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

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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 (Zebiak and Cane 1987), 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 (Chekroun et al. 2019) 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 Gaspard (2002) and made explicit in the second part of this contribution (Tantet et al. 2019). 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 Chekroun et al. (2019), 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.

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    If peaks in the power spectra are continuous, the decay of correlations may be slower at first but still be exponentially fast for infinite times (as formalized by the Paley–Wiener theorem). In general, however, peaks may be discontinuous and prevent the exponential decay of correlations. Such behavior is not visible in this study.

  2. 2.

    The matrix C(t) is in fact the covariance matrix of a periodic Ornstein-Uhlenbeck process with a drift given by the Jacobian matrix A(t) generating the fundamental matrix M(t) and with the diffusion matrix D(t), both evaluated along the limit cycle.


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This work has been partially supported by the Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI) Grant N00014-12-1-0911 and N00014-16-1-2073 (MDC), by the National Science Foundation Grant DMS-1616981(MDC) and AGS-1540518 (JDN), by the LINC Project (No. 289447) funded by EC’s Marie-Curie ITN (FP7-PEOPLE-2011-ITN) Program (AT and HD) and by the Utrecht University Center for Water, Climate and Ecosystems (AT).

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Correspondence to Alexis Tantet.

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Communicated by Valerio Lucarini.

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Tantet, A., Chekroun, M.D., Neelin, J.D. et al. Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part III: Application to the Cane–Zebiak Model of the El Niño–Southern Oscillation. J Stat Phys (2019).

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  • Ruelle–Pollicott resonances
  • Stochastic bifurcation
  • Markov matrix
  • ENSO