of two independent processes and thus its duration does not
follow an exponential distribution (coefficient of variation

Next, we investigated improvements of the consistency of the models with the
data by allowing their parameters to vary over time as linear combination of
two climate forcings. Choosing the best-fit linear combination of forcings,
we found the average time-varying indicators of both models to match very
well to the data curve. Thus, whereas the data were seen as a rather
out-lying realization consistent with a one-process model but not with a
two-process model, when introducing forcings the data become the
expected behavior of the models. The goodness of fit follows from the
correlation of the time-varying indicators and the forcings, which can be
seen in Fig.

Finally, we discuss the importance of ice volume and insolation in the
best-fit one- and two-process models. In the one-process model,
Eq. (

In the two-process model the warming and cooling transition rates are
influenced by the forcing in opposite ways, as can be seen from
Eq. (

An exhaustive investigation of whether our model description and subsequent
findings are consistent with governing mechanisms for DO-type variability
inferred from detailed data and realistic model studies is beyond the scope
of this paper. Nevertheless we conclude the discussion with some
interpretations which are more speculative in nature. We begin with
insolation control on stadial duration. Boreal summer insolation might
influence the occurrence frequency of warming transitions by modulating the
ice-ocean albedo feedback, which amplifies break-up or export of larger areas
of sea ice. Sea ice decrease could subsequently cause rapid warming through
subsurface ocean heat release

In conclusion, we show that the long-term variations in DO warming event frequency, often described as millennial climate activity, are consistent with a memory-less stationary random process. From the data at hand we cannot exclude the possibility that the long-term variations occurred by chance. If we however divide a DO cycle into two independent processes governing warming and cooling, this is not true anymore and significant time-varying structure is detected. We thus propose a model that incorporates long-term variations through forcing of the parameters with external climate factors. We find good agreement with the data in a model where the mean duration of interstadial phases of the DO cycle is controlled by global ice volume and the stadial phases by boreal summer insolation. This finding can help to differentiate the mechanisms proposed to cause DO events.

The principle data that are
used are the timings reported in a table in Rasmussen et al. (2014) (cited in
the article). The insolation data are publicly available as supporting online
material to Huybers (2006) (DOI:

The authors declare that they have no conflict of interest.

This project has received funding from the European Union's Horizon 2020 research and innovation Programme under the Marie Skłodowska-Curie grant agreement no. 643073. Edited by: Luke Skinner Reviewed by: Valerie N. Livina and Takahito Mitsui