In an earlier post, the observation was that ENSO models may not be unique due to the numerous possibilities provided by nonlinear math. This was supported by the fact that a tidal forcing model based on the Mf (13.66 day) tidal factor worked equally as well as a Mm (27.55 day) factor. This was not surprising considering that the aliasing against an annual impulse gave a similar repeat cycle — 3.8 years versus 3.9 years. But I have also observed that mixing the two in a linear fashion did not improve the fit much at all, as the difference created a long interference cycle which isn’t observed in the ENSO time series data. But then thinking in terms of the nonlinear modulation required, it may be that the two factors can be combined after the LTE solution is applied.

The non-linearity will act to fold (or break) the signal, a la Mach-Zehnder modulation, to destroy the long period and then the superposition will not necessarily lead to a predictable constructive interference. This also may make physical sense in that the Mf factor is a horizontal transverse (latitudinal) forcing applied to the equatorial toroidal (equatoroidal?) waveguide while the Mm factor is vertical. In other words, lunar perigee forcing (Mm) is orthogonal to lunar declination (Mf) forcing for the equatoroid and so the LTE modulation may in fact be independent in the two dimensions and thus the two resultants superposed as magnitudes — or at least that’s my latest ansatz.

Before breaking down the analysis, the chart below shows the initial excellent fit, and which converged surprisingly quickly.

This achieves a correlation coefficient of >0.9, which is quite good considering the low DOF afforded by the applied parameters. Individually, the models based on Mf and Mm alone are adequate but prone to overfitting

These two, each with a correlation coefficient of between 0.6 and 0.7, when combined give a much higher as shown in the first graph.

The presumptuous finality of this fit can be considered in the context of how constrained the tidal factors turned out to be. In 2018 at the AGU, we presented the model by constraining the tidal factors against known lunar ephemeris and also against the factors contributing to the Earth’s delta length-of-day (LOD). This is also described in the book and in a previous blog post.

Note that this agreement was not initially constrained but that the iterative solver fit arrived at the solution on its own accord, driven by the best fit to ENSO. There is a slight shift in alignment of perhaps a 1/2 a year, but this may be due to the solver not being able to move quickly to a better solution.

What does the forcing agreement look like on a finer temporal resolution scale? See below.

Ansatz that !