Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part III: Application to the Cane–Zebiak Model of the El Niño–Southern Oscillation

Abstract

The response of a low-frequency mode of climate variability, El Niño-Southern Oscillation, to stochastic forcing is studied in a high-dimensional model of intermediate complexity, the fully-coupled Cane-Zebiak model [ZC87], from the spectral analysis of Markov operators governing the decay of correlations and resonances in the power spectrum.Noise-induced oscillations excited before a supercritical Hopf bifurcation are examined by means of complex resonances, the reduced Ruelle-Pollicott (RP) resonances, via a numerical application of the reduction approach of the first part of this contribution [CTND19] to model simulations.The oscillations manifest themselves as peaks in the power spectrum which are associated with RP resonances organized along parabolas, as the bifurcation is neared. These resonances and the associated eigenvectors are furthermore well described by the small-noise expansion formulas obtained by [Gas02] and made explicit in the second part of this contribution [TCND19]. Beyond the bifurcation, the spectral gap between the imaginary axis and the real part of the leading resonances quantifies the diffusion of phase of the noise-induced oscillations and can be computed from the linearization of the model and from the diffusion matrix of the noise. In this model, the phase diffusion coefficient thus gives a measure of the predictability of oscillatory events representing ENSO. ENSO events being known to be locked to the seasonal cycle, these results should be extended to the non-autonomous case.More generally, the reduction approach theorized in [CTND19], complemented by our understanding of the spectral properties of reference systems such as the stochastic Hopf bifurcation, provides a promising methodology for the analysis of low-frequency variability in high-dimensional stochastic systems. KeywordsRuelle-Pollicott resonances · Stochastic Bifurcation · Markov matrix · ENSO IntroductionComplex and unpredictable behavior of trajectories is observed for many physical systems. This can be due to interactions with many degrees of freedom which can be modeled by a stochastic forcing or to nonlinear coupling resulting in chaotic trajectories. As a result, prediction beyond a certain horizon is hopeless and one focuses instead on the statistical evolution of the system. This loss of predictability manifests itself by the evolution of an ensemble of trajectories becoming independent on its initial condition after a given time, the mixing time. This notion of mixing in state space is in turn closely linked to the correlation function of a pair of observables, which assigns to any positive time lag the correlation between the first observable and the lagged version of the second, and thus gives a measure of the statistical dependence A. Tantet

Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory

“…At an intuitive level if a dominant recurrent behavior occurs within an irregular background, then T τ must still "feel" this recurrent behavior within V , in case this dominant behavior is reflected in V . As pointed out already in [CNK + 14] such a recurrent behavior is manifested by eigenvalues of T τ distributed evenly along an inner circle typically close to the unit circle, or by forming a parabola in the complex plane depending on the representation adopted; see Part II [TCND19a] and Part III [TCND19b] . An issue though, is that in general T τ does not come from a Markov semigroup (acting on functions of the reduced state space V ), and thus one cannot per se rely on the theory of RP resonances presented in Sec.…”

“…The RP resonances of this system, a Hopf normal form subject to an additive white noise, are analyzed in details in Part II [TCND19a]. This system is a paragon of nonlinear oscillations in presence of noise.…”

“…The usefulness of reduced RP resonance analysis is not limited to providing useful diagnosis for model reduction. The companions articles, Part II [TCND19a] and Part III [TCND19b], deal with further practical aspects of the theory presented in this article. One important byproduct consists of the diagnosis usefulness of stochastic dynamics that RP resonances themselves, offer.…”

A Minimal System Cost Minimization Model for Variable Renewable Energy Integration: Application to France and Comparison to Mean-Variance Analysis

“…This noise must thus represent factors affecting the energy systems that are sufficiently fast to be considered uncorrelated. On the other hand, the climate system, which is one of the main drivers of the load and the VRE generation, varies on a continuous range of time scales [42][43][44] and is perturbed by non-stationary forcing [45] such as the anthropogenic forcing. Therefore, the climate forcing cannot be treated as noise and cannot be considered stationary with respect to the probability distribution of the noise.…”

Introduction to the Special Issue on the Statistical Mechanics of Climate

“…The overall theoretical framework is presented in [21], while in [117] it is shown how to investigate Hopf bifurcations for a stochastic system and extract from data a clear fingerprint of nonlinear oscillations taking place within a stochastic background. Furthermore, the theoretical findings are exploited for studying the oscillations in the Cane-Zebiak stochastic ENSO model [116]. • The concept of natural variability requires using a different framework when considering nonautonomous systems.…”