Critical Transitions

Santos Gutiérrez, Manuel, Mickaël D. Chekroun, and Ilan Koren. Submitted. “Gibbs states and Brownian models for haze and cloud droplets”. arXiv version Abstract
Clouds microphysics describes the formation and evolution of cloud droplets, rain, and ice particles. It is among the most critical factors in determining the cloud's size, lifetime, precipitation, and radiative effect. Among all cloud types, the small clouds, characterized by weak updrafts, that are close to the haze-to-cloud transition pose challenges in measuring them and understanding their properties. They are superabundant but hard to capture by satellites and often falsely regarded as aerosols. Köhler's theory explains droplet activation and their growth by condensation at the earliest stages of cloud development. It fully describes the thermodynamic state of a single drop but falls short when explaining the collective behavior of large populations of particles. This is especially important when the supersaturation pool is limited. We present an analytical framework to extend Köhler's theory to coexisting cloud droplets. Our results suggest hysteresis and asymmetry in the process of droplet activation and deactivation. The turbulent nature of clouds is incorporated into our model formulation as Brownian noise to provide explicit droplet size distributions and activation timescales. The theoretical findings are confronted with experimental data stemming from laboratory clouds created in the Pi convection chamber, suggesting a new way of understanding haze-to-cloud transitions and small cloud formation processes.
Detection and attribution studies have played a major role in shaping contemporary climate science and have provided key motivations supporting global climate policy negotiations. Their goal is to associate unambiguously observed patterns of climate change with anthropogenic and natural forcings acting as drivers through the so-called optimal fingerprinting method. We show here how response theory for nonequilibrium systems provides the physical and dynamical foundations behind optimal fingerprinting for the climate change detection and attribution problem, including the notion of causality used for attribution purposes. We clearly frame assumptions, strengths, and potential pitfalls of the method. Additionally, we clarify the mathematical framework behind the degenerate fingerprinting method that leads to early warning indicators for tipping points. Finally, we extend the optimal fingerprinting method to the regime of nonlinear response. Our findings indicate that optimal fingerprinting for detection and attribution can be applied to virtually any stochastic system undergoing time-dependent forcing.
Koren, Ilan, Tom Dror, Orit Altaratz, and Mickaël D. Chekroun. 2024. “Cloud Versus Void Chord Length Distributions (LvL) as a Measure for Cloud Field Organization.” Geophysical Research Letters 51 (11): e2024GL108435. Publisher's Version Abstract
Cloud organization impacts the radiative effects and precipitation patterns of the cloud field. Deviating from randomness, clouds exhibit either clustering or a regular grid structure, characterized by the spacing between clouds and the cloud size distribution. The two measures are coupled but do not fully define each other. Here, we present the deviation from randomness of the cloud- and void-chord length distributions as a measure for both factors. We introduce the LvL representation and an associated 2D score that allow for unambiguously quantifying departure from well-defined baseline randomness in cloud spacing and sizes. This approach demonstrates sensitivity and robustness in classifying cloud field organization types. Its delicate sensitivity unravels the temporal evolution of a single cloud field, providing novel insights into the underlying governing processes.

Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics.

To identify these invariant sets we adopt an approach combining Galerkin-Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle-node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems.

These dynamical insights enable us in turn to design a stochastic model whose solutions—as the delay parameter drifts slowly through its critical values—produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping “points” beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena.

Chekroun, Mickaël D., H. Liu, K. Srinivasan, and James C. McWilliams. 2024. “The High-Frequency and Rare Events Barriers to Neural Closures of Atmospheric Dynamics.” Journal of Physics: Complexity 5: 025004. Publisher's version Abstract
Recent years have seen a surge in interest for leveraging neural networks to parameterize small-scale or fast processes in climate and turbulence models. In this short paper, we point out two fundamental issues in this endeavor. The first concerns the difficulties neural networks may experience in capturing rare events due to limitations in how data is sampled. The second arises from the inherent multiscale nature of these systems. They combine high-frequency components (like inertia-gravity waves) with slower, evolving processes (geostrophic motion). This multiscale nature creates a significant hurdle for neural network closures. To illustrate these challenges, we focus on the atmospheric 1980 Lorenz model, a simplified version of the Primitive Equations that drive climate models. This model serves as a compelling example because it captures the essence of these difficulties.
Chekroun, Mickaël D., Honghu Liu, and James C. McWilliams. 2023. “Optimal parameterizing manifolds for anticipating tipping points and higher-order critical transitions.” Chaos: An Interdisciplinary Journal of Nonlinear Science 33 (9): 093126. Publisher's Version Abstract
A general, variational approach to derive low-order reduced models from possibly non-autonomous systems is presented. The approach is based on the concept of optimal parameterizing manifold (OPM) that substitutes more classical notions of invariant or slow manifolds when the breakdown of “slaving” occurs, i.e., when the unresolved variables cannot be expressed as an exact functional of the resolved ones anymore. The OPM provides, within a given class of parameterizations of the unresolved variables, the manifold that averages out optimally these variables as conditioned on the resolved ones. The class of parameterizations retained here is that of continuous deformations of parameterizations rigorously valid near the onset of instability. These deformations are produced through the integration of auxiliary backward–forward systems built from the model’s equations and lead to analytic formulas for parameterizations. In this modus operandi, the backward integration time is the key parameter to select per scale/variable to parameterize in order to derive the relevant parameterizations which are doomed to be no longer exact away from instability onset due to the breakdown of slaving typically encountered, e.g., for chaotic regimes. The selection criterion is then made through data-informed minimization of a least-square parameterization defect. It is thus shown through optimization of the backward integration time per scale/variable to parameterize, that skilled OPM reduced systems can be derived for predicting with accuracy higher-order critical transitions or catastrophic tipping phenomena, while training our parameterization formulas for regimes prior to these transitions takes place.
Lucarini, Valerio, and Mickaël D. Chekroun. 2023. “Theoretical tools for understanding the climate crisis from Hasselmann’s programme and beyond.” Nature Review Physics 5: 744-765 . Publisher's link Abstract
Klaus Hasselmann’s revolutionary intuition in climate science was to use the stochasticity associated with fast weather processes to probe the slow dynamics of the climate system. Doing so led to fundamentally new ways to study the response of climate models to perturbations, and to perform detection and attribution for climate change signals. Hasselmann’s programme has been extremely influential in climate science and beyond. In this Perspective, we first summarize the main aspects of such a programme using modern concepts and tools of statistical physics and applied mathematics. We then provide an overview of some promising scientific perspectives that might clarify the science behind the climate crisis and that stem from Hasselmann’s ideas. We show how to perform rigorous and data-driven model reduction by constructing parameterizations in systems that do not necessarily feature a timescale separation between unresolved and resolved processes. We outline a general theoretical framework for explaining the relationship between climate variability and climate change, and for performing climate change projections. This framework enables us seamlessly to explain some key general aspects of climatic tipping points. Finally, we show that response theory provides a solid framework supporting optimal fingerprinting methods for detection and attribution.
Chekroun, Mickaël D., Honghu Liu, James C. McWilliams, and Shouhong Wang. 2023. “Transitions in Stochastic Non-equilibrium Systems: Efficient Reduction and Analysis.” Journal of Differential Equations 346 (10): 145-204. Publisher's version Abstract

A central challenge in physics is to describe non-equilibrium systems driven by randomness, such as a randomly growing interface, or fluids subject to random fluctuations that account e.g. for local stresses and heat fluxes in the fluid which are not related to the velocity and temperature gradients. For deterministic systems with infinitely many degrees of freedom, normal form and center manifold theory have shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns, associated with genuine physical regimes. However, in presence of random fluctuations, the underlying reduction principle to the center manifold is seriously challenged due to large excursions caused by the noise, and the approach needs to be revisited.

In this study, we present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds, on one hand, and energy estimates measuring the defect of parameterization of the high-modes, on the other. To operate for fluid problems subject to stochastic stirring forces, these error estimates are derived under assumptions regarding dissipation effects brought by the high-modes in order to suitably counterbalance the loss of regularity due to the nonlinear terms. As a result, the approach enables us to analyze, from reduced equations of the stochastic fluid problem, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation, as long as the noise’s intensity and the eigenvalue’s magnitude of the mildly unstable mode scale accordingly.

In the case of SPDEs forced by a multiplicative noise in the orthogonal subspace of e.g. its mildly unstable mode, our parameterization formulas show that the noise gets transmitted to this mode via non-Markovian coefficients, and that the reduced equation is only stochastically driven by the latter.  These coefficients depend explicitly on the noise path's history, and their memory content is self-consistently determined by the intensity of the random force and its interaction through the SPDE's nonlinear terms. Applications to a stochastic Rayleigh-B\'enard problem  are detailed, for which conditions for a stochastic pitchfork bifurcation (in large probability) to occur, are clarified.




Chekroun, Mickaël D., Ilan Koren, Honghu Liu, and Huan Liu. 2022. “Generic generation of noise-driven chaos in stochastic time delay systems: Bridging the gap with high-end simulations.” Science Advances 8 (46): eabq7137. Publisher's Version Abstract

Nonlinear time delay systems produce inherently delay-induced periodic oscillations, which are, however, too idealistic compared to observations. We exhibit a unified stochastic framework to systematically rectify such oscillations into oscillatory patterns with enriched temporal variabilities through generic, nonlinear responses to stochastic perturbations. Two paradigms of noise-driven chaos in high dimension are identified, fundamentally different from chaos triggered by parameter-space noise. Noteworthy is a low-dimensional stretch-and-fold mechanism, leading to stochastic strange attractors exhibiting horseshoe-like structures mirroring turbulent transport of passive tracers. The other is high-dimensional , with noise acting along the critical eigendirection and transmitted to deeperstable modes through nonlinearity, leading to stochastic attractors exhibiting swarm-like behaviors with power-law and scale break properties. The theory is applied to cloud delay models to parameterize missing physics such as intermittent rain and Lagrangian turbulent effects. The stochastically rectified model reproduces with fidelity complex temporal variabilities of open-cell oscillations exhibited by high-end cloud simulations.

Chekroun, Mickaël D., Henk A. Dijkstra, Taylan Şengül, and Shouhong Wang. 2022. “Transitions of zonal flows in a two- layer quasi-geostrophic ocean model.” Nonlinear Dynamics . Publisher's version Abstract
We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that two modes become critical and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to S^3. The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains
(stable or unstable) time-periodic solutions and a quasi-periodic solution.
Chekroun, Mickaël D., Honghu Liu, and James C. McWilliams. 2021. “Stochastic rectification of fast oscillations on slow manifold closures.” Proc. Natl. Acad. Sci. 118 (48): E2113650118. Publisher's Version Abstract
The problems of identifying the slow component (e.g., for weather forecast initialization) and of characterizing slow–fast interactions are central to geophysical fluid dynamics. In this study, the related rectification problem of slow manifold closures is addressed when breakdown of slow-to-fast scales deterministic parameterizations occurs due to explosive emergence of fast oscillations on the slow, geostrophic motion. For such regimes, it is shown on the Lorenz 80 model that if 1) the underlying manifold provides a good approximation of the optimal nonlinear parameterization that averages out the fast variables and 2) the residual dynamics off this manifold is mainly orthogonal to it, then no memory terms are required in the Mori–Zwanzig full closure. Instead, the noise term is key to resolve, and is shown to be, in this case, well modeled by a state-independent noise, obtained by means of networks of stochastic nonlinear oscillators. This stochastic parameterization allows, in turn, for rectifying the momentum-balanced slow manifold, and for accurate recovery of the multiscale dynamics. The approach is promising to be further applied to the closure of other more complex slow–fast systems, in strongly coupled regimes.
Chekroun, Mickaël D., Ilan Koren, and Honghu Liu. 2020. “Efficient reduction for diagnosing Hopf bifurcation in delay differential systems: Applications to cloud-rain models.” Chaos: An Interdisciplinary Journal of Nonlinear Science 30: 053130 . Publisher's Version Abstract

By means of Galerkin-Koornwinder (GK) approximations, an efficient reduction approach to the Stuart-Landau (SL) normal form and center manifold is presented for a broad class of nonlinear systems of delay differential equations (DDEs) that covers the cases of discrete as well as distributed delays. The focus is on the Hopf bifurcation as the consequence of the critical equilibrium's destabilization resulting from an eigenpair crossing the imaginary axis. The nature of the resulting Hopf bifurcation (super- or subcritical) is then characterized by the inspection of a Lyapunov coefficient easy to determine based on the model's coefficients and delay parameter.  We believe that our approach, which does not rely too much on functional analysis considerations but more on analytic calculations, is suitable  to concrete situations arising in physics applications.

Thus, using this GK approach to the Lyapunov coefficient and SL normal form, the occurrence of Hopf bifurcations in the cloud-rain delay models of Koren and Feingold (KF) on one hand, and Koren, Tziperman and Feingold (KTF), on the other, are analyzed. Noteworthy is the existence for the KF model of large regions of the parameter space for which subcritical and supercritical Hopf bifurcations coexist. These regions are determined in particular by the intensity of the KF model's nonlinear effects. ``Islands'' of supercritical Hopf bifurcations are shown to exist within a subcritical Hopf bifurcation ``sea;'' these islands being bordered by double-Hopf bifurcations occurring when the linearized dynamics at the critical equilibrium exhibit two pairs of purely imaginary eigenvalues. 

Tantet, Alexis, Mickaël D. Chekroun, Henk A. Dijkstra, and J. David Neelin. 2020. “Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation.” Journal of Statistical Physics 179: 1403–1448. Publisher's Version Abstract

The spectrum of the generator (Kolmogorov operator) of a diffusion process, referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed characterization of correlation functions and power spectra of stochastic systems via decomposition formulas in terms of RP resonances; see Part I of this contribution (Chekroun et al. in Theory J Stat., 2020). Stochastic analysis techniques relying on the theory of Markov semigroups for the study of the RP spectrum and a rigorous reduction method is presented in Part I Chekroun et al. (2020). This framework is here applied to study a stochastic Hopf bifurcation in view of characterizing the statistical properties of nonlinear oscillators perturbed by noise, depending on their stability. In light of the Hörmander theorem, it is first shown that the geometry of the unperturbed limit cycle, in particular its isochrons, i.e., the leaves of the stable manifold of the limit cycle generalizing the notion of phase, is essential to understand the effect of the noise and the phenomenon of phase diffusion. In addition, it is shown that the RP spectrum has a spectral gap, even at the bifurcation point, and that correlations decay exponentially fast. Explicit small-noise expansions of the RP eigenvalues and eigenfunctions are then obtained, away from the bifurcation point, based on the knowledge of the linearized deterministic dynamics and the characteristics of the noise. These formulas allow one to understand how the interaction of the noise with the deterministic dynamics affect the decay of correlations. Numerical results complement the study of the RP spectrum at the bifurcation point, revealing useful scaling laws. The analysis of the Markov semigroup for stochastic bifurcations is thus promising in providing a complementary approach to the more geometric random dynamical system (RDS) approach. This approach is not limited to low-dimensional systems and the reduction method presented in Chekroun et al. (2020) is applied to a stochastic model relevant to climate dynamics in the third part of this contribution (Tantet et al. in J Stat Phys., 2019).

Tantet, Alexis, Mickaël D. Chekroun, J. David Neelin, and Henk A. Dijkstra. 2020. “Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part III: Application to the Cane–Zebiak Model of the El Niño–Southern Oscillation.” Journal of Statistical Physics 179: 1449–1474. Publisher's Version Abstract

The response of a low-frequency mode of climate variability, El Niño–Southern Oscillation, to stochastic forcing is studied in a high-dimensional model of intermediate complexity, the fully-coupled Cane–Zebiak model (Zebiak and Cane 1987), from the spectral analysis of Markov operators governing the decay of correlations and resonances in the power spectrum. Noise-induced oscillations excited before a supercritical Hopf bifurcation are examined by means of complex resonances, the reduced Ruelle–Pollicott (RP) resonances, via a numerical application of the reduction approach of the first part of this contribution (Chekroun et al. 2019) to model simulations. The oscillations manifest themselves as peaks in the power spectrum which are associated with RP resonances organized along parabolas, as the bifurcation is neared. These resonances and the associated eigenvectors are furthermore well described by the small-noise expansion formulas obtained by Gaspard (2002) and made explicit in the second part of this contribution (Tantet et al. 2019). Beyond the bifurcation, the spectral gap between the imaginary axis and the real part of the leading resonances quantifies the diffusion of phase of the noise-induced oscillations and can be computed from the linearization of the model and from the diffusion matrix of the noise. In this model, the phase diffusion coefficient thus gives a measure of the predictability of oscillatory events representing ENSO. ENSO events being known to be locked to the seasonal cycle, these results should be extended to the non-autonomous case. More generally, the reduction approach theorized in Chekroun et al. (2019), complemented by our understanding of the spectral properties of reference systems such as the stochastic Hopf bifurcation, provides a promising methodology for the analysis of low-frequency variability in high-dimensional stochastic systems.

Chekroun, Mickaël D. 2018. “Topological instabilities in families of semilinear parabolic problems subject to nonlinear perturbations.” Discrete and Continuous Dynamical Systems B, doi: 10.3934/dcdsb.2018075. Publisher's version Abstract

In this article it is proved that the dynamical properties of a broad class of semilinear parabolic problems are sensitive to arbitrarily small but smooth perturbations of the nonlinear term, when the spatial dimension is either equal to one or two. This topological instability is shown to result from a local deformation of the global bifurcation diagram associated with the corresponding elliptic problems. Such a deformation is shown to systematically occur via the creation of either a multiple-point or a new fold-point on this diagram when an appropriate small perturbation is applied to the nonlinear term. More precisely, it is shown that for a broad class of nonlinear elliptic problems, one can always find an arbitrary small perturbation of the nonlinear term, that generates a local S on the bifurcation diagram whereas the latter is e.g. monotone when no perturbation is applied; substituting thus a single solution by several ones. Such an increase in the local multiplicity of the solutions to the elliptic problem results then into a topological instability for the corresponding parabolic problem.
The rigorous proof of the latter instability result requires though to revisit the classical concept of topological equivalence to encompass important cases for the applications such as semi-linear parabolic problems for which the semigroup may exhibit non-global dissipative properties, allowing for the coexistence of blow-up regions and local attractors in the phase space; cases that arise e.g. in combustion theory. A revised framework of topological robustness is thus introduced in that respect within which the main topological instability result is then proved for continuous, locally Lipschitz but not necessarily C1 nonlinear terms, that prevent in particular the use of linearization techniques, and for which the family of semigroups may exhibit non-dissipative properties.

Chekroun, Mickaël D., Honghu Liu, and James C. McWilliams. 2017. “The emergence of fast oscillations in a reduced primitive equation model and its implications for closure theories.” Computers & Fluids 151: 3-22. Publisher's Version Abstract

The problem of emergence of fast gravity-wave oscillations in rotating, stratified flow is reconsidered. Fast inertia-gravity oscillations have long been considered an impediment to initialization of weather forecasts, and the concept of a “slow manifold” evolution, with no fast oscillations, has been hypothesized. It is shown on a reduced Primitive Equation model introduced by Lorenz in 1980 that fast oscillations are absent over a finite interval in Rossby number but they can develop brutally once a critical Rossby number is crossed, in contradistinction with fast oscillations emerging according to an exponential smallness scenario such as reported in previous studies, including some others by Lorenz. The consequences of this dynamical transition on the closure problem based on slow variables is also discussed. In that respect, a novel variational perspective on the closure problem exploiting manifolds is introduced. This framework allows for a unification of previous concepts such as the slow manifold or other concepts of “fuzzy” manifold. It allows furthermore for a rigorous identification of an optimal limiting object for the averaging of fast oscillations, namely the optimal parameterizing manifold (PM). It is shown through detailed numerical computations and rigorous error estimates that the manifold underlying the nonlinear Balance Equations provides a very good approximation of this optimal PM even somewhat beyond the emergence of fast and energetic oscillations.


Dijkstra, Henk A, Alexis Tantet, Jan Viebahn, Erik Mulder, Mariët Hebbink, Daniele Castellana, Henri van den Pol, et al. 2016. “A numerical framework to understand transitions in high-dimensional stochastic dynamical systems.” Dynamics and Statistics of the Climate System 1 (1): 1-27. Publisher's Version Abstract

Dynamical systems methodology is a mature complementary approach to forward simulation which can be used to investigate many aspects of climate dynamics. With this paper, a review is given on the methods to analyze deterministic and stochastic climate models and show that these are not restricted to low-dimensional toy models, but that they can be applied to models formulated by stochastic partial differential equations. We sketch the numerical implementation of these methods and illustrate these by showing results for two canonical problems in climate dynamics.

Roques, Lionel, and Mickaël D. Chekroun. 2011. “Probing chaos and biodiversity in a simple competition model.” Ecological Complexity 8 (1): 98 - 104. Publisher's Version Abstract
Recent theoretical work has reported that chaos facilitates biodiversity. In this paper, we study the lowest-dimensional Lotka–Volterra competition model that exhibits chaotic trajectories, a model with four species. We observe that interaction and growth parameters leading respectively to extinction of three species, or coexistence of two, three or four species, are for the most part arranged in large regions with clear boundaries. Small islands of parameters that lead to chaos are also found. These regions where chaos occurs are, in the three cases presented here, situated at the interface between a non-chaotic four-species region and a region where extinction occurs. This implies a high sensitivity of biodiversity with respect to parameter variations in the chaotic regions. Additionally, in regions where extinction occurs which are adjacent to chaotic regions, the computation of local Lyapunov exponents reveals that a possible cause of extinction is the overly strong fluctuations in species abundances induced by local chaos at the beginning of the interval of study. For this model, we conclude that biodiversity is a necessary condition for chaos rather than a consequence of chaos, which can be seen as a signal of a high extinction risk.