  The response of a low-frequency mode of climate variability, El
Ni{\~n}o-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.
