This work proposes a general framework for analyzing noise-driven transitions in  spatially extended non-equilibrium systems and explains the emergence of coherent patterns beyond the instability onset. The framework relies on stochastic parameterization formulas to reduce the complexity of the original equations while preserving the essential dynamical effects of unresolved scales. The approach is flexible and operates for both Gaussian noise and non-Gaussian noise with jumps.

Our stochastic parameterization formulas offer two key advantages. First, they can approximate stochastic invariant manifolds when these manifolds exist. Second, even when such manifolds break down, our formulas can be adapted through a simple optimization of its constitutive parameters. This allows us to handle scenarios with weak time-scale separation where the system has undergone multiple transitions, resulting in large-amplitude solutions not captured by invariant manifolds or other time-scale separation methods.

The optimized stochastic parameterizations capture then how small-scale noise impacts larger scales through the system's nonlinear interactions. This effect is achieved by the very fabric of our parameterizations incorporating non-Markovian (memory-dependent) coefficients into the reduced equation. These coefficients account for the noise's past influence, not just its current value, using a finite memory length that is selected for optimal performance. The specific "memory" function, which determines how this past influence is weighted, depends on both the strength of the noise and how it interacts with the system's nonlinearities.

Remarkably, training our theory-guided reduced models on a single noise path effectively learns the optimal memory length for out-of-sample predictions. This approach retains indeed good accuracy in predicting noise-induced transitions, including rare events, when tested against a large ensemble of different noise paths. This success stems from our ``hybrid" approach, which combines analytical understanding with data-driven learning. This combination avoids a key limitation of purely data-driven methods: their struggle to generalize to unseen scenarios, also known as the "extrapolation problem."
 

 

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.

Cloud microphysics studies include how tiny cloud droplets grow and become rain. This is crucial for understanding cloud properties like size, life span, and impact on climate through radiative effects. Small weak-updraft clouds near the haze-to-cloud transition are especially difficult to measure and understand. They are abundant but hard to capture by satellites. Köhler’s theory explains initial droplet growth but struggles with large particle groups. Here, we present a stochastic, analytical framework building on Köhler’s theory to account for (monodisperse) aerosols and cloud droplet interaction through competitive growth in a limited water vapor field. These interactions are modeled by sink terms, while fluctuations in supersaturation affecting droplet growth are modeled by nonlinear white noise terms. Our results identify hysteresis mechanisms in the droplet activation and deactivation processes. Our approach allows for multimodal cloud’s droplet size distributions supported by laboratory experiments, offering a different perspective on haze-to-cloud transition and small cloud formation.

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.

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.

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.

 

 

 

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.

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. 

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.

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.

We study the pullback attractor (PBA) of a seasonally forced delay differential model for the El Ni\~no--Southern Oscillation (ENSO); the model has two delays, associated with a positive and a negative feedback. The control parameter is the intensity of the positive feedback and the PBA undergoes a crisis that consists of a chaos-to-chaos transition. Since  the PBA is dominated by chaotic behavior, we refer to it as a strange PBA. Both chaotic regimes correspond to an overlapping of resonances but the two differ by the properties of this overlapping. The crisis manifests itself by a brutal change not only in the size but also in the shape of the PBA. The change is associated with the sudden disappearance of the most extreme warm (El Ni\~no) and cold (La Ni\~na) events, as one crosses the critical parameter value from below.  The analysis reveals that regions of the strange PBA that survive the crisis are those populated by the most probable states of the system. These regions are those that exhibit robust foldings with respect to perturbations.  The effect of noise on this phase-and-paramater space behavior is then discussed. It is shown that the chaos-to-chaos crisis may or may not survive the addition of small noise to the evolution equation, depending on how the noise enters the latter.

 

 

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.

 

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.

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.