Mickaël D. Chekroun

This work investigates a three-dimensional slow-fast stochastic system with quadratic nonlinearity and additive noise, inspired by fluid dynamics.
The deterministic counterpart exhibits a periodic orbit and a slow manifold.
We demonstrate that, under specific parameter regimes, this deterministic slow manifold can serve as an approximate parameterization of the fast variable by the slow variables within the stochastic system.

Building upon this parameterization, we derive a two-dimensional reduced model, a stochastic Hopf normal form, that captures the essential dynamics of the original system. Both the original and the reduced systems possess ergodic invariant measures, characterizing their long-term behavior.

We quantify the discrepancy between the original system and its slow approximation by deriving error estimates involving the Wasserstein distance between the marginals of these invariant measures along the radial component. These error bounds are shown to be controlled by a parameterization defect, which measures the quality of the fast-slow variable parameterization.

A key technical innovation lies in the application of Girsanov's theorem to obtain these error estimates in the presence of oscillatory instabilities.  Furthermore, we extend our analysis to regimes exhibiting an "inverted" timescale separation, where the variable to be parameterized evolves on a slower timescale than the resolved variables. To address these more challenging scenarios, we introduce path-dependent coefficients in the parameterizing manifold, enabling the derivation of robust error bounds for the corresponding reduced model. Numerical simulations complement our theoretical findings, providing insights into the model's behavior and exploring parameter regimes beyond the scope of our analytical results.

 

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.