[mathjax]After having some very good success at modeling the ENSO via the Southern Oscillation Index Model (SOIM) but discovering a few loose ends, this post provides some puzzle pieces that may ultimately determine the source and synchronization of the ENSO forcing.

The significant finding with regards to the SOIM was that the input forcing had a period very close to the fundamental frequency of the Quasi-Biennial Oscillation (QBO) of stratospheric winds. Since measurements began in 1953, the fundamental period of the QBO has varied about a period close to 28 months.  And this value is close to what the SOIM uses as a forcing input — fit with a few Fourier series terms culled from the long-term QBO time series.  But the open question was whether a mutual connection exists between the SOI and the QBO, or whether perhaps they share a common forcing input, external to both the ocean and stratosphere.

As part of the verification process, I routinely use the Eureqa exploratory software to evaluate solutions with respect to what an “ignorant” smart automaton would find. This tool is ignorant in the sense that it doesn’t understand the physics of the problem, only the possible mathematical formulations describing a waveform.  In this particular case, I applied Eureqa to the QBO time-series and let it crank away for awhile.   What it found was initially disturbing, see Figure 1

Because the Eureqa software seemed to violate the Nyquist frequency criteria for finding valid Fourier series components, I was concerned about the usefulness of the results.  Since the QBO data was only available as a monthly time series, the smallest period we could reliably isolate is two months, yet the Eureqa software finds components of less than 1 month in period. Essentially, what Eureqa is finding is a waveform that looks like the top view in Figure 2 but it clearly wants you to believe that it is fitting the lower waveform.  That’s the problem with discrete sampling aliasing — the higher frequencies can fold over and imitate the lower actual frequencies.

Before trying to understand why Eureqa is finding this set of potentially aliased frequency components, I have to remark that one of the periods reported is very close to the lunar synodic month corresponding to 29.53 days. In Figure 1, the selection highlighted finds a 77.72 rad/year frequency which is within an eyelash of the 77.715 rad/year frequency corresponding to the lunar synodic month value calculated by 2π/(29.53/365.25) . In the same set, it finds a 153 rad/year component corresponding to a 15 day fortnightly period, which is half the monthly synodic value 2π/(15/365.25).

The aliasing can be decomposed by applying the following formula to the aliased frequency value , recovering the un-aliased value.

$$ omega_a = 2 pi n 12 + omega_u $$

For an n value of 1, the aliased value of 77.72 rad/year converts to an unaliased value of 2.32 rad/year or a period of 2.7 years, which is somewhat above the mean QBO period of 2.33 years (28 months).  Another frequency is found corresponding to 72.73 rad/year which is actually  n = -1  unaliased beat frequency of 2.668 rad/year or a period of  2.35 years, bringing the average closer to the mean QBO period.

The implication of this finding is that these components may represent an actual lunar forcing on the QBO behavior.  The lower frequency corresponding to a synodic component and the slightly higher frequency the sidereal component (with anomalistic and draconian lunar months potentially in the mix as well).  What I think Eureqa is determining is the lowest complexity waveform, with the monthly period bleeding through the sampling rate barrier (especially if there is any dispersion)  and thus providing the characteristics of both the monthly component and of slower beat frequency that results from the longer term lunar modulation. In other words, it is picking up a monthly ripple on the bottom view of Figure 2, but relating the results as a badly aliased value.

By the way, Eureqa does also find the lower frequency QBO period along its Pareto frontier of possible solutions. In a sense the Eureqa machine algorithm is not confused about the content but is remaining impartial about whether the results show the higher frequency monthly aliased components or the lower frequency 2-3 year real periods in the spectrum. It is giving the user the choice of what to believe is in the content of the waveform. So, unless Eureqa is buggy about providing higher frequency harmonics of the underlying waveform where the fundamental will do, it is likely actually picking out real monthly lunar effects that show up in the QBO as precursor remnants of the longer period beat frequency. It this rolls these two together to generate a lower-complexity formulation — and that is what gives rise to the Pareto frontier of possible mathematical solutions. The user’s job is to pick out the most plausible solutions amongst the frontier of potential solutions.

In a couple of ways, the Eureqa aliasing is either (1) an annoying  bug or a (2) benefiting feature in that it may be revealing that the QBO is being stimulated by the lunar modulation — and that inductively we can perhaps assert that this same effect is stimulating the related ENSO behavior.

So, as a first iteration of the Quasi-Biennial Oscillation Model (QBOM) we propose  a simple synodic/sidereal lunar forcing stimulating or exciting the oscillations observed.  Unlike ENSO, the Mathieu modulation of the underlying wave equation for the much less dense stratosphere is non-existent or can be ignored.   At most a dispersive and drag effect needs to likely be considered which could lead to a moderate resonance if the characteristic frequency of the stratospheric medium matches that of the lunar forcing.

This link is an animation describing the difference between the synodic and sidereal lunar months. The difference in frequencies between the two generates the yearly calendar period of 365.25 days. The addition of the two generates a value corresponding to a fortnightly period. Part of the reason that a yearly signal emerges from these monthly periodicities is that multiplying interactions can “square” the signal, thus providing a result that often doesn’t average to zero and also providing a mechanism for frequency doubling. For example, the following pairing of synodic and sidereal frequencies will generate a yearly beat superimposed with a fortnightly ripple.

$$ sin(omega_{sy} t) sin(omega_{sr} t) $$

This would simply explain why the Eureqa could detect the fortnightly 15-day period in the time series.

The next obvious question is whether anything in the literature is addressing the possibility of the QBO (and ENSO, and the Chandler Wobble for that matter) being stimulated by the same lunar forces that give rise to tides.

What I have found so far is that there are a couple schools of thought surrounding QBO.   The AGW skeptic Lindzen was one of the first to provide a theory for QBO by applying a fluid dynamics model of a rotating earth to extract a characteristic frequency of the oscillating behavior [1].  More recently there has been some research directed towards simplifying the QBO model with regards to the atmosphere’s intrinsic wave properties [2]. More radical is the work of Sidorenkov [3] who asserts that the fundamental driving force is indeed the same synodic lunar tidal force that I am postulating. Sidorenkov places the synodic forcing as a yearly period of 355 days (0.97 years) corresponding to 13 synodic months of 29.55 days apiece.

Sidorenkov’s basic mechanism determines the QBO period:

$$ frac{1}{2} ( frac{1}{0.97} – frac{1}{8.85} – frac{1}{18.61} ) = frac{1}{2.3} $$

which is “equal to a linear combination of the frequencies corresponding to the doubled periods of the tidal year (0.97 year), of the node motion (18.6 years), and of the perigee (8.85 years) of the Earth’s monthly orbit”.

Note that the node/perigee combination alone is essentially the 5.997 year perigean eclipse cycle

$$ frac{1}{8.85} + frac{1}{18.61} = frac{1}{6} $$

and the difference is the longer term 16.9 year beat frequency:

$$ frac{1}{8.85} – frac{1}{18.61} = frac{1}{16.88} $$

I don’t quite understand how Siderenkov’s doubling period comes about, because typically these periods are halved when interactions are considered. His more comprehensive book [4] which is published by Wiley, and which I referenced in a comment on a previous post, is a very detailed investigation of many of the possible lunar interactions that can impact the climate and geophysical processes such as the Chandler Wobble. He gives some compelling evidence that these dynamics are all related somehow — for example, the effective atmospheric angular momentum (QBO included) and oceanic angular momentum (ENSO sloshing included) account for up to 90% of the required Chandler wobble excitation.

The pieces are continuing to fit together. As Liess and Geller [5] found, there is no clear linear connection between ENSO and QBO, but the idea that a separate forcing term, possible lunar in origin, is generating a linear response with respect to QBO and a non-linear Mathieu-equation response with respect to ENSO remains very plausible.

 References

[1]  Lindzen, Richard S., and James R. Holton. “A theory of the quasi-biennial oscillation.” Journal of the Atmospheric Sciences 25.6 (1968): 1095-1107.
[2] DOBRYSHMAN E, M. “On modeling some peculiarities of the tropical atmosphere circulation.” ATMOSFERA 2.1.
[3]  N. Sidorenkov and T. Zhigailo, “Geophysical Effects of the Earth’s Monthly Motion,” Odessa Astronomical Publications, vol. 26, no. 2, p. 285, 2013.
[4] N. S. Sidorenkov, The Interaction Between Earth’s Rotation and Geophysical Processes. Wiley, 2009. http://books.google.ch/books?id=qw4YWTDpk1IC
[5] S. Liess and M. A. Geller, “On the relationship between QBO and distribution of tropical deep convection,” Journal of Geophysical Research: Atmospheres (1984–2012), vol. 117, no. D3, 2012.