# Biennial Connection to Seasonal Aliasing

I found an interesting mathematical simplification relating seasonal aliasing of short-period cycles with a biennial signal.

If we start with a signal of an arbitrary frequency $\omega_L = 2\pi/T_L$

$L(t) = k \cdot sin(\omega_L t + \phi)$

and then modulate it with a delta function array of one spike per year

$s(t) = \sum\limits_{i=1}^n a_i sin(2 \pi t i +\theta_i)$

This is enough to create a new aliased cycle that is simply the original frequency $\omega_L$ summed with an infinite series of that frequency shifted by multiples of $2\pi$ .

$f(t) = k/2 \sum\limits_{i=1}^n a_i sin((\omega_L - 2 \pi i)t +\psi_i) + ...$

This was derived in a previous post. So as a concrete example, the following figure is a summed series of f(t) -- specifically what we would theoretically see for an anomalistic lunar month cycle of 27.5545 days aliased against a yearly delta.

Fig 1: Series expansion of aliasing

If you count the number of cycles in the span of 100 years, it comes out to a little less than 26 cycles, or an approximately 3.9 year aliased period. If a low-pass filter is applied to this time series, which is likely what would happen in the lagged real world, a sinusoid of period 3.9 years would emerge.

The interesting simplification is that the series above can also be expressed exactly as a biennial + odd-harmonic expansion

$[cos(\pi t) + cos(3 \pi t) + cos(5 \pi t) + ...] \cdot sin(\frac{2\pi}{T} t)$

where T is given by

$1/T = 1/2 + int(1/T_L) - 1/T_L$

where the function "int" truncates to the integer part of the period reciprocal. Since $T_L$ is shorter than the yearly period of 1, then the reciprocal is guaranteed to be greater than one.

As a check, if we take only the first biennial term of this representation (ignoring the higher frequency harmonics) we essentially recover the same time series.

Fig 2: Biennial modulation term

What is interesting about this factoring is that a biennial modulation may naturally emerge as a result of yearly aliasing, which is possibly related to what we are seeing with the ENSO model in its biennial mode. Independent of how many lunar gravitational terms are involved, the biennial modulation would remain as an invariant multiplicative factor.

For the ENSO signal, an anomalistic term corresponding to a biennial modulation operating on a 4.085 year sinusoid appears significant in the model

1/4.085 = 1/2 + int(365.242/27.5545) - 365.242/27.5545

which is expanded as this pair of biennially split factors (see ingredient #5 in the ENSO model)

1/4.085 ~ 2/3.91 - 2/1.34

# Deterministically Locked on the ENSO Model

After several detours and dead-ends, it looks as if I have locked on a plausible ENSO model, parsimonious with recent research.  The sticky widget almost from day 1 was the odd behavior in the ENSO time-series that occurred starting around 1980 and lasting for 16 years. This has turned out to be a good news/bad news/good news opportunity.  Good in the fact that applying a phase inversion during that interval allowed a continuous fit across the entire span.  Yet it's bad in that it gives the impression of applying a patch, without a full justification for that patch. But then again good in that it explains why other researchers never found the deterministic behavior underlying ENSO -- applying conventional tools such as the Fourier transform aren't much help in isolating the phase shift (accepting Astudillo's approach).

Having success with the QBO model, I wasn't completely satisfied with the ENSO model as it stood at the beginning of this year. It didn't quite click in place like the QBO model did. I had been attributing my difficulties to the greater amount of noise in the ENSO data, but I realize now that's just a convenient excuse -- the signal is still in there.  By battling through the inversion interval issue, the  model has improved significantly.  And once the correct forcing and Mathieu modulation is applied, the model locks in place to the data with the potential to work as well as a deterministic tidal prediction algorithm.

# Crucial recent citations for ENSO

I have found the following research articles vital to formulating a basic model for ENSO.

The first citation finds the disturbance after 1980 leading to the identification of a phase reversal in the ENSO behavior. They apply Takens embedding theorem (which works for linear and non-linear systems such as Mathieu and Hill) to the time series, reconstructing current and future behavior from past behavior.

H. Astudillo, R. Abarca-del-Rio, and F. Borotto, “Long-term non-linear predictability of ENSO events over the 20th century,” arXiv preprint arXiv:1506.04066, 2015.

# Tidal Locking and ENSO

As I have been formulating a model for ENSO, I always try to relate it to a purely physical basis. The premise I have had from the beginning is that some external factor is driving the forcing of the equatorial Pacific thermocline. This forcing stimulus essentially causes a sloshing in the ocean volume due to small changes in the angular momentum of the rotating earth. I keep thinking that the origin is lunar as the success of the QBO model in relating lunisolar forcing to the oscillatory behavior of the QBO winds is enough motivation to keep on a lunar path.

Yet, I am finding that the detailed mechanism for ENSO differs from that of QBO. An interesting correlation I found is in the tidal-locking of the Earth to the moon. I think this is a subset of the more general case of spin-orbit resonance, where the rotation rate of a satellite is an integral ratio of the main body. In the case of the moon and the earth, it explains why the same moon face is always directed at the earth -- as they spin at the same rate during their mutual orbit, thus compensating via a kind of counter-rotation as shown in the left figure below.

Fig 1: Tidal Locking (left) results in the spin of the moon

# Biennial Mode of SST and ENSO

This recent 2016 paper [1] by Kim is supporting consensus to my model of a modulated biennial forcing to ENSO. I had read some of Kim's earlier papers [2] where he introduced the idea of cyclostationary behavior.

The insight that they and I share is that the strictly biennial oscillation is modulated by longer frequencies such that +/- sideband frequencies are created around the 2-year period.

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

This is not aliasing but essentially a non-aliased frequency modulation of the base cycle. The insight is clarified by Kim with respect to Meehl's [3] tropospheric biennial oscillation (TBO).

From [1]

That biennial mode locks the base frequency in place to a seasonal cycle, with the modulation creating what looks like a more chaotic pattern. That's why ENSO has been so stubborn to analysis, in that the number of Fourier frequencies doubles with each modulating term. Yet in reality it's likely half as complicated as most scientists have been lead to believe.

# Validating ENSO cyclostationary deterministic behavior

I tend to write a more thorough analysis of research results, but this one is too interesting not to archive in real-time.

First, recall that the behavior of ENSO is a cyclostationary yet metastable standing-wave process, that is forced primarily by angular momentum changes. That describes essentially the physics of liquid sloshing. Setting input forcings to the periods corresponding to the known angular momentum changes from the Chandler wobble and the long-period lunisolar cycles, it appears trivial to capture the seeming quasi-periodic nature of ENSO effectively.

The key to this is identifying the strictly biennial yet metastable modulation that underlies the forcing. The biennial factor arises from the period doubling of the seasonal cycle, and since the biennial alignment (even versus odd years)  is arbitrary, the process is by nature metastable (not ergodic in the strictest sense).  By identifying where a biennial phase reversal occurs, the truly cyclostationary arguments can be isolated.

The results below demonstrate multiple regression training on 30 year intervals, applying only known factors of the Chandler and lunisolar forcing (no filtering applied to the ENSO data, an average of NINO3.4 and SOI indices). The 30-year interval slides across the 1880-2013 time series in 10-year steps, while the out-of-band  fit maintains a significant amount of coherence with the data:

# QBO Model Validation

The strongly periodic Quasi-Biennial Oscillation (QBO) of stratospheric winds is externally forced by a cyclic mechanism. This is obvious in spite of what the current scientific literature says about the nature of QBO;  the general consensus being that the QBO emerges as a complex natural resonant response of other atmospheric factors [1]. Yet, an experienced experimental scientist should not consider that as the only plausible premise. It is in fact exceedingly rare for a phenomena of that global a scale to be the result of a natural resonance, and for the QBO to be a resonance is clearly an untenable hypothesis based on historical precedent. There simply aren't any climate phenomena that behave similarly and are not reliant on lunar or solar periods.

What causes the oscillation then?

# Common Origins of Climate Behaviors

I have been on a path to understand ENSO via its relationship to QBO and the Chandler wobble (along with possible TSI contributions, which is fading) for awhile now. Factors such as QBO and CW have all been considered as possible forcing mechanisms, or at least as correlations, to ENSO in the research literature.

Over time, I got sidetracked into trying to figure out the causes of QBO and the Chandler wobble hoping that it might shed light into how they could be driving ENSO.

But now that we see how the QBO and the Chandler wobble both derive from the seasonally aliased lunar Draconic cycle, it may not take as long to piece the individual bits of evidence together.

I am optimistic based on how simple these precursor models are. As far as both QBO and the Chandler wobble are concerned, one can't ask for a simpler explanation than applying the moon's Draconic orbital cycle as a common forcing mechanism.

As a possible avenue to pursue, the post on Biennial Connection from QBO to ENSO seems to be the most promising direction, as it allows for a plausible phase reversal mechanism in the ENSO standing wave. I'll keep on kicking the rocks to see if anything else pops out.

# 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

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.