Research Interests

From Chekroun et al. (2011), Physica D, 240 (21): 1685-1700:

 

Vimeo movie: https://vimeo.com/240039610

 

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Recent Publications

Chekroun, Mickaël D., Honghu Liu, James C. McWilliams, and Shouhong Wang. Submitted. “Transitions in Stochastic Non-equilibrium Systems: Efficient Reduction and Analysis”. 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., Jeroen S.W. Lamb, Christian J. Pangerl, and Martin Rasmussen. Submitted. “A Girsanov approach to slow parameterizing manifolds in the presence of noise”. arXiv's link Abstract
We consider a three-dimensional slow-fast system with quadratic nonlinearity and additive noise. The associated deterministic system of this stochastic differential equation (SDE) exhibits a periodic orbit and a slow manifold. The deterministic slow manifold can be viewed as an approximate parameterization of the fast variable of the SDE in terms of the slow variables. In other words the fast variable of the slow-fast system is approximately "slaved" to the slow variables via the slow manifold. We exploit this fact to obtain a two dimensional reduced model for the original stochastic system, which results in the Hopf-normal form with additive noise. Both, the original as well as the reduced system admit ergodic invariant measures describing their respective long-time behaviour. We will show that for a suitable metric on a subset of the space of all probability measures on phase space, the discrepancy between the marginals along the radial component of both invariant measures can be upper bounded by a constant and a quantity describing the quality of the parameterization. An important technical tool we use to arrive at this result is Girsanov's theorem, which allows us to modify the SDEs in question in a way that preserves transition probabilities. This approach is then also applied to reduced systems obtained through stochastic parameterizing manifolds, which can be viewed as generalized notions of deterministic slow manifolds.
 
Dror, Tom, Mickaël D. Chekroun, Ilan Koren, and Orit Altaratz. 2021. “Deciphering organization of GOES-16 green cumulus through the empirical orthogonal function (EOF) lens.” Atmospheric Chemistry and Physics 21: 12261–12272. Publisher's Version Abstract
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.
Chekroun, Mickaël D., Honghu Liu, and James C. McWilliams. 2021. “Stochastic rectification of fast oscillations on slow manifold closures.” Proceedings of the National Academy of Sciences 118 (48). 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.
Charó, Gisela D., Mickaël D. Chekroun, Denisse Sciamarella, and Michael Ghil. 2021. “Noise-driven topological changes in chaotic dynamics.” Chaos 31 (10): 103115. Publisher's Version Abstract

Noise modifies the behavior of chaotic systems in both quantitative and qualitative ways. To study these modifications, the present work compares the topological structure of the deterministic Lorenz (1963) attractor with its stochastically perturbed version. The deterministic attractor is well known to be “strange” but it is frozen in time. When driven by multiplicative noise, the Lorenz model’s random attractor (LORA) evolves in time. Algebraic topology sheds light on the most striking effects involved in such an evolution. In order to examine the topological structure of the snapshots that approximate LORA, we use branched manifold analysis through homologies—a technique originally introduced to characterize the topological structure of deterministically chaotic flows—which is being extended herein to nonlinear noise-driven systems. The analysis is performed for a fixed realization of the driving noise at different time instants in time. The results suggest that LORA’s evolution includes sharp transitions that appear as topological tipping points.

Santos Gutiérrez, Manuel, Valerio Lucarini, Mickaël D. Chekroun, and Michael Ghil. 2021. “Reduced-order models for coupled dynamical systems: Data-driven methods and the Koopman operator.” Chaos 31: 053116. Publisher's Version Abstract
Providing efficient and accurate parameterizations for model reduction is a key goal in many areas of science and technology. Here, we present a strong link between data-driven and theoretical approaches to achieving this goal. Formal perturbation expansions of the Koopman operator allow us to derive general stochastic parameterizations of weakly coupled dynamical systems. Such parameterizations yield a set of stochastic integrodifferential equations with explicit noise and memory kernel formulas to describe the effects of unresolved variables. We show that the perturbation expansions involved need not be truncated when the coupling is additive. The unwieldy integrodifferential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation. This connection helps setting up a parallelism between the top-down, equation-based methodology herein and the well-established empirical model reduction (EMR) methodology that has been shown to provide efficient dynamical closures to partially observed systems. Hence, our findings, on the one hand, support the physical basis and robustness of the EMR methodology and, on the other hand, illustrate the practical relevance of the perturbative expansion used for deriving the parameterizations.
Parameterizations aim to reduce the complexity of high-dimensional dynamical systems. Here, a theory-based and a data-driven approach for the parameterization of coupled systems are compared, showing that both yield the same stochastic multilevel structure. The results provide very strong support to the use of empirical methods in model reduction and clarify the practical relevance of the proposed theoretical framework.
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