If we don't have enough evidence that the forcing of ENSO is due to lunisolar cycles, this piece provides another independent validating analysis. What we will show is how well the forcing used in a model fit to an ENSO time series — that *when isolated* — agrees precisely with the forcing that generates the slight deviations in the earth's rotational speed, i.e. the earth's angular momentum. The latter as measured via precise measurements of the earth's length of day (LOD). The implication is that the gravitational forcing that causes slight variations in the earth's rotation speed will also cause the sloshing in the Pacific ocean's thermocline, leading to the cyclic ENSO behavior.

One can anticipate that this alignment likely exists via a rough LOD analysis, as I did earlier here. But with a more refined LOD analysis as described in this open-access review by Na [1], we can see the exact profile to compare against. The following excerpt lays out the comparison interval.

So the contention (*case A*) is that if we compare the forcing that we use in the ENSO model, isolated for a short training interval here, to the forcing due to LOD above, the two time-series should align precisely. If they don't (*case B*), the ENSO model may need to be reconsidered, if not rejected.

First, to get the maximum training we use the entire ENSO interval for fitting:

**Figure 2** uses the following lunar forcing stimuli.

Next, we take a straightforward additive combination of the three factors from **Figure 3** and superimpose them on the LOD factors of **Figure 1**. The dashed RED line remarkably aligns quite precisely with the LOD variations in BLUE, after the signal is inverted and a slight shift of 0.025 years (~9 days) is applied.

Because it is hard to tell the agreement at this scale due to the temporal compression, a portion of the fit is expanded and reproduced below.

The first takeaway is the alternation in the monthly lunar signal with the fortnightly (i.e. ~2 week) lunar signal. From the Na paper [1], they could not derive this from calculated tidal forces, yet it clearly exists. The other takeaway is the overall undulation. Taken altogether, this is enough agreement to solidly reject the claim (*case B*) that ENSO *can't* be forced by the lunisolar gravitational signal. What is left is a greatly strengthened *case A*, that ENSO is overwhelmingly likely driven by the lunisolar signal.

The only remaining *caveat* is that at least some of the LOD variation could be caused by the actual ENSO signal (e.g. the first figure here), as ocean sloshing and atmospheric changes will cause a change in the earth's angular momentum (Newton's 1st Law as a conservation of momentum argument), but this slower variation is easily isolated from the rapid lunar tidal signal. We should note that this mutual smearing of cause and effect for these compliantly interacting yet conserving systems is one of those issues we should always be cognizant of.

## Conclusion

The bottom-line key to this validation is how independent these two measurements are. The lunar tidal forcing as produced by a fit to the ENSO model had **no prior** **knowledge** of the quantitative LOD variation, yet the alignment between the two was within 10 days. I don't have a correlation coefficient between the two as the referenced article doesn't appear to provide raw data, but I wouldn't be surprised if it approaches 0.9. The metaphor here is that we are starting from opposite ends of a rope and finding that it indeed meets at the halfway point.

So this analysis becomes another piece of evidence demonstrating how well the ENSO model works as a metrology tool to reveal the underlying geophysics. And when you see a piece of scientific evidence fit so precisely into the jigsaw puzzle it sure gives one some encouragement that this is the right track. I am thinking that just trying to unwind all this agreement will take years.

## References

[1] S.-H. Na, “Earth Rotation–Basic Theory and Features,” Intech, 2013. http://dx.doi.org/10.5772/54584

**Added 6/4**

Digitized Figure 1 with the ENSO model forcing as comparison

**Added 6/5**

This is the flow chart for modeling ENSO, alongside the LOD variation. Note that the conjecture is that both ENSO and LOD are forced by long-period tidal cycles. What this analysis finds is that the two tidal forcings are nearly identical despite being arrived at by independent means -- via working backwards from the fitted data sets for ENSO and LOD. This validates the hypothesis that ENSO is indeed a lunar-forced behavior.

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This is how poorly the theory of tidal forcing on LOD works, which is Fig 1 digitized and compared against the LOD data (upper and lower charts superimposed)

Note that the fortnightly tide is completely missed, which the ENSO analysis gets completely right

The correlation coefficient of theory against LOD is close to 0 (since the fortnightly cancels whatever positive correlation exists), while that of ENSO forcing against LOD is about 0.7, which in fact may be higher subject to the vagaries of the digitization scheme used.

What this infers is that current geophysics models only gets the LOD forcing

qualitativelyright. This is no great shakes since that amounts to saying that the Draconic and Anomalistic cycles have a definite effect. Yet they are not able to discern the nonlinear effects of the fortnightly tides, nor the proportional effects. And so what is astounding is how well the ENSO analysis does in extracting the same forcing that the LOD data shows.This is essentially a gold mine for understanding the geophysics of the earth's dynamic interaction with the sun and moon's forcing.

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This is the physics -- imparting a 1 millisecond slowdown (or speedup) on the rotation of the earth with a surface velocity of almost 500 meters/second over the course of a couple of weeks (a fortnight) will result in an inertial lateral movement of ~ 1/2 a meter in the volume of the Pacific ocean due to Newton's first law.

This does not seem like a big deal until you realize that the thermocline can absorb this inertial impulse as a vertical sloshing, since the effective gravity is reduced by orders of magnitude due to the slight density differences above and below the thermocline. This is reflected as an Atwood number and shows up in Rayleigh-Taylor instability experiments, e.g. SEE THIS PAPER

With an Atwood number less than 0.001 which is ~0.1% density differences in a stratified fluid, the 0.5 meter displacement that occurs over two weeks now occurs effectively over half an hour. That's just an elementary scaling exercise.

So intuitively, one has to ask to the question of what would happen if the ocean was translated laterally by 1/2 a meter over the course of a 1/2 an hour? We know what happens with earthquakes in something as simple as a swimming pool

or as threatening as a tsunami. But this is much more subtle because we can't obviously see it, and why it has likely been overlooked as a driver of ENSO.

Good paper by Chao

Has various scales for LOD and this showing straightforward transfer function:

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