Multistability and intermediate tipping of the Atlantic Ocean circulation.

Tipping points (TP) in climate subsystems are usually thought to occur at a well-defined, critical forcing parameter threshold, via destabilization of the system state by a single, dominant positive feedback. However, coupling to other subsystems, additional feedbacks, and spatial heterogeneity may promote further small-amplitude, abrupt reorganizations of geophysical flows at forcing levels lower than the critical threshold. Using a primitive-equation ocean model, we simulate a collapse of the Atlantic Meridional Overturning Circulation (AMOC) due to increasing glacial melt. Considerably before the collapse, various abrupt, qualitative changes in AMOC variability occur. These intermediate tipping points (ITP) are transitions between multiple stable circulation states. Using 2.75 million years of model simulations, we uncover a very rugged stability landscape featuring parameter regions of up to nine coexisting stable states. The path to an AMOC collapse via a sequence of ITPs depends on the rate of change of the meltwater input. This challenges our ability to predict and define safe limits for TPs.

Probabilistic measures for biological adaptation and resilience.

This paper introduces an approach to quantifying ecological resilience in biological systems, particularly focusing on noisy systems responding to episodic disturbances with sudden adaptations. Incorporating concepts from nonequilibrium statistical mechanics, we propose a measure termed "ecological resilience through adaptation," specifically tailored to noisy, forced systems that undergo physiological adaptation in the face of stressful environmental changes. Randomness plays a key role, accounting for model uncertainty and the inherent variability in the dynamical response among components of biological systems. Our measure of resilience is rooted in the probabilistic description of states within these systems and is defined in terms of the dynamics of the ensemble average of a model-specific observable quantifying success or well-being. Our approach utilizes stochastic linear response theory to compute how the expected success of a system, originally in statistical equilibrium, dynamically changes in response to a environmental perturbation and a subsequent adaptation. The resulting mathematical derivations allow for the estimation of resilience in terms of ensemble averages of simulated or experimental data. Finally, through a simple but clear conceptual example, we illustrate how our resilience measure can be interpreted and compared to other existing frameworks in the literature. The methodology is general but inspired by applications in plant systems, with the potential for broader application to complex biological processes.

A punctuated equilibrium analysis of the climate evolution of cenozoic exhibits a hierarchy of abrupt transitions.

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, characterize, and rank past critical transitions in terms of importance, thus possibly yielding a more thorough perspective on climatic history. To illustrate such an approach, which is inspired by the punctuated equilibrium perspective 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 separate 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 describe a hierarchy of metastable states and associated tipping points.

Decomposing the dynamics of the Lorenz 1963 model using unstable periodic orbits: Averages, transitions, and quasi-invariant sets.

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 neighborhood of the various UPOs. Each UPO and its neighborhood 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.

Fingerprinting Heatwaves and Cold Spells and Assessing Their Response to Climate Change Using Large Deviation Theory.

Extreme events provide relevant insights into the dynamics of climate and their understanding is key for mitigating the impact of climate variability and climate change. By applying large deviation theory to a state-of-the-art Earth system model, we define the climatology of persistent heatwaves and cold spells in key target geographical regions by estimating the rate functions for the surface temperature, and we assess the impact of increasing CO_{2} concentration on such persistent anomalies. Hence, we can better quantify the increasing hazard due 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 atmospheric patterns that are exceptional compared to the typical ones but typical compared to the climatology of extremes. Their dynamics is encoded in the natural variability of the climate. Finally, we propose and test an approximate formula for the return times of large and persistent temperature fluctuations from easily accessible statistical properties.

Reduced-order models for coupled dynamical systems: Data-driven methods and the Koopman operator.

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.

Spectroscopy of phase transitions for multiagent systems.

In this paper, we study phase transitions for weakly interacting multiagent systems. By investigating the linear response of a system composed of a finite number of agents, we are able to probe the emergence in the thermodynamic limit of a singular behavior of the susceptibility. We find clear evidence of the loss of analyticity due to a pole crossing the real axis of frequencies. Such behavior has a degree of universality, as it does not depend on either the applied forcing or on the considered observable. We present results relevant for both equilibrium and nonequilibrium phase transitions by studying the Desai-Zwanzig and Bonilla-Casado-Morillo models.

Interrupting vaccination policies can greatly spread SARS-CoV-2 and enhance mortality from COVID-19 disease: The AstraZeneca case for France and Italy.

Several European countries have suspended the inoculation of the AstraZeneca vaccine out of suspicion that it causes 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.

Dynamical landscape and multistability of a climate model.

We apply two independent data analysis methodologies to locate stable climate states in an intermediate complexity climate model and analyse their interplay. First, drawing from the theory of quasi-potentials, 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, and 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 versions of PLASIM, the climate model used in this study. 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.

Rough basin boundaries in high dimension: Can we classify them experimentally?

We show that a known condition for having rough basin boundaries in bistable 2D maps holds for high-dimensional bistable systems that possess a unique nonattracting chaotic set embedded in their basin boundaries. The condition for roughness is that the cross-boundary Lyapunov exponent λx on the nonattracting set is not the maximal one. Furthermore, we provide a formula for the generally noninteger co-dimension of the rough basin boundary, which can be viewed as a generalization of the Kantz-Grassberger formula. This co-dimension that can be at most unity can be thought of as a partial co-dimension, and, so, it can be matched with a Lyapunov exponent. We show in 2D noninvertible- and 3D invertible-minimal models, that, formally, it cannot be matched with λx. Rather, the partial dimension D0 (x) that λx is associated with in the case of rough boundaries is trivially unity. Further results hint that the latter holds also in higher dimensions. This is a peculiar feature of rough fractals. Yet, D0 (x) cannot be measured via the uncertainty exponent along a line that traverses the boundary. Consequently, one cannot determine whether the boundary is a rough or a filamentary fractal by measuring fractal dimensions. Instead, one needs to measure both the maximal and cross-boundary Lyapunov exponents numerically or experimentally.

Beyond Forcing Scenarios: Predicting Climate Change through Response Operators in a Coupled General Circulation Model.

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. We perform our study using 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 CO2 increase on a vast range of temporal scales, from inter-annual to centennial, and for very diverse climatic variables. 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 partial recovery. The ACC strength initially increases due to 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 North Atlantic.

Can we use linear response theory to assess geoengineering strategies?

Geoengineering can control only some climatic variables but not others, resulting in side-effects. We investigate in an intermediate-complexity climate model the applicability of linear response theory (LRT) to the assessment of a geoengineering method. This application of LRT is twofold. First, our objective (O1) is to assess only the best possible geoengineering scenario by looking for a suitable modulation of solar forcing that can cancel out or otherwise modulate a climate change signal that would result from a rise in carbon dioxide concentration [CO2] alone. Here, we consider only the cancellation of the expected global mean surface air temperature Δ⟨[Ts]⟩. It is in fact a straightforward inverse problem for this solar forcing, and, considering an infinite time period, we use LRT to provide the solution in the frequency domain in closed form as fs(ω)=(Δ⟨[Ts]⟩(ω)-χg(ω)fg(ω))/χs(ω), where the χ's are linear susceptibilities. We provide procedures suitable for numerical implementation that apply to finite time periods too. Second, to be able to utilize LRT to quantify side-effects, the response with respect to uncontrolled observables, such as regional averages ⟨Ts⟩, must be approximately linear. Therefore, our objective (O2) here is to assess the linearity of the response. We find that under geoengineering in the sense of (O1), i.e., under combined greenhouse and required solar forcing, the asymptotic response Δ⟨[Ts]⟩ is actually not zero. This turns out not to be due to nonlinearity of the response under geoengineering, but rather a consequence of inaccurate determination of the linear susceptibilities χ. The error is in fact due to a significant quadratic nonlinearity of the response under system identification achieved by a forced experiment. This nonlinear contribution can be easily removed, which results in much better estimates of the linear susceptibility, and, in turn, in a fivefold reduction in Δ⟨[Ts]⟩ under geoengineering practice. This correction dramatically improves also the agreement of the spatial patterns of the predicted linear and the true model responses. However, considering (O2), such an agreement is not perfect and is worse in the case of the precipitation patterns as opposed to surface temperature. Some evidence suggests that it could be due to a greater degree of nonlinearity in the case of precipitation.

Response theory and phase transitions for the thermodynamic limit of interacting identical systems.

We study the response to perturbations in the thermodynamic limit of a network of coupled identical agents undergoing a stochastic evolution which, in general, describes non-equilibrium conditions. All systems are nudged towards the common centre of mass. We derive Kramers-Kronig relations and sum rules for the linear susceptibilities obtained through mean field Fokker-Planck equations and then propose corrections relevant for the macroscopic case, which incorporates in a self-consistent way the effect of the mutual interaction between the systems. Such an interaction creates a memory effect. We are able to derive conditions determining the occurrence of phase transitions specifically due to system-to-system interactions. Such phase transitions exist in the thermodynamic limit and are associated with the divergence of the linear response but are not accompanied by the divergence in the integrated autocorrelation time for a suitably defined observable. We clarify that such endogenous phase transitions are fundamentally different from other pathologies in the linear response that can be framed in the context of critical transitions. Finally, we show how our results can elucidate the properties of the Desai-Zwanzig model and of the Bonilla-Casado-Morillo model, which feature paradigmatic equilibrium and non-equilibrium phase transitions, respectively.

Effects of stochastic parametrization on extreme value statistics.

Extreme geophysical events are of crucial relevance to our daily life: they threaten human lives and cause property damage. To assess the risk and reduce losses, we need to model and probabilistically predict these events. Parametrizations are computational tools used in the Earth system models, which are aimed at reproducing the impact of unresolved scales on resolved scales. The performance of parametrizations has usually been examined on typical events rather than on extreme events. In this paper, we consider a modified version of the two-level Lorenz'96 model and investigate how two parametrizations of the fast degrees of freedom perform in terms of the representation of extreme events. One parametrization is constructed following Wilks [Q. J. R. Meteorol. Soc. 131, 389-407 (2005)] and is constructed through an empirical fitting procedure; the other parametrization is constructed through the statistical mechanical approach proposed by Wouters and Lucarini [J. Stat. Mech. Theory Exp. 2012, P03003 (2012); J. Stat. Phys. 151, 850-860 (2013)]. The two strategies show different advantages and disadvantages. We discover that the agreement between parametrized models and true model is in general worse when looking at extremes rather than at the bulk of the statistics. The results suggest that stochastic parametrizations should be accurately and specifically tested against their performance on extreme events, as usual optimization procedures might neglect them.

Transitions across Melancholia States in a Climate Model: Reconciling the Deterministic and Stochastic Points of View.

The Earth is well known to be, in the current astronomical configuration, in a regime where two asymptotic states can be realized. The warm state we live in is in competition with the ice-covered snowball state. The bistability exists as a result of the positive ice-albedo feedback. In a previous investigation performed on a intermediate complexity climate model we identified the unstable climate states (melancholia states) separating the coexisting climates, and studied their dynamical and geometrical properties. The melancholia states are ice covered up to the midlatitudes and attract trajectories initialized on the basin boundary. In this Letter, we study how stochastically perturbing the parameter controlling the intensity of the incoming solar radiation impacts the stability of the climate. We detect transitions between the warm and the snowball state and analyze in detail the properties of the noise-induced escapes from the corresponding basins of attraction. We determine the most probable paths for the transitions and find evidence that the melancholia states act as gateways, similarly to saddle points in an energy landscape.

Stochastic resonance for nonequilibrium systems.

Stochastic resonance (SR) is a prominent phenomenon in many natural and engineered noisy systems, whereby the response to a periodic forcing is greatly amplified when the intensity of the noise is tuned to within a specific range of values. We propose here a general mathematical framework based on large deviation theory and, specifically, on the theory of quasipotentials, for describing SR in noisy N-dimensional nonequilibrium systems possessing two metastable states and undergoing a periodically modulated forcing. The drift and the volatility fields of the equations of motion can be fairly general, and the competing attractors of the deterministic dynamics and the edge state living on the basin boundary can, in principle, feature chaotic dynamics. Similarly, the perturbation field of the forcing can be fairly general. Our approach is able to recover as special cases the classical results previously presented in the literature for systems obeying detailed balance and allows for expressing the parameters describing SR and the statistics of residence times in the two-state approximation in terms of the unperturbed drift field, the volatility field, and the perturbation field. We clarify which specific properties of the forcing are relevant for amplifying or suppressing SR in a system and classify forcings according to classes of equivalence. Our results indicate a route for a detailed understanding of SR in rather general systems.

Equivalence of nonequilibrium ensembles in turbulence models.

Understanding under what conditions it is possible to construct equivalent ensembles is key to advancing our ability to connect microscopic and macroscopic properties of nonequilibrium statistical mechanics. In the case of fluid dynamical systems, one issue is to test whether different models for viscosity lead to the same macroscopic properties of the fluid systems in different regimes. Such models include, besides the standard choice of constant viscosity, cases where the time symmetry of the evolution equations is exactly preserved, as it must be in the corresponding microscopic systems, when available. Here a time-reversible dynamics is obtained by imposing the conservation of global observables. We test the equivalence of reversible and irreversible ensembles for the case of a multiscale shell model of turbulence. We verify that the equivalence is obeyed for the mean values of macroscopic observables, up to an error that vanishes as the system becomes more and more chaotic.

Lessons on Climate Sensitivity From Past Climate Changes.

Over the last decade, our understanding of climate sensitivity has improved considerably. The climate system shows variability on many timescales, is subject to non-stationary forcing and it is most likely out of equilibrium with the changes in the radiative forcing. Slow and fast feedbacks complicate the interpretation of geological records as feedback strengths vary over time. In the geological past, the forcing timescales were different than at present, suggesting that the response may have behaved differently. Do these insights constrain the climate sensitivity relevant for the present day? In this paper, we review the progress made in theoretical understanding of climate sensitivity and on the estimation of climate sensitivity from proxy records. Particular focus lies on the background state dependence of feedback processes and on the impact of tipping points on the climate system. We suggest how to further use palaeo data to advance our understanding of the currently ongoing climate change.

Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems.

In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the chosen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan-Yorke dimension of the attractor. Preliminary numerical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

Extreme value theory for singular measures.

In this paper, we perform an analytical and numerical study of the extreme values of specific observables of dynamical systems possessing an invariant singular measure. Such observables are expressed as functions of the distance of the orbit of initial conditions with respect to a given point of the attractor. Using the block maxima approach, we show that the extremes are distributed according to the generalised extreme value distribution, where the parameters can be written as functions of the information dimension of the attractor. The numerical analysis is performed on a few low dimensional maps. For the Cantor ternary set and the Sierpinskij triangle, which can be constructed as iterated function systems, the inferred parameters show a very good agreement with the theoretical values. For strange attractors like those corresponding to the Lozi and Hènon maps, a slower convergence to the generalised extreme value distribution is observed. Nevertheless, the results are in good statistical agreement with the theoretical estimates. It is apparent that the analysis of extremes allows for capturing fundamental information of the geometrical structure of the attractor of the underlying dynamical system, the basic reason being that the chosen observables act as magnifying glass in the neighborhood of the point from which the distance is computed.