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In the climate sciences, highly simplified nonlinear models are useful tools for understanding and discussing tipping points. However, the economic models used to study their coupling to the economy, as in Integrated Assessment Models (IAMs), are typically linear and represent an inertia-free economy in equilibrium. This representation is challenged by persistent unemployment, recessions, and changing economic institutions. Therefore, we investigate the non-equilibrium dynamics of the economy and the corresponding tipping from equilibrium to so-called endogenous business cycles. To this end, we build a basic Solow-type equilibrium growth model that incorporates, in a highly simplified manner, frictions and delay in the labor system. When the delay exceeds a critical value of 3.4 days, business cycles with periodic unemployment and recessions arise in our minimal business cycle (MinBC) model. Given a dynamic investment mode, the MinBC's cyclic economy responds to external forcing asymmetrically throughout the cycle. Advanced time series analysis methods are applied to macroeconomic data sets to evaluate the realism of the model's response, with encouraging results.Our study is a step towards understanding the evolution of the sources of internal economic variability. Such an understanding is needed to represent the extent of coupling between the earth system and the economy.

Predicting the temporal and spatial patterns of South Asian monsoon rainfall within a season is of critical importance due to its impact on agriculture, water availability, and flooding. The monsoon intraseasonal oscillation (MISO) is a robust northward-propagating mode which determines the active and break phases of the monsoon, and much of the regional distribution of rainfall. However, dynamical atmospheric forecast models predict this mode poorly. Data-driven methods for MISO prediction have shown more skill, but only predict the rainfall portion corresponding to MISO.Here, we combine state-of-the-art ensemble precipitation forecasts from a high-resolution atmospheric model with data-driven forecasts of MISO using a novel method. The ensemble members of the detailed atmospheric model are projected onto a lower-dimensional subspace corresponding to the MISO dynamics, and are then weighted according to their distance from the data-driven MISO forecast in this subspace. We thereby achieve significant improvements in rainfall forecasts over India, as well as the broader monsoon region, at 10–30 day lead times, an interval that is generally considered as a predictability gap. Our results demonstrate the potential of leveraging the predictability of intraseasonal oscillations to improve extended-range forecasts; more generally, they point towards a future of combining dynamical and data-driven forecasts for Earth system prediction.

There is a history of simple error growth models designed to capture the key properties of error growth in operational numerical weather prediction models. We propose here such a scalar model that relies on the previous ones, but captures the effect of small scales on the error growth via additive noise in a nonlinear stochastic differential equation (SDE). We nondimensionalize the equation and study its behavior with respect to the error saturation value, the growth rate of small errors, and the magnitude of noise. We show that the addition of noise can change the curvature of the error growth curve. The SDE model seems to improve substantially the fit to operational error growth curves, compared to the deterministic counterparts.

The definition of climate itself cannot be given without a proper understanding of the key ideas of long-term behavior of a system, as provided by dynamical systems theory. Hence, it is not surprising that concepts and methods of this theory have percolated into the climate sciences as early as the 1960s. The major increase in public awareness of the socio-economic threats and opportunities of climate change has led more recently to two major developments in the climate sciences: (i) the Intergovernmental Panel on Climate Change's successive Assessment Reports; and (ii) an increasing understanding of the interplay between natural climate variability and anthropogenically driven climate change. Both of these developments have benefitted from remarkable technological advances in computing resources, in throughput as well as storage, and in observational capabilities, regarding both platforms and instruments. Starting with the early contributions of nonlinear dynamics to the climate sciences, we review here the more recent contributions of (a) the theory of nonautonomous and random dynamical systems to an understanding of the interplay between natural variability and anthropogenic climate change; and (b) the role of algebraic topology in shedding additional light on this interplay. The review is thus a trip leading from the applications of classical bifurcation theory to multiple possible climates to the tipping points associated with transitions from one type of climatic behavior to another in the presence of time-dependent forcing.

For many of us, the Covid-19 pandemic brought long-time scientific interest in epidemiology to the point of involvement. An important aspect of the evolution of acute respiratory epidemics is their seasonal character. Our toolkit for handling seasonal phenomena in the geosciences has increased in the last dozen years or so with the development and application of concepts and methods from the theory of nonautonomous and random dynamical systems (NDSs and RDSs). In this talk, I will briefly: Introduce some elements of these two closely related theories. Illustrate the two with an application to seasonal effects within a chaotic model of the El Niño–Southern Oscillation (ENSO). Introduce to a geoscientific audience a simple epidemiological “box” model of the Susceptible–Exposed–Infectious–Recovered (SEIR) type. Summarize NDS results for a chaotic SEIR model with seasonal effects. Mention the utility of data assimilation (DA) tools in the parameter identification and prediction of an epidemic’s evolution References - Chekroun, M D, Ghil M, Neelin J D (2018) Pullback attractor crisis in a delay differential ENSO model, in Nonlinear Advances in Geosciences, A. Tsonis (Ed.), Springer, pp. 1–33, doi: 10.1007/978-3-319-58895-7 - Crisan D, Ghil, M (2022) Asymptotic behavior of the forecast–assimilation process with unstable dynamics, Chaos, in preparation - Faranda D, Castillo I P, Hulme O, Jezequel A, Lamb J S, Sato Y, Thompson E L (2020) Asymptotic estimates of SARS-CoV-2 infection counts and their sensitivity to stochastic perturbation - Ghil, M (2019) A century of nonlinearity in the geosciences. Earth & Space Science 6:1007–1042, doi:10.1029/2019EA000599 - Kovács, T (2020) How can contemporary climate research help understand epidemic dynamics? Ensemble approach and snapshot attractors. J. Roy. Soc. Interface, 17(173):20200648, doi: 10.1098/rsif.2020.0648

Paleoclimate proxy records contain abrupt transitions that may represent former instances of the climate system crossing a tipping point (TP). Properly identifying these TPs in the Earth’s past helps determine critical thresholds in present-day climate and better understand the climate system’s underlying bifurcation mechanisms.Information contained in paleoclimate proxy records is often ambiguous because of the complexity of the system, which includes both deterministic and stochastic processes. Furthermore, paleoclimate time series differ in their time spans and periodicities, and often have high levels of noise and a nonuniform resolution. These combined sources of uncertainty highlight the need for using advanced statistical methods for robustly identifying and comparing TPs.A recently developed method that uses an augmented Kolmogorov-Smirnov test has been shown to be highly effective for transition detection in different types of records. Here, we apply this method to a set of high-quality paleoproxy records exhibiting centennial-to millennial-scale variability that have been compiled in the PaleoJump database. We thereby detect previously unrecognized transitions in these records and identify potential TPs. Furthermore, we investigate regime changes with recurrence analysis and spectral analysis. This study is supported by the H2020-funded Tipping Points in the Earth System (TiPES) project.

The topology of the branched manifold associated with the Lorenz model’s random attractor (LORA) evolves in time. LORA’s time-evolving branched manifold robustly supports the point cloud associated with the system’s invariant measure at each instant in time. This manifold undergoes not only continuous deformations — with branches that bend, stretch or compress — but also discontinuous deformations, with branches that intersect at discrete times. These discontinuities in the system's invariant measure manifest themselves in the decrease or increase of the number of 1-holes, thus producing abrupt changes in the branched manifold’s topology.Topological tipping points (TTPs) are defined as abrupt changes in the topology of a random attractor’s branched manifold. Branched Manifold Analysis through Homologies(BraMAH) is a robust method that allows one to detect these fundamental changes. The existence of such TTPs is being confirmed by careful statistical analysis of LORA’s time-evolving branched manifold, following up on Charó et al. (Chaos, 2021, doi:10.1063/5.0059461). Research is being pursued on early warning signals for these TTPs, concentrating on local fluctuations in the system’s invariant measure.

The dynamics of systems with time-dependent forcing or coefficients has become a matter of considerable interest in the last couple of decades in general and in the last dozen years or so in the climate sciences in particular (Ghil, 2019; Ghil & Lucarini, 2020; Ghil, 2021; Tel et al., 2021; and references therein). We shall provide a general introduction to the topic and illustrate it with several paleoclimate-related examples (Crucifix, 2012; Riechers et al., 2022; Rousseau et al., 2022). Perspectives for further applications of the concepts and methods of the theory of pullback and random attractors and of their tipping points to paleoclimate will also be provided.ReferencesCrucifix, M.: Oscillators and relaxation phenomena in Pleistocene climate theory, PTRSA, 370, 1140–1165, 2012. Ghil, M., 2019: A century of nonlinearity in the geosciences, Earth & Space Science, 6, 1007–1042, doi: 1029/2019EA000599. Ghil, M., 2020: Review article: Hilbert problems for the climate sciences in the 21st century – 20 years later, Nonlin. Processes Geophys., 27, 429–451, https://doi.org/10.5194/npg-27-429-2020. Ghil, M., and V. Lucarini, 2020: The physics of climate variability and climate change, Mod. Phys., 92(3), 035002, doi: 10.1103/RevModPhys.92.035002. Riechers, K., T. Mitsui, N. Boers, and M. Ghil, 2022: Orbital insolation variations, intrinsic climate variability, and Quaternary glaciations, Clim. Past Discuss. [preprint], https://doi.org/10.5194/cp-2021-136, in review. Rousseau, D.-D., W. Bagnewski, and M. Ghil, 2021: Abrupt climate changes and the astronomical theory: are they related?, Clim. Past, accepted, doi: 10.5194/cp-2021-103 . Tél, T., Bódai, T., Drótos, G., Haszpra, T., Herein, M., Kaszás, B. and Vincze, M., 2020. The theory of parallel climate realizations. Journal of Statistical Physics, 179(5), 1496–1530.

Data assimilation (DA) aims to optimally combine model forecasts and noisy observations. Multi-model DA generalizes the variational or Bayesian formulation of the Kalman filter, and we prove here that it is also the minimum variance linear unbiased estimator. However, previous implementations of this approach have not estimated the model error, and have therewith not been able to correctly weight the separate models and the observations. Here, we show how multiple models can be combined for both forecasting and DA by using an ensemble Kalman filter with adaptive model error estimation. This methodology is applied to the Lorenz-96 model and it results in significant error reductions compared to the best model and to an unweighted multi-model ensemble.

The Earth’s climate has experienced numerous abrupt and critical transitions during its long history. Such transitions are evidenced in precise, high-resolution records at different timescales. This type of evidence suggests the possibility of identifying a hierarchy of past critical events, which would yield a more complex perspective on climatic history of the than the classical saddle-node two-dimension representation of tipping points. Such a context allows defining a tipping, or dynamical, landscape (Lucarini and Bódai, 2020), similar to the epigenetic landscape of Waddington (1957).To illustrate a richer structure of critical transitions, we have analyzed 3 key high-resolution datasets covering the past 66 Ma and provided evidences of abrupt transitions detected with the augmented Kolmogorov-Smirnov test and a recurrence analysis (Bagniewski et al., 2021). These time series are the CENOGRID benthic d18O and d13C (Westerhold et al., 2020), the U1308 benthic d18O, d13C and the d18bulk carbonate (Hodell and Channell, 2016), and the NGRIP d18O (Rasmussen et al., 2014) records. The aim was to examine objectively the observed visual evidence of abrupt transitions and to identify among them the key thresholds indicating regime changes that differentiate among major clusters of variability. This identification is followed by establishing a hierarchy in the observed thresholds organized through a domino-like cascade of abrupt transitions that shaped the Earth’s climate system over the past 66 Ma.This study is supported by the H2020-funded Tipping Points in the Earth System (TiPES) project.ReferencesBagniewski, W., Ghil, M., and Rousseau, D. D.: Automatic detection of abrupt transitions in paleoclimate records, Chaos, 31, https://doi.org/10.1063/5.0062543, 2021.Hodell, D. A. and Channell, J. E. T.: Mode transitions in Northern Hemisphere glaciation: co-evolution of millennial and orbital variability in Quaternary climate, Clim. Past, 12, 1805–1828, https://doi.org/10.5194/cp-12-1805-2016, 2016.Lucarini, V. and Bódai, T.: Global stability properties of the climate: Melancholia states, invariant measures, and phase transitions, Nonlinearity, 33, R59–R92, https://doi.org/10.1088/1361-6544/ab86cc, 2020.Rasmussen, S. O., Bigler, M., Blockley, S. P., et al.: A stratigraphic framework for abrupt climatic changes during the Last Glacial period based on three synchronized Greenland ice-core records: refining and extending the INTIMATE event stratigraphy, Quat. Sci. Rev., 106, 14–28, https://doi.org/10.1016/j.quascirev.2014.09.007, 2014.Waddington, C. H.: The strategy of the genes., Allen & Unwin., London, 1957.Westerhold, T., Marwan, N., Drury, A. J., et al.: An astronomically dated record of Earth’s climate and its predictability over the last 66 million years, Science, 369, 1383-+, https://doi.org/10.1126/science.aba6853, 2020.

The relative role of external forcing and of intrinsic variability is a key question of climate variability in general and of our planet's paleoclimatic past in particular. Over the last 100 years since Milankovic's contributions, the importance of orbital forcing has been established for the period covering the last 2.6 Myr and the Quaternary glaciation cycles that took place during that time. A convincing case has also been made for the role of several internal mechanisms that are active on timescales both shorter and longer than the orbital ones. Such mechanisms clearly have a causal role in Dansgaard–Oeschger and Heinrich events, as well as in the mid-Pleistocene transition. We introduce herein a unified framework for the understanding of the orbital forcing's effects on the climate system's internal variability on timescales from thousands to millions of years. This framework relies on the fairly recent theory of non-autonomous and random dynamical systems, and it has so far been successfully applied in the climate sciences for problems like the El Niño–Southern Oscillation, the oceans' wind-driven circulation, and other problems on interannual to interdecadal timescales. Finally, we provide further examples of climate applications and present preliminary results of interest for the Quaternary glaciation cycles in general and the mid-Pleistocene transition in particular.

We present a simple innovation-based model error covariance estimation method for Kalman filters. The method is based on Berry and Sauer (2013) and the simplification results from assuming known observation error covariance. We carry out experiments with a prescribed model error covariance using a Lorenz (1996) model and ensemble Kalman filter. The prescribed error covariance matrix is recovered with high accuracy.

Abrupt climate changes constitute a relatively new field of research, which addresses variations occurring in a relatively short time interval of tens to a hundred years. Such time scales do not correspond to the tens or hundreds of thousands of years that the astronomical theory of climate addresses. The latter theory involves parameters that are external to the climate system and whose multi-periodic variations are reliably known and almost constant for a large extent of Earth history. Abrupt changes, conversely, appear to involve fast processes that are internal to the climate system; these processes varied considerably during the past 2.6 Myr, and yielded more irregular fluctuations. In this paper, we re-examine the main climate variations determined from the U1308 North Atlantic marine record, which yields a detailed calving history of the Northern Hemisphere ice sheets over the past 3.2 Myr. The magnitude and periodicity of the ice-rafted debris (IRD) events observed in the U1308 record allow one to determine the timing of several abrupt climate changes, the larger ones corresponding to the massive iceberg discharges labeled Heinrich events (HEs). In parallel, abrupt warmings, called Dansgaard-Oeschger (DO) events, have been identified in the Greenland records of the last glaciation cycle. Combining the HE and DO observations, we study a complex mechanism that may lead to the observed millennial-scale variability corresponding to the abrupt climate changes of last 0.9 Myr. This mechanism relies on amended Bond cycles, which group DO events and the associated Greenland stadials into a trend of increased cooling, with IRD events embedded into every stadial, the latest of these being an HE. These Bond cycles may have occurred during the last 0.9 Ma when Northern Hemisphere ice sheets reached their maximum extent and volume, thus becoming a major player in this time interval’s climate dynamics. Since the waxing and waning of ice sheets during the Quaternary period are orbitally paced, we conclude that the abrupt climate changes observed during the Mid and Upper Pleistocene are therewith indirectly linked to the astronomical theory of climate.

Tipping points (TPs) in the Earth system have been studied with growing interest and concern in recent years due to the potential risk of anthropogenic forcing causing abrupt, and possibly irreversible, climate transitions. Paleoclimate records are essential for identifying TPs in the Earth’s past and to properly understand the climate system’s underlying bifurcation mechanisms. Due to their varying quality, resolution, and dating methods, it is often necessary to select the records that give the best representation of past climate. Furthermore, as paleoclimate records vary in their origin, time spans, and periodicities, an objective, automated methodology is crucial for identifying and comparing TPs. To reach this goal, here we present the PaleoJump database of carefully selected, high-resolution records originating in ice, marine sediments, speleothems, loess, and lake sediments. These records, which include tipping elements, cover long time intervals and represent a global distribution from all continents and ocean basins. For every record, a transition detection methodology based on an augmented Kolmogorov-Smirnov test is applied to identify abrupt transitions. The PaleoJump database highlights these automatically detected transitions for every record together with other essential information, including location, temporal scale and resolution, as well as temporal plots; it therefore represents a valuable resource for researchers investigating TPs in past climates. This study is supported by the H2020-funded TiPES project.

The Pleistocene climate was dominated by alternating retreat and regrowth of massive ice sheets accompanied by large variations in the global mean temperature and sea level. Partial agreement between the power spectra of global ice volume proxies and high-latitude summer insolation provides evidence that quasi-periodic changes in the earth’s orbital configuration affect the timing of glaciations and deglaciations. It remains, however, a topic of active debate whether the main cause of glacial cycles is an internal self-sustained oscillation of the climate system that merely phased locked, more or less, to orbital forcing or whether glacial cycles could not exist at all in the absence of orbital forcing. Furthermore, it is unclear whether past ice volume records should be regarded as the result of a purely deterministic process or as a randomly selected trajectory of a stochastic process. To study plausible paths of the earth’s climate system given the orbital forcing, we compute the pullback attractors of several conceptual Pleistocene models. The results are confronted with the power spectra, as well as the time series of proxy records and conclusions will be drawn about the role of internal vs. forced variability and the possible contribution of stochastic processes to the mix of causes. We argue, moreover, that the explanatory power of either a deterministically chaotic or a dynamic-stochastic model cannot be assessed by comparing the model output to observations in the time domain alone.

Early evidence of abrupt transitions in Camp Century and Dye 3 Greenland ice cores (Dansgaard et al. 1982) has recently been reinforced by the identification of additional abrupt transitions in the NGRIP ice core (Rasmussen et al. 2014). These additional events correspond to changes of either short duration or amplitude of d18O that visual or statistical inspections do not necessarily validate. Abrupt transitions have been described for marine (Bond et al. 1992) and continental (Wang et al. 2001) records as well, and they provide a broader spatial perspective. Finally, abrupt transitions have also been documented over much deeper timescales (Zachos et al., 2001, Hodell & Channel, 2016, Westerhold et al. 2020). In spite of the variable time resolution of all these records, the abrupt transitions seem to reflect the individual impact of external forcing, of internal climate variability, or a combination of the two on Earth’s climate system. To illustrate this, we have analyzed 4 reference datasets with timescales ranging from one glacial cycle — i.e., the last 130,000 years — to the last 70 Ma. We show patterns that repeat within a single glacial cycle and seem to be related to internal variability, along with patterns associated with longer time periods and possibly related to external forcing; such forcing may arise from changes in either Earth’s orbit or its dynamics. This study is supported by the H2020-funded Tipping Points in the Earth System (TiPES) project.

The impact of the El Niño-Southern Oscillation (ENSO) on the extratropics is investigated in an idealized, reduced-order model that has a tropical and an extratropical module. Unidirectional forcing is used to mimic the atmospheric bridge between the tropics and the extratropics. The variability of the coupled ocean--atmosphere extratropical module is then investigated through the analysis of its pullback attractors (PBA). This analysis focuses on two ENSO-type forcings generated by the tropical module, one periodic and one aperiodic. For a substantial range of coupling parameters, multiple chaotic PBAs are found to coexist for the same set of parameter values. Different types of extratropical low-frequency variability are associated with each PBA over the parameter ranges explored. For periodic ENSO forcing, the coexisting PBAs are nonlinearly stable, while for the chaotic forcing, they are unstable and certain extratropical perturbations induce transitions between the PBAs. These distinct stability properties may have profound consequences for extratropical climate predictions, provided they are confirmed by studies using comprehensive climate models. Thus, for instance, ensemble averaging may no longer be a valid approach to isolate the low-frequency variability signal.

A quasi-geostrophic, low-order model of the wind-driven ocean circulation is used to illustrate tipping points induced by time-dependent forcing in excitable chaotic systems. When the wind stress amplitude G is constant in time, our model has a bifurcation from low-amplitude oscillations to high-amplitude relaxation oscillations (ROs) at a wind intensity value Gc. In the presence of time-dependent wind stress, the corresponding tipping point time ttp is defined as the time at which ROs arise. Numerical experiments are carried out using ensemble simulations in the presence of different drift rates of monotonically increasing forcing. Additional experiments include small periodic perturbations of such forcing. The results indicate substantial sensitivity of ttp and G(ttp) Rate-induced tipping, coexisting pullback attractors and total independence from initial states are found for subsets of parameter space. Besides, nonlinear resonance occurs in the presence of periodic perturbations for periods comparable to the RO time scale. The small periodic perturbation can be thought of as the seasonal-to-interannual variability in the wind stress, while the monotonically increasing component stands for the effect of amplification in the midlatitude winds due to anthropogenic warming.

The present work addresses two persistent quandaries of the climate sciences: (i) the existence of global oscillatory modes in the coupled ocean–atmosphere system; and (ii) solar effects on this coupled system. Interannual oscillatory modes, atmospheric and oceanic, are present in several large regions of the globe. We examine here interannual-to-decadal variability over the entire globe in the Community Earth System Model (CESM) and in the NCEP-NCAR reanalysis, and apply multichannel singular spectrum analysis (MSSA) to these two datasets.In the fully coupled CESM1.1 model, with its resolution of 0.1 × 0.1 degrees in the ocean and 0.25 × 0.25 degrees in the atmosphere, the fields analyzed are surface temperatures, sea level pressures and the 200-hPa geopotential. The simulation is 100-yr long and the last 66 yr are used in the analysis. The two statistically significant periodicities in this IPCC-class model are 11 and 3.4 yr.In the reanalysis, the fields of sea level pressure and of 200-hPa geopotential are analyzed at its resolution of 2.5 × 2.5 degrees over the 68-yr interval 1949–2016. Oscillations with periods of 12 and 3.6 yr are found to be statistically significant in this dataset. The spatio-temporal patterns of the oscillations in the two datasets are quite similar. The spatial pattern of these global oscillations over the North Pacific and North Atlantic resemble the Pacific Decadal Oscillation and the interannual variability found in the western North Atlantic, respectively.The two global modes, with their 11–12-yr and 3.4–3.6-yr periodicities, are quite robust, suggesting potential contributions of both to predictability at 1–3-yr horizons. On the other hand, the CESM run has no year-to-year changes in the prescribed insolation, excluding any role of the solar cycle in the model’s 11-yr mode. The solar cycle is present, however, in the reanalysis, since it is present in nature and hence it does affect the observations. We speculate, therefore, that regional oscillations — with their distinct near-periodicities and spatial patterns — are synchronized over the globe, thus yielding both the global oscillatory modes found in CESM. In nature, the decadal mode could be further synchronized with the solar cycle, but that does not seem to be the case, given the slight difference in period — 12 yr for the reanalysis and 11 yr for the solar cycle, which makes them drift in and out of phase.The work’s tentative conclusion is, therefore: (i) yes, there are global oscillatory modes in the climate system, especially a decadal mode; but (ii) no, this mode has little or nothing to do with the solar cycle.

This contribution introduces the dynamics of shallow and rotating flows that characterizes large-scale motions of the atmosphere and oceans. It then focuses on an important aspect of climate dynamics on interannual and interdecadal scales, namely the wind-driven ocean circulation. Studying the variability of this circulation and slow changes therein is treated as an application of the theory of nonautonomous dynamical systems. The contribution concludes by discussing the relevance of these mathematical concepts and methods for the highly topical issues of climate change and climate sensitivity.

Interannual oscillatory modes, atmospheric and oceanic, are present in several large regions of the globe. We examine here low-frequency variability (LFV) over the entire globe in the Community Earth System Model (CESM) and in the NCEP-NCAR and ECMWF ERA5 reanalyses. Multichannel singular spectrum analysis (MSSA) is applied to these three datasets. In the fully coupled CESM1.1 model, with its resolution of $$0.1 \times 0.1$$ 0.1 × 0.1 degrees in the ocean and $$0.25 \times 0.25$$ 0.25 × 0.25 degrees in the atmosphere, the fields analyzed are surface temperatures, sea level pressures and the 200-hPa geopotential. The simulation is 100-year long and the last 66 yr are used in the analysis. The two statistically significant periodicities in this IPCC-class model are 11 and 3.4 year. In the NCEP-NCAR reanalysis, the fields of sea level pressure and of 200-hPa geopotential are analyzed at the available resolution of $$2.5 \times 2.5$$ 2.5 × 2.5 degrees over the 68-years interval 1949–2016. Oscillations with periods of 12 and 3.6 years are found to be statistically significant in this dataset. In the ECMWF ERA5 reanalysis, the 200-hPa geopotential field was analyzed at its resolution of $$0.25 \times 0.25$$ 0.25 × 0.25 degrees over the 71-years interval 1950–2020. Oscillations with periods of 10 and 3.6 years are found to be statistically significant in this third dataset. The spatio-temporal patterns of the oscillations in the three datasets are quite similar. The spatial pattern of these global oscillations over the North Pacific and North Atlantic resemble the Pacific Decadal Oscillation and the LFV found in the Gulf Stream region and Labrador Sea, respectively. We speculate that such regional oscillations are synchronized over the globe, thus yielding the global oscillatory modes found herein, and discuss the potential role of the 11-year solar-irradiance cycle in this synchronization. The robustness of the two global modes, with their 10–12 and 3.4–3.6 years periodicities, also suggests potential contributions to predictability at 1–3 years horizons.

Comment: 22 pages, 5 figures, 3 tables, We propose a methodology for evaluating the performance of climate models based on the use of the Wasserstein distance. This distance provides a rigorous way to measure quantitatively the difference between two probability distributions. The proposed approach is flexible and can be applied in any number of dimensions; it allows one to rank climate models taking into account all the moments of the distributions. Furthermore, by selecting the combination of climatic variables and the regions of interest, it is possible to highlight the deficiencies of each of the models under study. The Wasserstein distance thus enables a comprehensive evaluation of climate model skill. We apply this approach to a selected number of physical fields, ranking the models in terms of their performance in simulating them, as well as pinpointing their weaknesses in the simulation of some of the selected physical fields in specific areas of the Earth.

Comment: Electronic copy of final peer-reviewed manuscript accepted for publication in the Journal of Statistical Physics, Arnold's standard circle maps are widely used to study the quasi-periodic route to chaos and other phenomena associated with nonlinear dynamics in the presence of two rationally unrelated periodicities. In particular, the El Nino-Southern Oscillation (ENSO) phenomenon is a crucial component of climate variability on interannual time scales and it is dominated by the seasonal cycle, on the one hand, and an intrinsic oscillatory instability with a period of a few years, on the other. The role of meteorological phenomena on much shorter time scales, such as westerly wind bursts, has also been recognized and modeled as additive noise. We consider herein Arnold maps with additive, uniformly distributed noise. When the map's nonlinear term, scaled by the parameter $\epsilon$, is sufficiently small, i.e. $\epsilon < 1$, the map is known to be a diffeomorphism and the rotation number $\rho_{\omega}$ is a differentiable function of the driving frequency $\omega$. We concentrate on the rotation number's behavior as the nonlinearity becomes large, and show rigorously that $\rho _{\omega }$ is a differentiable function of $\omega $, even for $\epsilon \geq 1$, at every point at which the noise-perturbed map is mixing. We also provide a formula for the derivative of the rotation number. The reasoning relies on linear-response theory and a computer-aided proof. In the diffeomorphism case of $\epsilon

The scientific problems posed by the Earth's atmosphere, oceans, cryosphere – along with the land surface and biota that interact with them – are central to major socio-economic and political concerns in the 21st century. It is natural, therefore, that a certain impatience should prevail in attempting to solve these problems. The point of a review paper published in this journal in 2001 was that one should proceed with all diligence, but not excessive haste: festina lente, i.e. hurry in a measured way. The earlier paper traced the necessary progress through the solutions of ten problems, starting with What can we predict beyond one week, for how long, and by what methods? and ending with Can we achieve enlightened climate control of our planet by the end of the century? A unified framework was proposed to deal with these problems in succession, from the shortest to the longest time scale, i.e. from weeks to centuries and millennia. The framework is that of dynamical systems theory, with an emphasis on successive bifurcations and the ergodic theory of nonlinear systems, on the one hand, and on pursuing this approach across a hierarchy of climate models, from the simplest, highly idealized ones, to the most detailed ones. Here, we revisit some of these problems, 20 years later, and extend the framework to coupled climate-economics modeling.

The time-dependent deformation of volcanic edifices can help one understand the dynamics of pre-eruptive overpressure build-up in magma chambers. Thus, geodetic time series recorded at the Okmok volcano, in Alaska, show a pattern of fast and short inflations — referred to as “pulses” — followed by either slower and longer deflations, or time intervals with no deformation. This pattern is superimposed onto a longer-period inflation. A rapid inflation occurred just before the 2008 eruption, which suggests that the underlying process may lead to eruptions. It is crucial to understand whether such a behavior is driven by external forcing, such as melt supply variations, or whether it can develop spontaneously within the volcano's plumbing system. Here we model the observed oscillations (2004–2008) as resulting from the hydraulic interaction between two shallow magma chambers fed by a deeper source region, a geometry that is consistent with geochemical, petrological, and geophysical data. The model shows that episodes of periodic fast inflations occur (i) when a viscosity gradient is present in the vertical pipe, for instance as the result of a temperature gradient; and (ii) when the flux supplying the shallower chamber lies between two bounds that we derive analytically. The deformation pulses observed at Okmok can, therewith, be fully explained by the internal variability of the magmatic system and do not require time-variable external forcing. The proposed model can also be seen as an improvement upon the classic hydraulic models regularly used to explain a commonly observed pattern of volcanic deformations, i.e. exponential inflation.

In this talk, we will attempt to cover, as time permits, issues pertaining to a self-consistent, unified treatment of the climate system’s natural and forced variability, i.e. climate change, sensitivity and intrinsic variability. To set the stage, key features of short-, intermediate, and long-term prediction will be sketched, followed by the effects of the system’s multiple scales of motion. After summarizing the main results and uncertainties of successive assessment reports of the International Panel on Climate Change (IPCC), time-dependent forcing will be introduced, in both its natural and anthropogenic forms. We will outline the generalization of strange attractors to this non-autonomous setting, namely pullback and random attractors (PBAs & RAs), as well as the generalization of the bifurcations known from classical, autonomous dynamical systems to the tipping points (TPs) of non-autonomous ones. The case of the Lorenz convection model with stochastic forcing and of its RA will be used as an illustrative example. The talk will conclude with a list of questions and a selected bibliography.

International audience; We study the influence of interannual climate variability on the economy of several countries in the Sahel region. In the agricultural sector, we are able to identify coupled climate-economic modes that are statistically significant on interannual time scales. In particular, precipitation is a key climatic factor for agriculture in this semi-arid region. Locality and diversity characterize the Sahel's climatic and economic system, with the coupled climate-economic patterns exhibiting substantial differences from country to country. Large-scale atmospheric patterns — like the El Niño–Southern Oscillation and its quasi-biennial and quasi-quadrennial oscillatory modes — have quite limited influence on the economies, while more location-specific rainfall patterns play an important role.

Providing efficient and accurate parameterizations for model reduction is a key goal in many areas of science and technology. Here, we present a strong link between data-driven and theoretical approaches to achieving this goal. Formal perturbation expansions of the Koopman operator allow us to derive general stochastic parameterizations of weakly coupled dynamical systems. Such parameterizations yield a set of stochastic integrodifferential equations with explicit noise and memory kernel formulas to describe the effects of unresolved variables. We show that the perturbation expansions involved need not be truncated when the coupling is additive. The unwieldy integrodifferential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation. This connection helps setting up a parallelism between the top-down, equation-based methodology herein and the well-established empirical model reduction (EMR) methodology that has been shown to provide efficient dynamical closures to partially observed systems. Hence, our findings, on the one hand, support the physical basis and robustness of the EMR methodology and, on the other hand, illustrate the practical relevance of the perturbative expansion used for deriving the parameterizations.\ud Parameterizations aim to reduce the complexity of high-dimensional dynamical systems. Here, a theory-based and a data-driven approach for the parameterization of coupled systems are compared, showing that both yield the same stochastic multilevel structure. The results provide very strong support to the use of empirical methods in model reduction and clarify the practical relevance of the proposed theoretical framework.

Noise modifies the behavior of chaotic systems in both quantitative and qualitative ways. To study these modifications, the present work compares the topological structure of the deterministic Lorenz (1963) attractor with its stochastically perturbed version. The deterministic attractor is well known to be "strange" but it is frozen in time. When driven by multiplicative noise, the Lorenz model's random attractor (LORA) evolves in time. Algebraic topology sheds light on the most striking effects involved in such an evolution. In order to examine the topological structure of the snapshots that approximate LORA, we use Branched Manifold Analysis through Homologies (BraMAH) -- a technique originally introduced to characterize the topological structure of deterministically chaotic flows -- which is being extended herein to nonlinear noise-driven systems. The analysis is performed for a fixed realization of the driving noise at different time instants in time. The results suggest that LORA's evolution includes sharp transitions that appear as topological tipping points., Comment: 12 pages and 4 figures