Research Interests

 

Turbulence Closure with Small, Neural Networks:

 

Stochastic Strange Attractor (Chekroun et al. (2011), Physica D, 240):

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

 

Cloud Physics and Stochastic Strange Attractors (Chekroun et al. (2022), Science Advances, 8 (46)):

    

Vimeo movie: https://vimeo.com/773696444

 

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Recent Publications

Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics.

To identify these invariant sets we adopt an approach combining Galerkin-Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle-node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems.

These dynamical insights enable us in turn to design a stochastic model whose solutions—as the delay parameter drifts slowly through its critical values—produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping “points” beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena.

 
 
Detection and attribution studies have played a major role in shaping contemporary climate science and have provided key motivations supporting global climate policy negotiations. Their goal is to associate unambiguously observed patterns of climate change with anthropogenic and natural forcings acting as drivers through the so-called optimal fingerprinting method. We show here how response theory for nonequilibrium systems provides the physical and dynamical foundations behind optimal fingerprinting for the climate change detection and attribution problem, including the notion of causality used for attribution purposes. We clearly frame assumptions, strengths, and potential pitfalls of the method. Additionally, we clarify the mathematical framework behind the degenerate fingerprinting method that leads to early warning indicators for tipping points. Finally, we extend the optimal fingerprinting method to the regime of nonlinear response. Our findings indicate that optimal fingerprinting for detection and attribution can be applied to virtually any stochastic system undergoing time-dependent forcing.

The emergence of organised multiscale patterns resulting from convection is ubiquitous, observed throughout different cloud types around the world. The nonlinear dynamics understanding of such cloud patterns by cloud-resolving models such as large eddy simulation models remains a grand challenge. In this work, we present an alternative approach based on conceptual stochastic delay differential models. We show that with the suitable stochastic parameterization accounting for the missing physics, the delay model's response to stochastic perturbations can indeed reproduces with fidelity the rich variability of cloud oscillations such as extracted from Lagrangian analysis of high-resolution satellite images.

Our approach employs Lagrangian attractors obtained by tracking oscillatory features from satellite images  that we confront to ensemble and pullback attractors from stochastic delay models experiencing a stochastic Hopf bifurcation.  Our analysis reveals that while the closed-cell dynamics corresponds to that of  a random steady state, the open-cell dynamics is much richer, as associated to that of a random limit cycle, and dominated by different types of phase-locked oscillations that unfold in the course of the day.

Chekroun, Mickaël D., H. Liu, K. Srinivasan, and James C. McWilliams. Submitted. “The High-Frequency and Rare Events Barriers to Neural Closures of Atmospheric Dynamics”. arXiv's link Abstract
Neural parameterizations and closures of climate and turbulent models have raised a lot of interest in recent years. In this short paper, we point out two fundamental problems in this endeavour, one tied to sampling issues due to rare events, and the other one tied to the high-frequency content of slow-fast solutions which constitute an intrinsic barrier to neural closure of such multiscale systems. We argue that the atmospheric 1980 Lorenz model, a truncated model of the Primitive Equations -- the fuel engine of climate models -- serves as a remarkable metaphor to illustrate these fundamental issues.
Chekroun, Mickaël D., Tom Dror, Orit Altaratz, and Ilan Koren. Submitted. “Equations discovery of organized cloud fields: Stochastic generator and dynamical insights”. arXiv's link Abstract

The emergence of organized multiscale patterns resulting from convection is ubiquitous, observed throughout different cloud types. The reproduction of such patterns by general circulation models remains a challenge due to the complex nature of clouds, characterized by processes interacting over a wide range of spatio-temporal scales. The new advances in data-driven modeling techniques have raised a lot of promises to discover dynamical equations from partial observations of complex systems.
This study presents such a discovery from high-resolution satellite datasets of continental cloud fields. The model is made of stochastic differential equations able to simulate with high fidelity the spatio-temporal coherence and variability of the cloud patterns such as the characteristic lifetime of individual clouds or global organizational features governed by convective inertia gravity waves. This feat is achieved through the model's lagged effects associated with convection recirculation times, and hidden variables parameterizing the unobserved processes and variables.

Srinivasan, Kaushik, Mickaël D. Chekroun, and James C. McWilliams. Submitted. “Turbulence closure with small, local neural networks: Forced two-dimensional and β-plane flows”. arXiv's link Abstract

We parameterize sub-grid scale (SGS) fluxes in sinusoidally forced two-dimensional turbulence on the beta-plane at high Reynolds numbers (Re~25000) 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 16X downscaling factors. Our methodology significantly improves training efficiency and speed of online Large Eddy Simulations (LES) 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. We minimize a loss between the SGS vorticity flux divergence (computed from the high-resolution solver) and that obtained from the CNN-modeled deviatoric stress tensor, 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 and generalizable because of their intrinsic low dimensionality.

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