This work sits on the fringe of the research map: its nearest neighbors touch on superconductivity, quantum paradoxes, and nanoresonators, and only one related work deals with Cooper pairs and superfluidity. Quantum measurements and numerical methods for the Dirac equation have been studied around it, but this study brings spectral analysis of Markov operators into a climate model. It is the first application of Ruelle-Pollicott reduction to a high-dimensional stochastic ocean-atmosphere system.
The Cane-Zebiak model with 930 variables was perturbed by a Wiener process in the wind stress field with a noise level of 0.01. Observations were projected onto a plane of eastern sea surface temperature and western thermocline depth, divided into 100×100 bins. From 6000-year simulations with a 5-day time step, transition matrices with a 4-year lag were estimated, and eigenvalues were converted into resonances via the logarithm of the ratio of matrices at lags of 4 years and 4 years plus half a month.
Before the bifurcation (coupling parameter ξ=2.80 and 2.85), resonances form a triangular lattice generated by the eigenvalues of the Jacobian at the stationary point. After the bifurcation (ξ=2.90 and 2.95), they align into parabolas, where the imaginary part is a multiple of the fundamental frequency and the real part grows quadratically with the harmonic number. The spectral gap between the leading resonances and the imaginary axis narrows as the bifurcation is approached, sharpening peaks in the power spectrum and slowing the decay of correlations. The eigenvectors take the form of a harmonic times Laguerre or Hermite polynomials, and their phase isolines are tilted due to a nonzero twist coefficient β.
Unlike neighboring works that analyze low-dimensional or quantum systems, here the reduction is applied to an intermediate-complexity climate model with thousands of degrees of freedom. This is the first test that parabolic resonance structures predicted for the Hopf normal form emerge in a realistic El Niño model. The work shows that phase diffusion, not just the stability of the limit cycle, governs the loss of predictability.
The method can be extended to non-autonomous systems to account for the seasonal cycle, which phases El Niño events toward December-January. It is also promising to compute phase diffusion coefficients directly from Floquet analysis of the Cane-Zebiak limit cycle, which would require numerical continuation of the cycle and the fundamental matrix.
The results will influence diagnostics of El Niño predictability in climate models and the theory of stochastic bifurcations in high-dimensional systems.
The next step is to extend the reduction to models with seasonal forcing and compare phase diffusion coefficients with observed El Niño indices.
For climate dynamics, an open question remains: to what extent phase diffusion caused by fast atmospheric processes limits the practical predictability of El Niño beyond a few months. This work links that problem to a measurable spectral gap, but does not solve it for the full non-autonomous system with a seasonal cycle.
Noise forcing with a level of just 0.01 (dimensionless wind stress amplitude) can excite oscillations with a period of 3-4 years in a model that, without noise, simply decays to a stationary state.