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  1. 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. 
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    Free, publicly-accessible full text available April 1, 2025
  2. Abstract

    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.

     
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  3. A general, variational approach to derive low-order reduced models from possibly non-autonomous systems is presented. The approach is based on the concept of optimal parameterizing manifold (OPM) that substitutes more classical notions of invariant or slow manifolds when the breakdown of “slaving” occurs, i.e., when the unresolved variables cannot be expressed as an exact functional of the resolved ones anymore. The OPM provides, within a given class of parameterizations of the unresolved variables, the manifold that averages out optimally these variables as conditioned on the resolved ones. The class of parameterizations retained here is that of continuous deformations of parameterizations rigorously valid near the onset of instability. These deformations are produced through the integration of auxiliary backward–forward systems built from the model’s equations and lead to analytic formulas for parameterizations. In this modus operandi, the backward integration time is the key parameter to select per scale/variable to parameterize in order to derive the relevant parameterizations which are doomed to be no longer exact away from instability onset due to the breakdown of slaving typically encountered, e.g., for chaotic regimes. The selection criterion is then made through data-informed minimization of a least-square parameterization defect. It is thus shown through optimization of the backward integration time per scale/variable to parameterize, that skilled OPM reduced systems can be derived for predicting with accuracy higher-order critical transitions or catastrophic tipping phenomena, while training our parameterization formulas for regimes prior to these transitions takes place.

     
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  4. A stochastic framework is exhibited to produce systematically a broadband response from periodic solutions of time-delay systems. 
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  5. null (Ed.)