For ocean tidal predictions, once an agreement is reached on the essential lunisolar terms, then the second-order terms are refined. Early in the last century Doodson catalogued most of these terms:

"Since the mid-twentieth century further analysis has generated many more terms than Doodson's 388. About 62 constituents are of sufficient size to be considered for possible use in marine tide prediction, but sometimes many fewer can predict tides to useful accuracy."

That's possibly the stage we have reached in the ENSO model. There are two primary terms for lunar forcing (the Draconic and Anomalistic) cycles, that when mixed with the annual and biannual cycles, will reproduce the essential ENSO behavior. The second-order effects are the modulation of these two lunar cycles with the Tropical/Synodic cycle. This is most apparent in the modification of the Anomalistic cycle. Although not as important as in the calculation of the Total Solar Eclipse times, the perturbation is critical to validating the ENSO model and to eventually using it to make predictions.

The variation in the Anomalistic period is described at the NASA Goddard eclipse page. They provide two views of the variation, a time-domain view and a histogram view.

Since NASA Goddard doesn't provide an analytical form for this variation, we can see if the ENSO Model solver can effectively match it via a best-fit search to the ENSO data. This is truly an indirect method.

First we start with a parametric approximation to the variation, described by a pair of successive frequency modulated (and full-wave rectified) terms that incorporate the Tropical-modified term, **wm**. The Anomalistic term is **wa**.

COS(wa*t+pa+c_1*ABS(SIN(wm*t+k_1*ABS(SIN(wm*t+k_2))+c_2)))

This can generate the cusped behavior observed, but the terms **pa, c_1, c_2, k_1**, and **k_2** need to be adjusted to align to the NASA model. The solver will try to do this indirectly by fitting to the 1880-1950 ENSO interval.

Plotting in **RED** the Anomalistic time series and the histogram of frequencies embedded in the ENSO waveform, we get:

This captures the histogram view quite well, and the time-domain view roughly (in other cases it gives a better cusped fit). The histogram view is arguably more important as it describes the frequency variation over a much wider interval than the 3-year interval shown.

What would be even more effective is to find the correct analytical representation of the Anomalistic frequency variation and then plug that directly into the ENSO model. That would provide another constraint to the solver, as it wouldn't need to spend time optimizing for a known effect.

Yet as a validation step, the fact that the solver detects the shape required to match the variation is remarkable in itself. The solver is obviously searching for the forcing needed to produce the ENSO waveform observed, and happens to use the precise parameters that also describe the second-order Anomalistic behavior. That could happen by accident but in that case there have been too many happy accidents already, i.e. period match, LOD match, Eclipse match, QBO match, etc.

I got this opinion from a climate scientist that I had previously cited and then engaged with on Twitter.

He thinks that if the lunar-forcing models for ENSO and QBO are true, then the currently accepted models for wave forcing theory are incorrect. That's a bit much because all I'm showing is the missing forcing. Everything that follows from physics should still hold.

I like to use the electrical circuit analogy. If there was some weird signal on an output that no one could decipher, and then it was finally revealed that it was due to a 60Hz leakage, then we wouldn't throw out Kirchoff's laws. No, we would say that the input to the circuit was being forced by a line voltage signal and it became amplified in the output.

In the case of QBO, we can simply say that the denier Richard Lindzen did not look hard enough for the forcing, couldn't find it, and so made up an elaborate theory to explain an emergent oscillation. Perhaps some variation of Lindzen's theory still holds to some degree, it just needs to be stimulated by the lunar forcing.

Same thing with the models of ENSO by the denier A.A.Tsonis, who thinks that variations of ENSO are likely the result of chaotic emergence. Well, one theory of chaos describes the Butterfly Effect leads to an apparent chaos, as changes in the initial conditions leads to different outcomes. But what happens if a forcing supercedes the initial conditions and leads to what is called the forced response? Of course, we don't throw away chaos theory as that can be used for something else. What we do is simply ignore Tsonis and use a non-chaotic forcing model. Then instead of the Butterfly effect, we look at the Hawkmoth effect instead.

I really don't think much will change in the foundational models, other than they will be firmed up and the parametric characteristics will be further refined, as this post demonstrates.