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In the analysis and interpretation of climate data, both from model simulations and observations, it is often of interest to establish relations between the responses of different observables to a global forcing. This problem in its generality is relevant in the context of the identification of emergent constraints for the climate system, detection and attribution studies, and the analysis of proxy data. Recently it has been discussed how in linear response theory it is possible to build proxy response operators, that allow to use the response of one observable to a forcing to predict the response of another observable. The spectral properties of the proxy response functions determine then the properties of statistical predictability at different time scales for the pair of observables. The skill and feasibility of this approach for complex climate data has however not been fully tested yet. In this work we analyse the properties of proxy response in experiments with the coupled general circulation model MPI-ESM v.1.2. We consider ensemble simulations of abrupt CO2 doubling and 1% per year CO2 increase scenarios. We study the response of different atmospheric and oceanic variables, and we compute proxy response functions for different pairs of observables. We analyse the predictive power for the different cases, and interpret differences in skills in terms of causal relations among observables. We also study the relation between statistical variability and long term sensitivity, and we discuss differences between ensemble and internal variability in unforced and forced states. We then link our results to the discussion on the interpretation of emergent constraints in climate change simulations.

The Asian summer monsoons are globally significant meteorological features, creating a strongly seasonal pattern of precipitation, with the majority of the annual precipitation falling between June and September. The stability of such a strongly seasonal hydrological cycle is of extreme importance for a vast range of ecosystems and for the livelihoods of a large share of the world’s population. Simulations are performed with an intermediate complexity climate model, PLASIM, in order to assess the future response of the Asian monsoons to changing concentrations of aerosols and greenhouse gases. The radiative forcing associated with aerosol loading consists of a mid-tropospheric warming and a compensating surface cooling, which is applied to India, Southeast Asia and East China, both concurrently and independently. The primary effect of increased aerosol loading is a decrease in summer precipitation in the vicinity of the applied forcing, although the regional responses vary significantly. The decrease in precipitation is only partially ascribable to a decrease in the precipitable water, and instead derives from a reduction of the precipitation efficiency, due to changes in the stratification of the atmosphere. When the aerosol loading is added in all regions simultaneously, precipitation in East China is most strongly affected, with a quite distinct transition to a low precipitation regime as the radiative forcing increases beyond 60 W/m2. The response is less abrupt as we move westward, with precipitation in South India being least affected. By applying the aerosol loading to each region individually, we are able to explain the mechanism behind the lower sensitivity observed in India, and attribute it to aerosol forcing over East China. Additionally, we note that the effect on precipitation is approximately linear with the forcing. The impact of doubling carbon dioxide levels is to increase precipitation over the region, whilst simultaneously weakening the circulation. When the carbon dioxide and aerosol forcings are applied at the same time, the carbon dioxide forcing partially offsets the surface cooling and reduction in precipitation associated with the aerosol response.

Extremes in extratropical meridional energy transports in the atmosphere are associated with the dynamics of the atmosphere at multiple spatial scales, from planetary to synoptic. This is related to the nature of amplifying baroclinic waves, that are fundamentally intermittent and sporadic, significantly affecting the net seasonal transport across latitudes. Here, we use the ERA5 reanalysis data to perform a wavenumber decomposition of meridional energy transports in the Northern Hemisphere mid-latitudes during winter and summer. Extreme transport events are linked to atmospheric circulation anomalies and dominant weather regimes, identified by clustering 500 hPa geopotential height fields. Partitioning the extreme events across zonal wavenumber highlights the different role of scales in different seasons and regions. In general, planetary-scale waves determine the strength and meridional position of the synoptic-scale baroclinic activity with their phase and amplitude. During winter, large wavenumbers (k = 2–3) are key drivers of the meridional-energy-transport extremes, and planetary- and synoptic-scale transport extremes virtually never co-occur. In summer, extremes are associated with higher wavenumbers (k = 4–6), identified as synoptic-scale motions. Focusing on recently occurred exceptionally strong summertime heat waves and wintertime cold spells, we notice that regime structures of these events are typical of extremely large poleward meridional energy transports.

Following Hasselmann’s ansatz, the climate system may be viewed as a multistable dynamical system internally driven by noise. Its long-term evolution will then feature noise-induced critical transitions between the competing attracting states. In the weak-noise limit, large deviation theory allows predicting the transition rate and most probable transition path of these tipping events. However, the limit of zero noise is never obtained in reality. In this work we show that, even for weak finite noise, sample transition paths may disagree with the large deviation prediction – the minimum action path, or instanton – if multiple timescales are at play. We illustrate this behavior in selected box models of the bistable Atlantic Meridional Overturning Circulation (AMOC), where different restoring times of temperature and salinity induce a fast-slow characteristic. While the minimum action path generally crosses the basin boundary at a saddle point, we demonstrate cases in which ensembles of sample transition paths cross far away from the saddle. We discuss the conditions for saddle avoidance and relate this to the flatness of the quasipotential, a central object of large deviation theory. We further probe the vicinity of the weak-noise limit by applying a pathspace method that generates transition samples for arbitrarily weak noise. Our results highlight that predictions by large deviation theory must be treated cautiously in multiscale dynamical systems.

Eddy saturation describes the nonlinear mechanism in geophysical flows whereby, when average conditions are considered, direct forcing of the zonal flow increases the eddy kinetic energy, while the energy associated with the zonal flow does not increase. We present a minimal baroclinic model that exhibits complete eddy saturation. Starting from Phillips’ classical quasi-geostrophic two-level model on the beta channel of the mid-latitudes, we derive a reduced order model comprising of six ordinary differential equations including parameterised eddies. This model features two physically realisable steady state solutions, one a purely zonal flow and one where, additionally, finite eddy motions are present. As the baroclinic forcing in the form of diabatic heating is increased, the zonal solution loses stability and the eddy solution becomes attracting. After this bifurcation, the zonal components of the solution are independent of the baroclinic forcing, and the excess of heat in the low latitudes is efficiently transported northwards by finite eddies, in the spirit of baroclinic adjustment.

The South and East Asian summer monsoons are globally significant meteorological features, creating a strongly seasonal pattern of precipitation. The stability of such a strongly seasonal hydrological cycle is of extreme importance for a vast range of ecosystems and for the livelihoods of a large share of the world's population. Simulations are performed with an intermediate complexity climate model, PLASIM, to assess the future response of the monsoons to changing concentrations of aerosols and greenhouse gases. The aerosol loading consists of a mid-tropospheric warming and a surface cooling, which is applied to India, Southeast Asia and East China, both concurrently and independently. The primary effect of increased aerosol loading is a decrease in summer precipitation in the vicinity of the applied forcing, although the regional response varies significantly. The decrease in precipitation is only partially ascribable to a decrease in the precipitable water, and instead derives from a reduction of the precipitation efficiency, due to changes in the stratification of the atmosphere. When the aerosol loading is added in all regions simultaneously, precipitation in East China is most strongly affected, with a quite distinct transition to a low precipitation regime as the radiative forcing increases beyond 60 W/m2. The response is less abrupt as we move westward, with precipitation in South India being least affected. This lower sensitivity in South India is attributed to aerosol forcing over East China. Additionally, the effect on precipitation is approximately linear with the forcing. The impact of doubling carbon dioxide levels is to increase precipitation over the regions and weaken the circulation. When the carbon dioxide and aerosol forcings are applied at the same time, the carbon dioxide forcing partially offsets the surface cooling and reduction in precipitation associated with the aerosol response.

The Earth's climate has experienced numerous critical transitions during its history, which have often been accompanied by massive and rapid changes in the biosphere. Such transitions are evidenced in various proxy records covering different timescales. The goal is then to identify, date, and rank past critical transitions in terms of importance, thus possibly yielding a more thorough perspective on climatic history. To illustrate such an angle, which inspired the punctuated equilibrium angle on the theory of evolution, we have analyzed 2 key high-resolution datasets: the CENOGRID marine compilation (past 66 Myr), and North Atlantic U1308 record (past 3.3 Myr). By combining recurrence analysis of the individual time series with a multivariate representation of the system based on the theory of the quasi-potential, we identify the key abrupt transitions associated with major regime changes that differentiate various clusters of climate variability. This allows interpreting the time-evolution of the system as a trajectory taking place in a dynamical landscape, whose multiscale features are associated with a hierarchy of tipping points., 27 pages, 5 figures plus supplementary material

The extratropical meridional energy transport in the atmosphere is fundamentally intermittent in nature, having extremes large enough to affect the net seasonal transport. Here, we investigate how these extreme transports are associated with the dynamics of the atmosphere at multiple spatial scales, from planetary to synoptic. We use the ERA5 reanalysis data to perform a wavenumber decomposition of meridional energy transport in the Northern Hemisphere mid-latitudes during winter and summer. We then relate extreme transport events to atmospheric circulation anomalies and dominant weather regimes, identified by clustering 500 hPa geopotential height fields. In general, planetary-scale waves determine the strength and meridional position of the synoptic-scale baroclinic activity with their phase and amplitude, but important differences emerge between seasons. During winter, large wavenumbers (k = 2–3) are key drivers of the meridional-energy-transport extremes, and planetary- and synoptic-scale transport extremes virtually never co-occur. In summer, extremes are associated with higher wavenumbers (k = 4–6), identified as synoptic-scale motions. We link these waves and the transport extremes to recent results on exceptionally strong and persistent co-occurring summertime heat waves across the Northern Hemisphere mid-latitudes. We show that the weather regime structures associated with these heat wave events are typical for extremely large poleward-energy-transport events.

The perturbation theory of operator semigroups is used to derive response formulas for a variety of combinations of acting forcings and reference background dynamics. In the case of background stochastic dynamics, we decompose the response formulas using the Koopman operator generator eigenfunctions and the corresponding eigenvalues, thus providing a functional basis towards identifying relaxation timescales and modes and towards relating forced and natural fluctuations in physically relevant systems. To leading order, linear response gives the correction to expectation values due to extra deterministic forcings acting on either stochastic or chaotic dynamical systems. When considering the impact of weak noise, the response is linear in the intensity of the (extra) noise for background stochastic dynamics, while the second order response given the leading order correction when the reference dynamics is chaotic. In this latter case we clarify that previously published diverging results can be brought to common ground when a suitable interpretation—Stratonovich vs Itô—of the noise is given. Finally, the response of two-point correlations to perturbations is studied through the resolvent formalism via a perturbative approach. Our results allow, among other things, to estimate how the correlations of a chaotic dynamical system changes as a results of adding stochastic forcing.

We study the impact of applying stochastic forcing to the Ghil-Sellers energy balance climate model in the form of a fluctuating solar irradiance. Through numerical simulations, we explore the noise-induced transitions between the competing warm and snowball climate states. We consider multiplicative stochastic forcing driven by Gaussian and $\alpha$-stable L\'evy - $\alpha\in(0,2)$ - noise laws, examine the statistics of transition times, and estimate most probable transition paths. While the Gaussian noise case has been carefully studied in a plethora of investigations on metastable systems, much less is known about the L\'evy case, especially in the case of high- and infinite-dimensional systems. In the weak noise limit, the expected residence time in each metastable state scales in a fundamentally different way in the Gaussian vs. L\'evy noise case with respect to the intensity of the noise. In the former case, the classical Kramers-like exponential law is recovered. In the latter case, power laws are found, with the exponent equal to $-\alpha$, in apparent agreement with rigorous results obtained for additive noise in a related - yet different - reaction-diffusion equation as well as in simpler models. This can be better understood by treating the L\'evy noise as a compound Poisson process. The transition paths are studied in a projection of the state space and remarkable differences are observed between the two different types of noise. The snowball-to-warm and the warm-to-snowball most probable transition path cross at the single unstable edge state on the basin boundary. In the case of L\'evy noise, the most probable transition paths in the two directions are wholly separated, as transitions apparently take place via the closest basin boundary region to the outgoing attractor. This property can be better elucidated by considering singular perturbations to the solar irradiance., Comment: 36 pages, 8 figures

The multistability of the Atlantic Meridional Overturning Circulation (AMOC) challenges the predictability of long-term climate evolution. In light of an observed weakening in AMOC strength, it is crucial to study the probabilities of noise-induced transitions between the different competing flow regimes. From a dynamical systems perspective, the phase space of a multistable system can be characterised as a non-equilibrium potential landscape, with valleys corresponding to the different basins of attraction. Knowing the potential, one can infer the statistics and pathways of noise-induced transitions. Particularly, in the weak-noise limit, transition paths lead through special regions of the basin boundaries, called Melancholia states. Recent studies have applied these concepts to climate models of low and intermediate complexity. Here, we investigate the quasi-potential landscape of a three-box model of the AMOC, based on the popular model by Rooth. We analyse noise-induced transitions between the two stable circulation states and elucidate the role of the Melancholia state. Forcing the model with different noise laws, which represent fluctuations caused by different physical processes, we discuss how the properties of transitions change when considering non-Gaussian processes, specifically Lévy noise. Simulated transition rates are related to their theoretical values using the quasi-potential landscape. Our results yield a comprehensive picture of the dynamical properties of an inter-hemispheric three-box AMOC model under stochastic forcing. By relating the deterministic structure of this simple model to the statistics of critical transitions, we hope to build a basis for transferring this approach to more complex models of the AMOC.

The Indian summer monsoon is a key meteorological event in the Indian calendar, bringing approximately 80% of India’s annual rainfall over the months of June—September. How the monsoon system will react to a changing future climate is of crucial importance for India’s agriculture, industry and economy. Uncertainty remains in the future climate projections of the Indian monsoon, primarily due to uncertainties in the amount and composition of aerosols over the Asian region. A further complication is that greenhouse gases, expected to increase over the next 50 years and dominate over aerosol forcing, have a directly competing effect on the monsoon. Generally, greenhouse gases act to heat the Earth’s surface, enabling greater moisture uptake and enhancing monsoonal precipitation. In contrast, aerosols have a cooling effect at the surface. The presence of aerosols is analogous to anomalous heating in the mid-troposphere, increasing the static stability of the atmosphere, which is associated with a weakening of the large-scale circulation and thus a weakening of the monsoon. We use the Planet Simulator (PlaSim), an intermediate complexity climate model, to investigate the interplay between varying aerosol and greenhouse gas forcing, in relation to the Indian summer monsoon. The model is modified to include anomalous heating in the mid-troposphere, which represents the presence of aerosols and effectively cools the surface. Varying the intensity and location of aerosol forcing, as well doubling the amount of carbon dioxide, alters the spatial pattern of precipitation over the Asian region. Increasing the anomalous heating to 150 Watts and applying over the regions of India, East China and Southeast Asia, significantly weakens the large-scale circulation and reduces the summer precipitation to 10-20% of a normal year; essentially a breakdown of the monsoon system. Through modelling sensitivity studies, we define a safe operating space of future climate conditions, where the Indian and East Asian monsoons retain their current regimes. We can also show early warning signals in the precipitation intensity, indicating a phase change of the Indian or the East Asian monsoon.

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.

For a wide range of values of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in the past our planet flipped between these two states. The main physical mechanism responsible for such an instability is the ice-albedo feedback. In a previous work, we defined the Melancholia states that sit between the two climates. Such states are embedded in the boundaries between the two basins of attraction and feature extensive glaciation down to relatively low latitudes. Here, we explore the global stability properties of the system by introducing random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attraction. In the weak noise limit, large deviation laws define the invariant measure, the statistics of escape times, and typical escape paths called instantons. By constructing the instantons empirically, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first-order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. Finally, we put forward a new method for constructing Melancholia states from direct numerical simulations, black which provides a possible alternative with respect to the edge-tracking algorithm., 48 Pages, 10 Figures

We study the teleconnection between El Niño–Southern Oscillation (ENSO) and the Indian summer monsoon (IM) in large ensemble simulations, the Max Planck Institute Earth System Model (MPI-ESM), and the Community Earth System Model (CESM1). We characterize ENSO by the June–August Niño-3 box-average SST and the IM by the June–September average precipitation over India, and define their teleconnection in a changing climate as an ensemble-wise correlation. To test robustness, we also consider somewhat different variables that can characterize ENSO and the IM. We utilize ensembles converged to the system’s snapshot attractor for analyzing possible changes in the teleconnection. Our main finding is that the teleconnection strength is typically increasing on the long term in view of appropriately revised ensemble-wise indices. Indices involving a more western part of the Pacific reveal, furthermore, a short-term but rather strong increase in strength followed by some decrease at the turn of the century. Using the station-based Southern Oscillation index (SOI) as opposed to area-based indices leads to the identification of somewhat more erratic trends, but the turn-of-the-century “bump” is well detectable with it. All this is in contrast, if not in contradiction, to the discussion in the literature of a weakening teleconnection in the late twentieth century. We show here that this discrepancy can be due to any of three reasons: 1) ensemble-wise and temporal correlation coefficients used in the literature are different quantities; 2) the temporal moving correlation has a high statistical variability but possibly also persistence; or 3) MPI-ESM does not represent the Earth system faithfully., The simulations for the MPI-RCP8.5E were supported by the H2020 grant for the CRESCENDO project (Grant 641816). MH is thankful for the great support of the DFG Cluster of Excellence CliSAP. This research was supported by the National Research, Development and Innovation Office, NKFIH, under Grants PD124272, K125171, and FK124256. GD was supported by the Ministerio de Economía, Industria y Competitividad of the Spanish Government under Grant LAOP CTM2015-66407-P (AEI/FEDER, EU), and by the European Social Fund under CAIB grant PD/020/2018 “Margalida Comas”. TB, VL, and FL were supported by the H2020 grants for the CRESCENDO (Grant 641816) and Blue-Action (Grant 727852) projects. VL has been supported by the DFG Sfb/Transregio TRR181 project. TB was also supported by the Institute for Basic Science (IBS), under IBS-R028-D1.

The extratropical meridional energy transport in the atmosphere is fundamentally intermittent in nature, having extremes large enough to affect the net seasonal transport. Here, we investigate how these extreme transports are associated with the dynamics of the atmosphere at multiple scales, from planetary to synoptic. We use ERA5 reanalysis data to perform a wavenumber decomposition of meridional energy transport in the Northern Hemisphere mid-latitudes during winter and summer. We then relate extreme transport events to atmospheric circulation anomalies and dominant weather regimes, identified by clustering 500 hPa geopotential height fields. In general, planetary-scale waves determine the strength and meridional position of the synoptic-scale baroclinic activity with their phase and amplitude, but important differences emerge between seasons. During winter, large wavenumbers (k = 2 − 3) are key drivers of the meridional energy transport extremes, and planetary and synoptic-scale transport extremes virtually never co-occur. In summer, extremes are associated with higher wavenumbers (k = 4 − 6), identified as synoptic-scale motions. We link these waves and the transport extremes to recent results on exceptionally strong and persistent co-occurring summertime heat waves across the Northern Hemisphere mid-latitudes. We show that these events are typical, in terms of dominant regime patterns associated with extremely strong meridional energy transports.Link to pre-print: https://wcd.copernicus.org/preprints/wcd-2021-85/

Elucidating the statistical properties of extreme meteo-climatic events and capturing the physical processes responsible for their occurrence are key steps for improving our understanding of climate variability and climate change and for better evaluating the associated hazards. It has recently become apparent that large deviation theory is very useful for investigating persistent extreme events, and specifically, for flexibly estimating long return periods and for introducing a notion of dynamical typicality. Using a methodological framework based on large deviation theory and taking advantage of long simulations by a state-of-the-art Earth System Model, we investigate the 2021 North America Heatwave. Indeed, our analysis shows that the 2021 event can be seen as an unlikely but possible manifestation of climate variability, whilst its probability of occurrence is greatly amplified by the ongoing climate change. We also clarify the properties of spatial coherence of the 2021 heatwave and elucidate the role played by the Rocky Mountains in modulating hot, dry, and persistent extreme events in the Western Pacific region of North America., Comment: 11 pages, 4 figures plus supplementary material

Detection and attribution studies have played a major role in shaping contemporary climate science and have provided key motivations supporting global climate policy negotiations. The goal of such studies is to associate observed climatic patterns of climate change with acting forcings - both anthropogenic and natural ones - with the goal of making statements on the acting drivers of climate change. The statistical inference is usually performed using regression methods referred to as optimal fingerprinting. We show here how a fairly general formulation of linear response theory relevant for nonequilibrium systems provides the physical and mathematical foundations behind the optimal fingerprinting approach for the climate change detection and attribution problem. Our angle allows one to clearly frame assumptions, strengths and potential pitfalls of the method., Comment: 6 pages

We study the impact of applying stochastic forcing to the Ghil-Sellers energy balance climate model in the form of a fluctuating solar irradiance. Through numerical simulations, we explore the noise-induced transitions between the competing warm and snowball climate states. We consider multiplicative stochastic forcing driven by Gaussian and $α$-stable Lévy - $α\in(0,2)$ - noise laws, examine the statistics of transition times, and estimate most probable transition paths. While the Gaussian noise case has been carefully studied in a plethora of investigations on metastable systems, much less is known about the Lévy case, especially in the case of high- and infinite-dimensional systems. In the weak noise limit, the expected residence time in each metastable state scales in a fundamentally different way in the Gaussian vs. Lévy noise case with respect to the intensity of the noise. In the former case, the classical Kramers-like exponential law is recovered. In the latter case, power laws are found, with the exponent equal to $-α$, in apparent agreement with rigorous results obtained for additive noise in a related - yet different - reaction-diffusion equation as well as in simpler models. This can be better understood by treating the Lévy noise as a compound Poisson process. The transition paths are studied in a projection of the state space and remarkable differences are observed between the two different types of noise. The snowball-to-warm and the warm-to-snowball most probable transition path cross at the single unstable edge state on the basin boundary. In the case of Lévy noise, the most probable transition paths in the two directions are wholly separated, as transitions apparently take place via the closest basin boundary region to the outgoing attractor. This property can be better elucidated by considering singular perturbations to the solar irradiance., 36 pages, 8 figures

Unstable periodic orbits (UPOs) are a valuable tool for studying chaotic dynamical systems, as they allow one to distill their dynamical structure. We consider here the Lorenz 1963 model with the classic parameters' value. We investigate how a chaotic trajectory can be approximated using a complete set of UPOs up to symbolic dynamics' period 14. At each instant, we rank the UPOs according to their proximity to the position of the orbit in the phase space. We study this process from two different perspectives. First, we find that longer period UPOs overwhelmingly provide the best local approximation to the trajectory. Second, we construct a finite-state Markov chain by studying the scattering of the orbit between the neighbourhood of the various UPOs. Each UPO and its neighbourhood are taken as a possible state of the system. Through the analysis of the subdominant eigenvectors of the corresponding stochastic matrix we provide a different interpretation of the mixing processes occurring in the system by taking advantage of the concept of quasi-invariant sets., 13 pages, 7 figures

Climate can be interpreted as a complex, high dimensional non-equilibrium stationary system characterised by multiple time and space scales spanning various orders of magnitude. Statistical mechanics and dynamical system theory have been key mathematical frameworks in the study of the climate system. In particular, unstable periodic orbits (UPOs) have been proven to provide relevant insight in the understanding of its statistical properties. In a recent paper Lucarini and Gritsun [1] provided an alternative approach for understanding the properties of the atmosphere.In general, UPOs decomposition plays a relevant role in the study of chaotic dynamical systems. In fact, UPOs densely populate the attractor of a chaotic system, and can therefore be thought as building blocks to construct the dynamic of the system itself. Since they are dense in the attractor, it is always possible to find a UPO arbitrarily near to a chaotic trajectory: the trajectory will remain close to the UPO, but it will never follow it indefinitely, because of its instability. Loosely speaking, a chaotic trajectory is repelled between neighbourhoods of different UPOs and can thus be approximated in terms of these periodic orbits. The statistical properties of the system can then be reconstructed from the full set of periodic orbits in this fashion.The numerical study of UPOs thus represents a relevant problem and an interesting research topic for Climate Science and chaotic dynamical systems in general. In this presentation we address the problem of sampling UPOs for the paradigmatic Lorenz-63 model. First, we present results regarding the measure of the system, thus its statistical properties, using UPOs theory, namely with the trace formulas. Second, we introduce a more innovative approach, considering UPOs as global states of the system. We approximate the exact dynamics by a suitable Markov chain process, describing how the system hops on different UPOs, and we compare the two different approaches. [1] V. Lucarini and A. Gritsun, “A new mathematical framework for atmospheric blocking events,” Climate Dynamics, vol. 54, pp. 575–598, Jan 2020.

We analyse persistent temperature events, like heat waves or cold spells, by applying large deviation theory (LDT), and show that events with a long duration have also a substantial spatial extension. We point out that by using LDT one finds typical spatial patterns related to the persistent temperature extremes. Based on the output of a state-of-the-art climate model, we define the climatology of persistent heatwaves and cold spells in some key target regions of the planet by constructing empirically the corresponding rate functions for the surface temperature, and we assess the impact of increasing CO2 concentration on such persistent anomalies. In particular, we notice the increasing hazard associated to heatwaves in a warmer climate. We show that two 2010 high impact events - summer Russian heatwave and winter Dzud in Mongolia – are associated with extended atmospheric patterns that are exceptional compared to the typical ones, but typical compared to the climatology of extreme events.

The climate can be regarded as a stationary non-equilibrium statistical system (Gallavotti 2006): a continuous and spatially inhomogeneous input of solar energy enters at the top-of-atmosphere and compensates the action of non-conservative forces, mainly occurring at small scales, to give rise to a statistically steady state (or attractor) for the whole climate. Depending on the initial conditions and the range of forcing, all other parameters being the same, some climate models have the property to settle down on different attractors. Multi-stability reflects how energy, water mass and entropy can be re-distributed in multiple ways among the climate components, such as the atmosphere, the ocean or the ice, through a different balance between nonlinear mechanisms. Starting from a configuration where competing climate attractors occur under the same forcing, we have explored their robustness performing two kinds of numerical experiment. First, we have investigated the impact of frictional heating on the overall energy balance and we have shown that such contribution, generally neglected in the atmospheric component of climate models, has crucial consequences on conservation properties: it improves the energy imbalance at top-of-atmosphere, typically non negligible in coarse simulations (Wild et al. 2020), strengthens the hydrological cycle, mitigates the mechanical work associated to atmospheric circulation intensity and reduces the heat transport peaks in the ocean. Second, we have compared two bulk formulas for the cloud albedo, one where it is constant everywhere and the other where it increases with latitude, as implemented in the new version of the atmospheric module SPEEDY in order to improve comparisons with observational data (Kucharski 2013). We have checked that this new parameterization does not affect energy and water-mass imbalances, while reduces global temperature and water-mass transport on the attractor, giving rise to a larger conversion of heat into mechanical work in the atmosphere.In order to perform such studies, we have run the climate model MITgcm on coupled aquaplanets at 2.8 horizontal resolution until steady states are reached (Brunetti el al. 2019) and we have applied the Thermodynamic Diagnostic Tool (TheDiaTo, Lembo et al. 2019). References: Brunetti, Kasparian, Vérard, Climate Dynamics 53, 6293 (2019)Gallavotti, Math. Phys. 3, 530 (2006)Kucharski et al., Bulletin of the American Meteorological Society 94, 25 (2013)Lembo, Lunkeit, Lucarini, Geoscientific Model Development 12, 3805 (2019)Wild, Climate Dynamics 55, 553 (2020)

We use a simple yet Earth-like atmospheric model to propose a new framework for understanding the mathematics of blocking events, which are associated with low frequency, large scale waves in the atmosphere. Analysing error growth rates along a very long model trajectory, we show that blockings are associated with conditions of anomalously high instability of the atmosphere. Additionally, the lifetime of a blocking is positively correlated with the intensity of such an anomaly, against intuition. In the case of Atlantic blockings, predictability is especially reduced at the onset and decay of the blocking, while a relative increase of predictability is found in the mature phase, while the opposite holds for Pacific blockings, for which predictability is lowest in the mature phase. We associate blockings to a specific class of unstable periodic orbits (UPOs), natural modes of variability that cover the attractor of the system. The UPOs differ substantially in terms of instability, which explains the diversity of the atmosphere in terms predictability. The UPOs associated to blockings are indeed anomalously unstable, which leads to them being rarely visited. The onset of a blocking takes place when the trajectory of the system hops into the neighbourhood of one of these special UPOs. The decay takes place when the trajectory hops back to the neighbourhood of usual, less unstable UPOs associated with zonal flow. This justifies the classical Markov chains-based analysis of transitions between weather regimes. The existence of UPOs differing in the dimensionality of their unstable manifold indicates a very strong violation of hyperbolicity in the model, which leads to a lack of structural stability. We propose that this is could be a generic feature of atmospheric models and might be a fundamental cause behind difficulties in representing blockings for the current climate and uncertainties in predicting how their statistics will change as a result of climate change.References:V. Lucarini, A. Gritsun, A. A new mathematical framework for atmospheric blocking events. Climate Dynamics 54, 575–598 (2020). https://doi.org/10.1007/s00382-019-05018-2M. Ghil, V. Lucarini, The Physics of Climate Variability and Climate, Rev. Modern Physics, 92, 035002 (2020). https://link.aps.org/doi/10.1103/RevModPhys.92.035002 S. Schubert, V. Lucarini, Dynamical analysis of blocking events: spatial and temporal fluctuations of covariant Lyapunov vectors. Q. J. R. Meteorol. Soc. 142, 2143-2158 (2016). https://doi.org/10.1002/qj.2808

Several European countries have suspended the inoculation of the AstraZeneca vaccine out of suspicion of causing deep vein thrombosis. In this letter we report some Fermi estimates performed using a stochastic model aimed at making a risk-benefit analysis of the interruption of the delivery of the AstraZeneca vaccine in France and Italy. Our results clearly show that excess deaths due to the interruption of the vaccination campaign injections largely overrun those due to thrombosis even in worst case scenarios of frequency and gravity of the vaccine side effects., Comment: 15 pages, 3 figures, plus Supplementary Material

We propose a comprehensive framework able to address both the predictability of the first and of the second kind for high-dimensional chaotic models. For this purpose, we analyse the properties of a newly introduced multistable climate toy model constructed by coupling the Lorenz '96 model with a zero-dimensional energy balance model. First, the attractors of the system are identified with Monte Carlo Basin Bifurcation Analysis. Additionally, we are able to detect the Melancholia state separating the two attractors. Then, Neural Ordinary Differential Equations are applied in order to predict the future state of the system in both of the identified attractors., 17 pages, 7 figures. Eur. Phys. J. Spec. Top. (2021)

We apply two independent data analysis methodologies to locate stable climate states in an intermediate complexity climate model and analyze their interplay. First, drawing from the theory of quasipotentials, and viewing the state space as an energy landscape with valleys and mountain ridges, we infer the relative likelihood of the identified multistable climate states, and investigate the most likely transition trajectories as well as the expected transition times between them. Second, harnessing techniques from data science, specifically manifold learning, we characterize the data landscape of the simulation output to find climate states and basin boundaries within a fully agnostic and unsupervised framework. Both approaches show remarkable agreement, and reveal, apart from the well known warm and snowball earth states, a third intermediate stable state in one of the two climate models we consider. The combination of our approaches allows to identify how the negative feedback of ocean heat transport and entropy production via the hydrological cycle drastically change the topography of the dynamical landscape of Earth's climate., 28 pages, 12 figures plus Supplementary Material. Revised version

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.

We introduce the special issue on the Statistical Mechanics of Climate published on the Journal of Statistical Physics by presenting an informal discussion of some theoretical aspects of climate dynamics that make it a topic of great interest for mathematicians and theoretical physicists. In particular, we briefly discuss its nonequilibrium and multiscale properties, the relationship between natural climate variability and climate change, the different regimes of climate response to perturbations, and critical transitions., Comment: 24 pages

Global Climate Models are key tools for predicting the future response of the climate system to a variety of natural and anthropogenic forcings. Here we show how to use statistical mechanics to construct operators able to flexibly predict climate change for a variety of climatic variables of interest. We perform our study on a fully coupled model - MPI-ESM v.1.2 - and for the first time we prove the effectiveness of response theory in predicting future climate response to CO$_2$ increase on a vast range of temporal scales, from inter-annual to centennial, and for very diverse climatic quantities. We investigate within a unified perspective the transient climate response and the equilibrium climate sensitivity and assess the role of fast and slow processes. The prediction of the ocean heat uptake highlights the very slow relaxation to a newly established steady state. The change in the Atlantic Meridional Overturning Circulation (AMOC) and of the Antarctic Circumpolar Current (ACC) is accurately predicted. The AMOC strength is initially reduced and then undergoes a slow and only partial recovery. The ACC strength initially increases as a result of changes in the wind stress, then undergoes a slowdown, followed by a recovery leading to a overshoot with respect to the initial value. Finally, we are able to predict accurately the temperature change in the Northern Atlantic., 17 pages, 8 figures. Improved discussion of the results and expanded bibliography compared to the previous version

We study the teleconnection between the El Niño–Southern Oscillation (ENSO) and the Indian summer monsoon (IM) in large ensemble simulations, the Max Planck Institute Earth System Model (MPI-ESM) and the Community Earth System Model (CESM1). We characterize ENSO by the JJA Niño 3 box-average SST and the IM by the JJAS average precipitation over India, and define their teleconnection in a changing climate as an ensemble-wise correlation. To test robustness, we also consider somewhat different variables that can characterize ENSO and the IM. We utilize ensembles converged to the system’s snapshot attractor for analyzing possible changes in the teleconnection. Our main finding is that the teleconnection strength is typically increasing on the long term in view of appropriately revised ensemble-wise indices. Indices involving a more western part of the Pacific reveal, furthermore, a short-term but rather strong increase in strength followed by some decrease at the turn of the century. Using the station-based SOI as opposed to area-based indices leads to the identification of somewhat more erratic trends, but the turn-of-the-century “bump” is well-detectable with it. All this is in contrast, if not in contradiction, with the discussion in the literature of a weakening teleconnection in the late 20th century. We show here that this discrepancy can be due to any of three reasons: ensemble-wise and temporal correlation coefficients used in the literature are different quantities; the temporal moving correlation has a high statistical variability but possibly also persistence; MPI-ESM does not represent the Earth system faithfully.

We use large deviation theory to study persistent extreme events of temperature, like heat waves or cold spells. We consider the mid-latitudes of a simplified yet Earth-like general circulation model of the atmosphere and numerically estimate large deviation rate functions of near-surface temperature averages over different spatial scales. We find that, in order to represent persistent extreme events based on large deviation theory, one has to look at temporal averages of spatially averaged observables. The spatial averaging scale is crucial, and has to correspond with the scale of the event of interest. Accordingly, the computed rate functions indicate substantially different statistical properties of temperature averages over intermediate spatial scales (larger, but still of the order of the typical scale), as compared to the ones related to any other scale. Thus, heat waves (or cold spells) can be interpreted as large deviations of temperature averaged over intermediate spatial scales. Furthermore, we find universal characteristics of rate functions, based on the equivalence of temporal, spatial, and spatio-temporal rate functions if we perform a re-normalisation by the integrated auto-correlation.

We analyze linear perturbations of a coupled quasi-geostrophic atmosphere-ocean model based on Covariant Lyapunov Vectors (CLVs). CLVs reveal the local geometrical structure of the attractor, and point into the direction of linear perturbations applied to the trajectory. Thus they represent a link between the geometry of the attractor and basic dynamical properties of the system, and they are physically meaningful. We compute the CLVs based on the so-called Ginelli method using the tangent linear version of the quasi-geostrophic atmosphere-ocean model MAOOAM (Modular Arbitrary-Order Ocean-Atmosphere Model). Based on the CLVs, we can quantify the contribution of each model variable on each scale to the development of linear instabilities. We also study the changes in the structure of the attractor - and, consequently, in the basic dynamical properties of our system - as an effect of the ocean-atmopshere coupling strength and the model resolution.

Unstable periodic orbits (UPOs) have been proved to be a relevant mathematical tool in the study of Climate Science. In a recent paper Lucarini and Gritsun [1] provided an alternative approach for understanding the properties of the atmosphere. Climate can be interpreted as a non-equilibrium steady state system and, as such, statistical mechanics can provide us with tools for its study.UPOs decomposition plays a relevant role in the study of chaotic dynamical systems. In fact, UPOs densely populate the attractor of a chaotic system, and can therefore be thought as building blocks to construct the dynamic of the system itself. Since they are dense in the attractor, it is always possible to find a UPO arbitrarily near to a chaotic trajectory: the trajectory will remain close to the UPO, but it will never follow it indefinitely, because of its instability. Loosely speaking, a chaotic trajectory is repelled between neighbourhoods of different UPOs and can thus be approximated in terms of these periodic orbits. The characteristics of the system can then be reconstructed from the full set of periodic orbits in this fashion.The sampling of UPOs is therefore a relevant problem for describing chaotic dynamical systems and can represent an interesting topic for the study of Climate Science. In this work we address this problem and present an algorithm to numerically extract UPOs from the attractor of a simple Climate Model such as Lorenz-63.[1] V. Lucarini and A. Gritsun, “A new mathematical framework for atmospheric blocking events,” Climate Dynamics, vol. 54, pp. 575–598, Jan 2020.

The atmospheric meridional energy transport in the Northern Hemisphere midlatitudes is mainly accomplished by planetary and synoptic waves. A decomposition into wave components highlights the strong seasonal dependence of the transport, with both the total transport and the contributions from planetary and synoptic waves peaking in winter. In both winter and summer months, poleward transport extremes primarily result from a constructive interference between planetary and synoptic motions. The contribution of the mean meridional circulation is close to climatology. Equatorward transport extremes feature a mean meridional equatorward transport in winter, while the planetary and synoptic modes mostly transport energy poleward. In summer, a systematic destructive interference occurs, with planetary modes mostly transporting energy equatorward and synoptic modes again poleward. This underscores that baroclinic conversion dominates regardless of season in the synoptic wave modes, whereas the planetary waves can be either free or forced, depending on the season.

Global Climate Models are key tools for predicting the future response of the climate system to a variety of natural and anthropogenic forcings. Typically, an ensemble of simulations is performed considering a scenario of forcing, in order to analyse the response of the climate system to the specific forcing signal. Given that the the climate response spans a very large range of timescales, such a strategy often requires a dramatic amount of computational resources. In this paper we show how to use statistical mechanics to construct operators able to flexibly predict climate change for a variety of climatic variables of interest, going beyond the limitation of having to consider specific time patterns of forcing. We perform our study on a fully coupled GCM - MPI-ESM v.1.2 - and for the first time we prove the effectiveness of response theory in predicting future climate response to CO2 increase on a vast range of temporal scales. We specifically treat atmospheric (surface temperature) and oceanic variables (strength of the Atlantic Meridional Overturning Circulation and of the Antarctic Circumpolar Current), as well as the global ocean heat uptake.

Extremes are related to high impact and serious hazard events and hence their study and prediction have been and continue to be highly relevant for all kind of applications in geoscience and beyond. Extreme value theory is promising to be able to predict them reliably and robustly. In the last fifteen years the classical extreme value theory for stochastic processes has been extended to dynamical systems and has been related to properties of physical measure (statistical properties of the system), return and hitting times. We will review what one can say for highly dimensional perfectly chaotic systems. We will concentrate on relations between the index of the extreme distribution and invariants of the underlying dynamical system which are stable, in the sense that they will continuously depend on changing parameters in the dynamics. Furthermore, we explore whether there exists a response theory for extremes, that is, whether the change of extremes can be quantitatilvely expressed in terms of changing parameters.

The climate system can be seen as a thermal engine that generates entropy by irreversible processes and achieves a steady state by redistributing the input solar energy among its different components (ocean, atmosphere, etc) and by balancing the energy, water mass and entropy budgets over all the spatial scales. Biases in modern climate models are generally related to the fact that their statistical properties are not well represented, giving rise to imperfect closures of the energy cycle. Thus, a proper measurement of the efficiency of the thermal engine in each climate model is needed. Moreover, possible steady states (attractors) that can be approached at climate tipping-points are characterised by different feedbacks becoming dominant in the thermal engine.We apply the Thermodynamic Diagnostic Tool (TheDiaTo) [1] to the attractors recently obtained using the MIT general circulation model (MITgcm) in a coupled aquaplanet [2], a planet where the ocean covers the entire globe. Such coupled aquaplanets, where nonlinear interactions between atmosphere, ocean and sea ice are fully taken into account, provide a relevant framework to understand the role of the main feedbacks at play in the climate system. Five attractors have been found, ranging from snowball (where ice covers the entire planet) to hot state conditions (where ice completely disappears) [2].Using TheDiaTo, we analyse the five climate attractors by estimating: a) the energy budgets and meridional energy transports; b) the water mass and latent energy budgets and respective meridional transports; c) the Lorenz Energy Cycle; d) the material entropy production. We consider different coupled atmosphere-ocean-sea ice configurations and cloud parameterizations of MITgcm where the energy balance at the top of the atmosphere is progressively better closed in order to understand the occurrence of possible biases in the statistical properties of each attractor. Our contribution will help clarify the thermodynamic differences in climate attractors and their stability to external perturbations that could shift the climate from a steady state to the other.[1] Lembo V., Lunkeit F., Lucarini V., TheDiaTo (v1.0) – a new diagnostic tool for water, energy amd entropy budgets in climate models, Geosci. Model Dev. 12, 3805-3834 (2019)[2] Brunetti M., Kasparian J., Vérard C., Co-existing climate attractors in a coupled aquaplanet, Climate Dynamics 53, 6293-6308 (2019)