ENSO and Fourier analysis

Much of tidal analysis has been performed by Fourier analysis, whereby one can straightforwardly deduce the frequency components arising from the various lunar and solar orbital factors. In a perfectly linear world with only two ideal sinusoidal cycles, we would see the Fourier amplitude spectra of Figure 1.

Fig 1: Amplitude spectra for a signal with two sinusoidal Fourier components. To establish the phase, both a real value and imaginary value is plotted.

But in a non-linear world such as ENSO where the tidal forces interact with the seasonal cycle via modulated feedbacks the picture is quite different. What happens is that the cycles interact and get folded multiple times until what originally were three cycles (yearly plus draconic and anomalistic lunar periods) end up appearing as Figure 2.

Fig 2: The amplitude spectra of the ENSO time series shows an abundance of spectral peaks. The model of ENSO developed here can track the profile quite precisely, apart from possible excursions shown at the yellow highlighted regions. Those could be related to the 2.715 aliased Tropical lunar cycle and the 8.85 year anomalistic envelope modulated by the Tropical tide, but are definitely of second-order importance compared to the overall fit.

Probably over 90% of the scientists and engineers with any signal processing expertise will start off with the ENSO signal and do a Fourier analysis as a first step. Then they will try to make sense of the spectra and try to identify cyclic components from the peak locations. Eventually they will scratch their heads in not being able to make any headway due to the richness in the spectra, and possibly attribute it to a chaotic time series -- or perhaps describe it as a red noise subject to random wind burst stimuli. Unfortunately, that's a misguided or at least shortsighted view of how to do a comprehensive time-series analysis. It's well known that non-linear equations such as the Mathieu equation cause multiple splitting of the fundamental spectral peaks resulting in spectra with rich profiles and complex phase space trajectories. In particular see this recent Schroedinger equation analysis of harmonic generation [1].

Fortunately, I already knew of these issues from the solid-state world when I started working on the ENSO topic a few years ago -- yet it still didn't make it any easier to get to my current understanding.  As one can see from Figure 1, it's almost impossible to intuit that something as basic as three interacting sinusoidal factors can lead to such a rich spectra, but that's exactly what happens when solving the delayed/Mathieu differential equation. It just took this long to capture the dynamics, simply because there aren't a lot of crutches like a Fourier analysis to help out along the way. Everyone has the same problem, but it's partly luck and partly persistence of trying to find the right combination that creates the best fit.

References

[1] C. Zagoya, M. Bonner, H. Chomet, E. Slade, and C. F. de Morisson Faria, “Different time scales in plasmonically enhanced high-order-harmonic generation,” Physical Review A, vol. 93, no. 5, p. 053419, 2016.

 

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