# Biennial Connection from QBO to ENSO

I see these as loose-end issues in trying to tie the forcing of the Quasi-Biennial Oscillation (QBO) to the forcing behind the El Nino Southern Oscillation (ENSO).

1. The nature of the biennial oscillations in ENSO [1] — and specifically, what drives the differences in forcing between QBO and ENSO .
2. Why do the tides in the Southern Pacific have a more strictly biennial (i.e. =2 year) periodicity than the quasi-biennial (i.e. ~2.33 year) oscillations in atmospheric wind?
3. The tie-in to the Chandler wobble on the triaxial earth [2], which appears more significant for ENSO than for QBO.
4. Phase reversals in the ENSO standing wave, particularly in 1981.

While collectively trying to resolve these issues, I discovered an intriguing pattern in the wave-equation transformation of the ENSO signal.  This new pattern is based on defining precise sidebands +/- on each side of the exact biennial period. A pair of sinusoidal sidebands are formed when a primary frequency is modulated by another sinusoid.

$sin(pi t) cdot sin(omega_m t) = frac{1}{2} ( cos(pi t - omega_m t) - cos(pi t + omega_m t) )$

The sidebands appear to match the period of three identified wobbles in the angular momentum of the rotating triaxial earth [2]. These sidebands are sufficient to extrapolate most of the wave-equation transformed curve when fitting to either a large interval or to a short interval within the time series. The latter is simply a consequence of a shorter interval containing enough information to reconstruct the rest of the stationary time series.  See Figure 1 below for examples of the effectiveness of the fit across various cross-sectional intervals and how well the short interval sampling extrapolates over the rest of the time series. Even as short a training interval as 15 years results in a fairly effective extrapolation, since 15 years is comparable to the longest constituent modulation period.

Fig. 1: Examples of stationary model fits to the wave equation transformed  ENSO data. The top panel is a fit to the entire interval and the three below are extrapolated from training intervals of varying lengths. Click to enlarge.

This new pattern is essentially a refined extension of the sloshing formulation I started with — but now the symmetry and canonical form is becoming much more readily apparent. The identified side-bands have periods of 6.5, 14.3, and 18.6 years, which you can understand from reading the fractured English in reference [2]. These three periods are known modulations of the earth’s rotation (ala the Chandler wobble) and all fit in to the F(t) term of the biennial-modulated wave equation.

# Daily Double

[mathjax]A short piece that ties together the analysis of ENSO and QBO over the last year.

The premise has been that periodic changes in angular momentum applied to the earth’s rotation is enough of a forcing to steer the behavior of the El Nino Southern Oscillation (ENSO) in the equatorial Pacific ocean and of the Quasi-Biennial Oscillation (QBO) in upper atmospheric winds. Whoever would have you believe that these behaviors could be spontaneously generated is clearly not thinking straight. For every action there is a reaction, and both QBO and ENSO are likely reactions to the same forcing action.

Both this forum (and the Azimuth Project forum) has provided plenty of analysis to show exactly how that comes about, but in retrospect, it’s the machine learning (ML) experiments via Eureqa that has provided the most eye-opening evidence. Robots find what they find and since they are free from the vagaries of human nature, they can’t lie about what they discover.

The first two for QBO have a primary sinusoidal factor that are nearly identical, 2.66341033 and 2.663161 rads/year, and the ENSO has a value 2.64123448 rads/year. If the first two values are averaged and then that is averaged with the ENSO value, the result is 2.65226007 rads/year (the significant figures are as reported by Eureqa). That value is equivalent to a seasonally aliased 2.65226007 +13$$cdot$$2$$pi$$ rads/year, which is a period of 27.21195913 days — while the Draconic lunar month is 27.21222082 days. That’s an error of 0.00096%.

So the primary ENSO forcing period as determined by ML was a tiny bit shorter than the Draconic and the primary QBO forcing period was a wee bit longer than the Draconic period. Given that is partly due to noise in the fit, it’s reassuring to see that the average would get even closer to a plausible forcing value.

The entire premise of the lunar forcing driving both QBO and ENSO hinges on the precision of the modeled values; as the cycles of a lunisolar model can quickly get out of sync with the data unless enough precision is available to span 60 to 100 years.

Recall again these words by the professional contrarian scientist Richard Lindzen:

” 5. Lunar semidiurnal tide : One rationale for studying tides is that they are motion systems for which we know the periods perfectly, and the forcing almost as well (this is certainly the case for gravitational tides). Thus, it is relatively easy to isolate tidal phenomena in the data, to calculate tidal responses in the atmosphere, and to compare the two. Briefly, conditions for comparing theory and observation are relatively ideal. Moreover, if theory is incapable of explaining observations for such a simple system, we may plausibly be concerned with our ability to explain more complicated systems. Lunar tides are especially well suited to such studies since it is unlikely that lunar periods could be produced by anything other than the lunar tidal potential. The only drawback in observing lunar tidal phenomena in the atmosphere is their weak amplitude, but with sufficiently long records this problem can be overcome [viz. discussion in Chapman and Lindaen (1970)] at least in analyses of the surface pressure oscillation. ” — from Lindzen, Richard S., and Siu-Shung Hong. “Effects of mean winds and horizontal temperature gradients on solar and lunar semidiurnal tides in the atmosphere.” Journal of the Atmospheric Sciences 31.5 (1974): 1421-1446.

That bolded part is the monetary payoff. If Lindzen, who is known as the father of QBO theory, asserts that if measured periods aligning with lunar periods is a sufficient comparison, then he would be forced into agreeing with this current analysis. Nothing else will come close to the precision required.

And the payoff turns into the daily double as it also works for explaining ENSO. The combination of parsimony and plausibility is hard to argue with.

# Eureqa!

Whenever I do machine learning (ML) experiments, I save the results for posterity. If I can’t make sense of them at the time, I will revisit later.

The following is a set of results from a Eureqa symbolic regression ML experiment on ENSO data from May of last year. The surprise is that it shouldn’t be a surprise, especially based on the fascinating Quasi-Biennial Oscillation (QBO) results of the last few months.