Numerical climate simulations produce vast amounts of high-resolution data. This poses new challenges to the palaeoclimate community – and indeed to the broader climate community – in how to efficiently process and interpret model output. The palaeoclimate community also faces the additional challenge of having to characterise and compare a much broader range of climates than encountered in other subfields of climate science. Here we propose an analysis framework, grounded in dynamical systems theory, which may contribute to overcoming these challenges. The framework enables the characterisation of the dynamics of a given climate through a small number of metrics. These may be applied to individual climate variables or to several variables at once, and they can diagnose properties such as persistence, active number of degrees of freedom and coupling. Crucially, the metrics provide information on instantaneous states of the chosen variable(s). To illustrate the framework's applicability, we analyse three numerical simulations of mid-Holocene climates over North Africa under different boundary conditions. We find that the three simulations produce climate systems with different dynamical properties, such as persistence of the spatial precipitation patterns and coupling between precipitation and large-scale sea level pressure patterns, which are reflected in the dynamical systems metrics. We conclude that the dynamical systems framework holds significant potential for analysing palaeoclimate simulations. At the same time, an appraisal of the framework's limitations suggests that it should be viewed as a complement to more conventional analyses, rather than as a wholesale substitute.

Numerical climate models have enjoyed widespread use in palaeoclimate studies, from early investigations based on simple thermodynamic or general circulation models

A related, yet distinct, challenge faced by the palaeoclimate community are the large uncertainties often found in palaeo-simulations. These reflect the uncertainties in palaeo-archives and in our knowledge of the boundary conditions and forcings affecting past climates

Here, we propose an analysis framework which addresses the challenges of efficiently processing and interpreting large amounts of model output to compare different simulated palaeoclimates. The framework is grounded in dynamical systems theory and enables the characterisation of the dynamics of a given dynamical system – for example the atmosphere – through three one-dimensional metrics. The first metric estimates the persistence of instantaneous states of the system. We term this metric rather mundanely “persistence”. The second metric, which we term “local dimension”, provides information on how the system evolves to or from instantaneous states. Finally, the co-recurrence ratio is a metric applicable to two (or more) variables, which quantifies their instantaneous coupling. In other words, the dynamical information embedded in three-dimensional (latitude, longitude and time) or four-dimensional (latitude, longitude, pressure level and time) data, commonly produced by climate models, can be projected onto three metrics, which each provide a single value for every time step in the data. These may then be interpreted and compared with relative ease.

The rest of this technical note is structured as follows: in Sect.

In Sect.

The dynamical systems framework we propose rests on three indicators. All are instantaneous in time, meaning that given a long data series, they provide a value for each time step. For example, if we were to analyse daily latitude–longitude SLP in a given geographical region over 30 years, we would have 30

The first indicator, termed “local dimension” (

The second indicator, termed “persistence” (

Both indicators may be used to characterise the dynamics underlying complex systems, including the Earth's climate

Unlike the first two, the third metric we propose here, termed “co-recurrence ratio” (

The three dynamical systems metrics described above issue from the combination of extreme value theory with Poincaré recurrences

We next define a high-threshold

Schematic of the computation of the dynamical systems metrics for a state

Finally, we define the co-recurrence ratio by considering two trajectories

The above derivations all rely on the definition of recurrences relative to a threshold

The analytical derivation of the above framework makes a number of assumptions that are typically not realised for climate data. For example, one has to take into account both the finite length of the datasets and non-stationarities such as those issuing from internal low-frequency variability or varying external forcing. A formal justification of the applicability of the dynamical systems metrics to finite data issues from the results of

MATLAB code to compute

Today, the Sahara is the largest hot desert on Earth. Most of the precipitation in north-western Africa is associated with the West African Monsoon (WAM), which reaches to around 16–17

The most recent AHP peaked during the mid-Holocene (MH), approximately 9000–6000 years BP. It is thought to have coincided with an intensification and northward shift of the WAM, allowing the presence of vegetation, lakes and wetlands in areas that today are desert

Numerical climate simulations of the MH have struggled to reproduce the full extent of the monsoonal intensification suggested by the palaeo-archives, and commonly suffer from a dry bias

Here, we analyse the simulations used in

To illustrate the dynamical systems approach described in Sect.

We analyse 30 years of daily data of sea level pressure (SLP), 500 hPa geopotential height (Z500) and precipitation frequency (prp) for each simulation. Precipitation frequency is constructed by assigning a value of 1 to grid points and time steps with non-zero precipitation and a value of 0 otherwise. This is preferable to using raw precipitation data for estimating the dynamical systems metrics (and

The main interest in analysing the above simulations lies in understanding whether and why they reproduce different hydroclimates over the Sahelian–Saharan region. Our aim in this section is not to systematically investigate these two aspects but instead to illustrate how the dynamical systems framework proposed here can be used to characterise the individual simulations and provide a concise overview of the differences between them. We argue that such an approach can provide a valuable complement to conventional analyses, and we relate our results to those obtained in earlier studies

JJAS precipitation (mm d

A simple composite of JJAS average precipitation immediately highlights large differences in the precipitation regimes, with the MH

Seasonal cycle of median

We begin by studying the seasonality of

JJAS precipitation anomalies (mm d

The seasonal variations in

Seasonal cycle of median

We next try to understand the physical processes underlying the differences in precipitation in the three simulations, by computing the co-recurrence ratio

JJAS precipitation (colours, mm d

JJAS precipitation (colours, mm d

As for

The above results illustrate some of the strengths and limitations of the analysis framework we propose in this work, which we discuss further in Sect.

Palaeoclimate simulations of the same period and region may yield very different results, the understanding of which requires analysis tools that may efficiently distil the discrepancies and point to possible underlying drivers. In this technical note, we have outlined an analysis framework which can efficiently compare the salient dynamical features of different simulated palaeoclimates. The framework is grounded in dynamical systems theory and rests on computing three metrics: the local dimension

From a theoretical standpoint, the dynamical systems framework presents a number of advantages over other statistical approaches for the analysis of large amounts of climate data such as clustering, principal component analysis or canonical correlation analysis. The first two are often used to define climate variability modes or weather regimes. The

Because of these characteristics, the dynamical systems metrics can be particularly helpful when processing large datasets (see, e.g.

As a caveat, we note that our approach is more successful in providing insights into the changes between the control and each of the Green Sahara simulations than between the latter two simulations. Previous analyses of these same simulations and studies from other authors

Additionally, obtaining good estimates of

In this technical note, we aimed to give a taste of the dynamical systems framework's possible application to palaeoclimate simulations, as opposed to presenting a systematic analysis. We specifically wished to highlight its potential for comparing different palaeoclimates while also providing an appraisal of its limitations. To do so, we focussed on three existing simulations and on a small number of atmospheric variables. However, the approach is relevant to a very broad range of palaeoclimate applications and is thus not limited to the comparison of different climates or to the atmosphere. In particular, the co-recurrence coefficient could be used to study interactions between the different components of the climate system varying on different timescales, such as the hydrosphere and the atmosphere or the hydrosphere and the cryosphere (e.g. by comparing the response of different numerical models to the same forcing). As mentioned above,

In this appendix, we provide a schematic of the raindrop analogy for the dynamical systems metrics and figures illustrating the sensitivity of our results to the choice of geographical domain and season. The figures are discussed in the main text.

The raindrop analogy for the dynamical systems metrics.

The same as Fig.

The same as Fig.

The same as Fig.

The same as Fig.

The same as Fig.

The same as Fig.

The code to compute the three dynamical systems indicators used in this study is made freely available through the cloud storage of the Centre National de la Recherche Scientifique (CNRS) under a CC BY-NC 3.0 license:

The EC-Earth model data are stored as global 3-D or 4-D NetCDF files and exceed the size limitations of most online repositories. The files needed to reproduce the results presented in this study may be obtained upon request to the corresponding author.

GM conceived the study and performed the analysis. DF provided the publicly available code. Both authors contributed to drafting the manuscript.

The authors declare that they have no conflict of interest.

The authors thank Francesco Pausata, Qiong Zhang and Marco Gaetani for making the palaeoclimate simulations available. We also thank the two anonymous reviewers for the detailed and pertinent comments they provided.

Gabriele Messori has been partly supported by the Swedish Research Council Vetenskapsrådet (grant no. 2016-03724) and the Swedish Research Council for Sustainable Development FORMAS (grant no. 2018-00968). Davide Faranda was supported by a CNRS/INSU LEFE/MANU grant (DINCLIC project) and by an ANR-TERC grant (BOREAS project).The article processing charges for this open-access publication were covered by Stockholm University.

This paper was edited by Martin Claussen and reviewed by Christian Franzke and one anonymous referee.