The model fit to ENSO takes place in the time domain. However, the correlation coefficient between model and data of the corresponding power spectra is higher than in the time series. Below in Figure 1 the CC is 0.92, while the CC in the time series is 0.82.
Fig.1 : Power spectra of ENSO data against model
The model allows only 3 fundamental lunar frequencies along with the annual cycle, plus the harmonics caused by the non-linear orbital path and the seasonally impulsed modulation.
What this implies is that almost all the peaks in the power spectra shown above are caused by interactions of these 4 fundamental frequencies. Figure 2 shows a satellite view of peak splitting (also shown here).
Fig 2: Frequency sideband plot identifying components created by modulation of a biennial cycle with the lunar cycles (originally described here).
One of the reasons that the power spectrum gives a higher correlation coefficient — despite the fact that the spectrum wasn't used in the fit — is that the lunar tides are precisely determined and thus all the harmonics should align well in the frequency domain. And that's what is observed with the multiple-peak alignment.
Furthermore, according to Ref , this result is definitely not a characteristic of noise-driven system, and it also possesses a very low dimension of chaotic content. The same frequency content is observed largely independent of the prediction time profile, i.e. training interval.
1. Bhattacharya, Joydeep, and Partha P. Kanjilal. "Revisiting the role of correlation coefficient to distinguish chaos from noise." The European Physical Journal B-Condensed Matter and Complex Systems 13.2 (2000): 399-403.
Experiment to compare training runs from 1880 to 1980 of the ENSO model against both the NINO34 time-series data and the SOI data. The solid red-curves are the extrapolated cross-validation interval..
Many interesting inferences one can potentially draw from these comparisons. The SOI signal appears more noisy, but that could actually be signal. For example, the NINO34 extrapolation pulls out a split peak near 2013-2014, which does show up in the SOI data. And a discrepancy in the NINO34 data near 1934-1935 which predicts a minor peak, is essentially noise in the SOI data. The 1984-1986 flat valley region is much lower in NINO34 than in SOI, where it hovers around 0. The model splits the difference in that interval, doing a bit of both. And the 1991-1992 valley predicted in the model is not clear in the NINO34 data, but does show up in the SOI data.
Of course these are subjectively picked samples, yet there may be some better combination of SOI and NINO34 that one can conceive of to get a better handle on the true ENSO signal.
click to enlarge
In the last month, two of the great citizen scientists that I will be forever personally grateful for have passed away. If anyone has followed climate science discussions on blogs and social media, you probably have seen their contributions.
Keith Pickering was an expert on computer science, astrophysics, energy, and history from my neck of the woods in Minnesota. He helped me so much in working out orbital calculations when I was first looking at lunar correlations. He provided source code that he developed and it was a great help to get up to speed. He was always there to tweet any progress made. Thanks Keith
Kevin O'Neill was a metrologist and an analysis whiz from Wisconsin. In the weeks before he passed, he told me that he had extra free time to help out with ENSO analysis. He wanted to use his remaining time to help out with the solver computations. I could not believe the effort he put in to his spreadsheet, and it really motivated me to spending more time in validating the model. He was up all the time working on it because he was unable to lay down. Kevin was also there to promote the research on other blogs, right to the end. Thanks Kevin.
There really aren't too many people willing to spend time working analysis on a scientific forum, and these two exemplified what it takes to really contribute to the advancement of ideas. Like us, they were not climate science insiders and so will only get credit if we remember them.
Laplace developed his namesake tidal equations to mathematically explain the behavior of tides by applying straightforward Newtonian physics. In their expanded form, known as the primitive equations, Laplace's starting formulation is used as the basis of almost all detailed climate models. Since that's what they are designed to do, this post provides the details for solving Laplace's tidal equations in the context of the El Nino Southern Oscillation (ENSO) of the equatorial Pacific ocean. The derivation and results shown below essentially describe the framework of my presentation at this month's AGU meeting: Biennial-Aligned Lunisolar-Forcing of ENSO: Implications for Simplified Climate Models
The concise derivation for a model of ENSO depends on reducing Laplace's tidal equations along the equator. I could not find anyone taking a similar approach anywhere in the literature, even though it appears to be routinely obvious: (1) solve Laplace's tidal equations in a simplified context, then (2) apply the known tidal forcing and observe if the result correlates or matches the ENSO time series. In fact, it does, as I have shown before (and for QBO as well); but this is the first time that I have worked out the details in full for ENSO. Below is a two part solution.