When the model is able to discern values of fundamental physical constants to high precision, it has ceased to become of hypothetical interest and transformed into one of practical significance and of essentially able to determine ground truth.

The ENSO model is a Mathieu differential equation with a biennial modulation and an added delay differential of one year. The DiffEq is this :

This is solved with a straightforward differential expansion

The resultant *f(t)* is compared to the NINO34 time series by maximizing the correlation coefficient. All the DiffEq parameters are allowed to vary (constrained to the appropriate sign for feedback parameters) and also the two unknown forcing periods corresponding to what we hypothesize as the strongly seasonally-pulsed Draconic and Anomalistic cycles (the Tropical/Synodic/Sidereal cycle is not considered first-order as a global effect). With minimal filtering of the NINO34 signal over the range of 1880 to present day, the correlation coefficient reaches upwards of 0.65 with a clearly obvious peak matching.

The DiffEq iterative solver is only constrained such that the Draconic and Anomalistic periods are close to the known cycles of 27.21222082 and 27.55454988 days. So we constrain them to the following intervals and find out whether the solver homes in on the actual values.

This is respect to an average calendar year of 365.2422 days. (As an aside, this computation is so sensitive that knowledge of leap years has a real impact)

We also clamp the unknown phases to a known value of the node crossing (for Draconic) and of a known value of a perigee point (for Anomalistic). This does not influence the precise knowledge of the unknown cycle period but supplying this particular ansatz prevents the solver from wandering around too much and giving a strong hint as to where to phase align the two cycles.

Incredibly, the results after about an hour's worth of computation gives

Drac = 27.21178772 days

Anom = 27.55490106 days

This precision amount to predicting the Draconic lunar month period to within 37 seconds and the Anomalistic lunar month to within 30 seconds. This is with respect to a starting search window of 0.1 day or 2.4 hours, which is 8640 seconds, thus winnowing down the initial guess to a much finer resolution. So, if this was a random chance occurrence it would have a probability of (37/8640)*(30/8640) = 0.000015 to occur within that error margin. Remember that both periods match closely and are independent so the likelihoods are multiplied. That gives a 1 in over 60,000 chance of a random draw falling within that margin. Moreover, this value gives a strong peak in the correlation coefficient, falling off quickly away from these values. For a 30 second accumulated error per lunar month, this when propagated over ~1800 lunar months in a ~130 year interval will lead to a 15 hour error, or 15/24 out of 27 days, which is about a 0.15 radian propagated error. For comparison, 1.57 radians is needed to cause the phase to interfere destructively against the true value of the detected lunar month over this long interval. But cosine(0.15) is 0.99, which will give only a weak 1% of the worst case destructive interference.

As it stands the result is deterministically repeatable, as it will hone in on the same periods independent of a seed value. Evidently, the 1800 lunar months in the ENSO series is enough to provide a high resolution determination of its value -- this is the equivalent to taking sea-level height tidal readings over many months to extract the diurnal or semi-diurnal periods. The difference is that this ENSO analysis is detecting the monthly long-period tidal values and not the much more convenient daily tidal data.

The significance of the described model and the associated results should not be undersold. The ENSO time series record obviously tracks the primary lunar cycles to such an extent that the model becomes an extremely sensitive metrology tool for indirectly estimating the lunar periods. (The direct measurement is obviously done though visual or sensor-based lunar tracking techniques, the most rudimentary of which has been known for centuries)

And bottom-line, realistically anybody can do this computation -- as supplemental guide, a spreadsheet was supplied in the previous post . It's all data driven and there really is no hands-on manipulation except a nudge to point the computation in the right direction. Odds are in favor that everyone that tries will find the same result, much like a tidal analysis will generate an identical result for the primary constituent periodic factors.

Unfortunately, the perceptions of a tidal period contributing to a clearly regular and discernible pattern (see this) remains a powerful incentive to question the results. Yet, no one is under the pretense that a non-linear model will produce an obviously periodic pattern either. It's really a case of a cyclic attractor that is stationary and stable, and thus conducive to predictive modeling. Contrary to what the climate science deniers such as Judith Curry and Anastasios Tsonis assert, ENSO may not really be that complicated.

This is how the draconic and anomalistic values zone in on the true value

Fitting the DiffEq values progressively away from the true value for the lunar tidal cycle results in a smaller correlation coefficient.

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