Mickaël David Chekroun

Köhler theory predicts the supersaturation required for an aerosol particle to activate into a cloud droplet by considering curvature and solute effects. However, in a saturated environment containing numerous aerosols, the competition for water vapor and the subsequent droplet activation process become highly complex. This complexity is further amplified by the presence of turbulence, which introduces significant variability in the local supersaturation experienced by each particle. This study employs Brownian models to investigate the activation of polydisperse aerosols in a turbulent environment. Turbulence is represented by temporally correlated stochastic disturbances, resulting in diverse supersaturation histories for individual particles. Our findings demonstrate that turbulence can enhance aerosol activation, but this effect exhibits nonlinear dependence on aerosol number concentration and size distribution breadth. Furthermore, our simulations reveal a counterintuitive phenomenon: Smaller aerosols can activate at the expense of larger ones due to the influence of turbulent fluctuations. This effect becomes more pronounced with increasing turbulence intensity. Significance Statement: This study investigates how turbulence impacts cloud droplet activation beyond traditional Köhler theory. By incorporating stochastic supersaturation fluctuations, we demonstrate a complex interplay between turbulence, aerosol number concentration, and size distribution breadth. Our findings reveal that turbulence can enhance activation, but with nonlinear and sometimes counterintuitive effects. Notably, smaller aerosols can gain an activation advantage over larger ones in turbulent conditions. These results have significant implications for understanding cloud microphysics and precipitation formation, emphasizing the critical role of turbulence in shaping cloud droplet distributions and ultimately influencing climate.

We present a generalization of linear response theory(LRT) for mixed jump-diffusion models—which combine both Gaussian and Lévy noise forcings that interact with the nonlinear dynamics—by deriving a comprehensive response formulas that accounts for perturbations to both the drift term and the jumps law. This class of models is particularly relevant for parameterizing the effects of unresolved scales in complex systems. Our formulas help thus quantifying uncertainties in either what needs to be parameterized (e.g. the jumps law), or measuring dynamical changes due to perturbations of the drift term (e.g. parameter variations). By generalizing the concepts of Kolmogorov operators and Green’s functions, we obtain new forms of fluctuation-dissipation relations. The resulting response is decomposed into contributions from the eigenmodes of the Kolmogorov operator, providing a fresh look into the intimate relationship between a system’s natural and forced variability. We demonstrate the theory’s predictive power with two distinct climate-centric applications. First, we apply our framework to a paradigmatic El Niño-Southern Oscillation model subject to state-dependent jumps and additive white noise, showing how the theory accurately predicts the system’s response to perturbations and how Kolmogorov modes can be used to diagnose its complex time variability. In a second, more challenging application, we use our LRT to perform accurate climate change projections in the Ghil–Sellers energy balance climate model, which is a spatially-extended model forced here by a spatio-temporal α -stable process. This work provides a comprehensive approach to climate modeling and prediction that enriches Hasselmann’s program, with implications for understanding climate sensitivity, detection and attribution of climate change, and assessing the risk of climate tipping points. Our results may find applications beyond the realm of climate, and seem of relevance for epidemiology, biology, finance, and quantitative social sciences, among others.

This work proposes a general framework for analyzing noise-driven transitions in spatially extended non-equilibrium systems and explaining 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.’

Detection and attribution (DA) studies are cornerstones of climate science, providing crucial evidence for policy decisions. Their goal is to link observed climate change patterns to anthropogenic and natural drivers via the optimal fingerprinting method (OFM). We show that response theory for nonequilibrium systems offers the physical and dynamical basis for OFM, including the concept of causality used for attribution. Our framework clarifies the method’s assumptions, advantages, and potential weaknesses. We use our theory to perform DA for prototypical climate change experiments performed on an energy balance model and on a low-resolution coupled climate model. We also explain the underpinnings of degenerate fingerprinting, which offers early warning indicators for tipping points. Finally, we extend the OFM to the nonlinear response regime. Our analysis shows that OFM has broad applicability across diverse stochastic systems influenced by time-dependent forcings, with potential relevance to ecosystems, quantitative social sciences, and finance, among others. Published by the American Physical Society 2024

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.

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.

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.

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.