We introduce a generalization of linear response theory for mixed jump-diffusion models, combining both Gaussian and Lévy noise forcings that interact with the nonlinear dynamics. This class of models covers a broad range of stochastic chaos and complexity for which the jump-diffusion processes are a powerful tool to parameterize the missing physics or effects of the unresolved scales onto the resolved ones.
By generalizing concepts such as Kolmogorov operators and Green's functions to this context, we derive fluctuation-dissipation relationships for such models. The system response can then be interpreted in terms of contributions from the eigenmodes of the Kolmogorov operator (Kolmogorov modes) decomposing the time-lagged correlation functions of the unperturbed dynamics. The underlying formulas offer a fresh look on the intimate relationships between the system's natural variability and its forced variability.
We apply our theory to a paradigmatic El Niño-Southern Oscillation (ENSO) subject to state-dependent jumps and additive white noise parameterizing intermittent and nonlinear feedback mechanisms, key factors in the actual ENSO phenomenon. Such stochastic parameterizations are shown to produce stochastic chaos with an enriched time-variability. The Kolmogorov modes encoding the latter are then computed, and our Green's functions formulas are shown to achieve a remarkable accuracy to predict the system's response to perturbations.
This work enriches Hasselmann's program by providing a more comprehensive approach to climate modeling and prediction, allowing for accounting the effects of both continuous and discontinuous stochastic forcing. Our results have implications for understanding climate sensitivity, detection and attributing climate change, and assessing the risk of climate tipping points.

Systems far from equilibrium approach stability slowly due to "anti-mixing" characterized by regions of the phase-space that remain disconnected after prolonged action of the flow. We introduce the Optimal Growth Mode (OGM) to capture this slow initial relaxation. The OGM is calculated from Markov matrices approximating the action of the Fokker-Planck operator onto the phase space. It is obtained as the mode having the largest growth in energy before decay. Important nuances between the OGM and the more traditional slowest decaying mode are detailed in the case of the Lorenz 63 model. The implications for understanding how complex systems respond to external forces, are discussed.

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.

 

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.

We parameterize sub-grid scale (SGS) fluxes in sinusoidally forced two-dimensional turbulence on the β-plane at high Reynolds numbers (Re ∼25,000) using simple 2-layer convolutional neural networks (CNN) having only O(1000) parameters, two orders of magnitude smaller than recent studies employing deeper CNNs with 8–10 layers; we obtain stable, accurate, and long-term online or a posteriori solutions at 16× downscaling factors. Our methodology significantly improves training efficiency and speed of online large eddy simulations runs, while offering insights into the physics of closure in such turbulent flows. Our approach benefits from extensive hyperparameter searching in learning rate and weight decay coefficient space, as well as the use of cyclical learning rate annealing, which leads to more robust and accurate online solutions compared to fixed learning rates. Our CNNs use either the coarse velocity or the vorticity and strain fields as inputs, and output the two components of the deviatoric stress tensor, S_d. We minimize a loss between the SGS vorticity flux divergence (computed from the high-resolution solver) and that obtained from the CNN-modeled S_d, without requiring energy or enstrophy preserving constraints. The success of shallow CNNs in accurately parameterizing this class of turbulent flows implies that the SGS stresses have a weak non-local dependence on coarse fields; it also aligns with our physical conception that small-scales are locally controlled by larger scales such as vortices and their strained filaments. Furthermore, 2-layer CNN-parameterizations are more likely to be interpretable.

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.

Clouds play a key role in Earth's energy budget and water cycle. Their response to global warming contributes the largest uncertainty to climate prediction. Here, by performing an empirical orthogonal function analysis on 42 years of reanalysis data of global cloud coverage, we extract an unambiguous trend and El-Niño–Southern-Oscillation-associated modes. The trend mode translates spatially to decreasing trends in cloud coverage over most continents and increasing trends over the tropical and subtropical oceans. A reduction in near-surface relative humidity can explain the decreasing trends in cloud coverage over land. Our results suggest potential stress on the terrestrial water cycle and changes in the energy partition between land and ocean, all associated with global warming.

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.

 

 

 

One of the major sources of uncertainty in climate prediction results from the limitations in representing shallow cumulus (Cu) in models. Recently, a class of continental shallow convective Cu was shown to share distinct morphological properties and to emerge globally mostly over forests and vegetated areas, thus named greenCu. Using machine-learning supervised classification, we identify greenCu fields over three regions, from the tropics to mid- and higher-latitudes, and establish a novel satellite-based data set called greenCuDb, consisting of 1° × 1° sized, high-resolution MODIS images. Using greenCuDb in conjunction with ERA5 reanalysis data, we create greenCu composites for different regions and reveal that greenCu are driven by similar large-scale meteorological conditions, regardless of their geographical locations throughout the world's continents. These conditions include distinct profiles of temperature, humidity and large-scale vertical velocity. The boundary layer is anomalously warm and moderately humid, and is accompanied by a strong large-scale subsidence in the free troposphere.

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.

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.

A subset of continental shallow convective cumulus (Cu) cloud fields has been shown to have distinct spatial properties and to form mostly over forests and vegetated areas, thus referred to as “green Cu” (Dror et al., 2020). Green Cu fields are known to form organized mesoscale patterns, yet the underlying mechanisms, as well as the time variability of these patterns, are still lacking understanding. Here, we characterize the organization of green Cu in space and time, by using data-driven organization metrics and by applying an empirical orthogonal function (EOF) analysis to a high-resolution GOES-16 dataset. We extract, quantify, and reveal modes of organization present in a green Cu field, during the course of a day. The EOF decomposition is able to show the field's key organization features such as cloud streets, and it also delineates the less visible ones, as the propagation of gravity waves (GWs) and the emergence of a highly organized grid on a spatial scale of hundreds of kilometers, over a time period that scales with the field's lifetime. Using cloud fields that were reconstructed from different subgroups of modes, we quantify the cloud street's wavelength and aspect ratio, as well as the GW-dominant period.