Note that the maximum excursions (perigee and declination excursion) align with the occurrence of total solar eclipses. These are the first three that I looked at, which includes the latest August 21 eclipse in the center chart.
There are about 90 more of these stretching back to 1880. The best way to fit the calibration is to take the negative excursions of the two lunar forcings and multiply these together, i.e. use the effective Draconic*Anomalistic amplitudes (also only take the fortnightly cycle of the Draconic, as eclipses occur during both the ascending and descending node crossings). The main fitting factors are the phases of the two lunar months. To get the maximum alignment from the search solver, we maximize the sum of the effective amplitudes across the entire interval. This results in a phase difference between the two of about 0.74 radians based at the starting year of 1880 (i.e. year 0).
Here are all the listed total solar eclipses, maximized to Draconic*Anomalistic values:
There was one odd point here at August 21, 1914 and it turns out that was a bad key entry in mistakenly entering an ordinal month for Aug as 9 instead of 8.
That of course would create an anti-phase cancellation of the pair of monthly cycles at that point. With that corrected, then it fit as follows
The ENSO model can use this calibration to help align the phases of the Anomalistic and Draconic cycles, and then these are locked in during the ENSO fitting process, whereby only the forcing amplitudes are allowed to vary. The slight discrepancy is that while the eclipse calibration has a phase shift of +0.74 radians between the two cycles, the typical fit to an ENSO model has a phase shift anywhere from +0.9 to +1.1 radians. The difference between the two (worst case 0.36 radians) is a day and a half of extra lag in a cycle.
The other key is calibrating the Anomalistic period variation. This is what the total solar eclipse calibration generates, after supplying a variable two-factor harmonic composition necessary to match the shape :
The fit appears to amplify the cusped nature of the variation. This may be due to the sensitivity of the tropical/synodic cycle to the precise timing of a total solar eclipse. Although the NASA Goddard chart from where this is adapted does not provide an unambiguous Y-scale.
But for the ENSO fit, the amplitude is reduced to what appears to scale to the NASA model:
The full ENSO fit is here, using the training interval 1950 to 2016, and a fixed phase shift between Draconic and Anomalistic cycles of 0.9 radians.
These two terms along with the critical mixed lunar tidal terms are all that are required to model ENSO. I have found that everything you need to know about the lunar cycle is located on the following NASA Goddard pages (they recently remapped their web sites because of the recent interest, thus there is a new “eclipses” site and an older “eclipse” site):
- https://eclipses.gsfc.nasa.gov/eclipse.html (all eclipses)
- https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html (orbit info)
Below is a NASA Goddard histogram of the Anomalistic period variation over time, with the results from the ENSO model overlaid. Not sure why the model is offset shifted relative to the NASA calculation on the positive side, but the model mean does align with the Anomalistic period. Likely the issue is that the ENSO model is using a balanced (+/-) frequency variation about the mean while the NASA variation is unbalanced around the mean. Could use the actual analytic calculation for the Anomalistic variation rather than the empirical parametric form guessed for the ENSO model.
With this kind of calibration available, along with the calibration info from LOD measurements, it becomes increasingly difficult to overfit the ENSO model. In this case, it’s due to the phase of the forcing being known more precisely and that the phase remains stationary to the accuracy of the known periods. Yet, it’s still tricky as the lunar orbit is deceptively complex in it’s details — and that’s what calculations of behaviors behind the total solar eclipse and ocean tides (and now apparently ENSO) are sensitive to.