Research Interests

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

 

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

 

Recent Publications

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.
Wang, Peng, James C. McWilliams, Yusuke Uchiyama, Mickaël D. Chekroun, and Daling Li Yi. 2020. “Effects of wave streaming and wave variations on nearshore wave-driven circulation.” J. Phys. Oceanograhy 50 (10): 3025-3041. Publisher's Version Abstract

Wave streaming is a near-bottom mean current induced by the bottom drag on surface gravity waves. Wave variations include the variations in wave heights, periods, and directions. Here we use numerical simulations to study the effects of wave streaming and wave variations on the circulation that is driven by incident surface waves. Wave streaming induces an inner-shelf Lagrangian overturning circulation, which links the inner shelf with the surf zone. Wave variations cause along shore-variable wave breaking that produces surf eddies; however, such eddies can be suppressed by wave streaming. Moreover, with passive tracers we show that wave streaming and wave variations together enhance the cross- shelf material transport.

More