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.


If the glove don't fit ...

To finish off that sentence, "If the glove don't fit, you must acquit". One of the most common complaints that I've received whenever I describe fitting models to data, is that "correlation does not equal causation". I especially get this at Daily Kos with respect to any comment I make to science posts. One knee jerk follows me around and reminds me of this bit of wisdom like clockwork.  I can't really argue the assertion out of hand because it's indeed true that there are plenty of coincidental correlations that don't imply anything significant in terms of causation.

Fig 1:  Correlation of the QBO 30 hPa time-series data.  Does this excellent correlation of data to a model of aliased tidal periods lead to a causation conclusion?  Or is it just a coincidence of the fit chosen?

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Possible Luni-Solar Tidal Mechanism for the Chandler Wobble

Thanks to KO, I have started subscribing to Robert Grumbine's blog. The topic currently is on what provides the pacemaker for the Chandler wobble.

My own opinion, which I hinted at before (and Keith Pickering also pointed out), is the same general mechanism as for the model of the QBO -- but twice as fast. This idea essentially relates that the modulation of the yearly solar orbit (365.242 days) with the draconic (or nodal) lunar month of 27.21222 days sets up the perfect cyclic forcing for the 433 +/- 1.1 days Chandler wobble period.

Think in terms of the maximum declination of (1) the moon with respect to the equator along with (2) the maximum declination of the sun with respect to the equator. For (1) this happens once every ½ of the 27.2122 day nodal cycle or 13.60611 days and for (2) this happens twice a year (once for the southern hemisphere summer and once for the northern hemisphere summer).

Calculating this out, the closest aliased value is 2\pi (365.242/13.606) - 52\pi = 5.303 rads/year, equivalent to a period of 432.77 days. That is close to the generally accepted value of 433 days for the wobble [1]. One can also produce this value graphically by sampling a sinusoid of period 13.60611 days every half a year (see below). That's enough to reveal the solar-lunar declination synchronization pretty clearly. In 120 years, I count a little over 101 complete cycles, which is close to the 433 day period.


Sampling of the Draconic sinusoidal signal (period 13.6 days) with a sampling of twice per year.

The Earth has a large inertia compared to the moon, so it is essentially picking up differential changes in gravitational mass forcing, which is enough to get the wobble in motion.

So that's my simple explanation for what drives the Chandler wobble, yet this differs from Grumbine's idea, and from many other theories, many of which refer to it as a free nutation stemming from a resonance [2]. I am closer to Grumbine, who thinks it is planetary-solar while I think it is luni-solar. Without getting into the plausibility of the gravitational dynamics, it's just too simple and parsimonious to pass up without posting a comment. It is also in line with my general thinking that very few natural processes follow a natural resonance, and that external forcing should always be the initial hypothesis. That works for the QBO in particular; in that case, the primary forcing period is the full Draconic cycle, likely due to the stronger asymmetry in lithosphere response for the two hemispheres.

Model of the second derivative of the QBO, featuring factors related only to lunar and solar gravitational forcing cycles.

And besides the QBO, this potential mechanism likely holds more clues to the behavior behind the ENSO model.  As I commented at Grumbine's blog:

@whut said...The Chandler Wobble, QBO, LOD variation, ENSO, Angular Atmospheric Momentum, and oceanic and atmospheric tides all have varying degrees of connection, likely ultimately tied to luni-solar origins.

Keep plugging away at what you are doing because there are likely common origins to much of the behavior. If you can figure it all out, it will be useful in establishing the natural variability in climate, and thereafter the GCMs can then use this information in their models.



[1] 435 days also happens to be a commonly cited number (see here), but I get the smaller 433 days when I look at the data myself . And Nastula and Gross recently came up with a value of 430.9 days from newly available satellite measurements.
[2] "On the maintenance of the Chandler wobble" Alejandro Jenkins

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.

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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.

tides courtesy of NOAA

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.


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.

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QBO Model Sensitivity

I have been concentrating on modeling the quasi-biennial oscillation (QBO) recently because the results I am getting are a sure bet in my opinion, and the ENSO model will mature and follow in due time.

There are no adjustable parameters in my QBO model, as it relies completely on the known lunar tidal periods. So it's actually conceptually difficult to do a real sensitivity analysis. Those are set in stone, so moving them from their nominal values is artificially changing the physical foundation for the model. Still it would be useful to consider how to move the periods in unison away from the nominal values, so that the artificiality is minimized.

One clever way of doing this is to change a common factor used in the aliasing calculation for each of these periods. Obviously the best candidate is the solar year. To get the correct aliasing with respect to each of the periods, the value of the year in days must be set ( see this article ). And because of the aliased signal's period sensitivity to the mixing frequency, any change in the year value will magnify the error in a fit. So if the theoretically predicted aliased lunar months form an optimal fit, then any change in the solar year away from the nominal value will degrade this fit. And it should degrade it rapidly, because of the sensitivity of the aliasing calculation.

In the following chart, the nominal value for the solar year is 365.242 days, and the x-axis shows what happens when the value is modified away from this value, based on the multiple linear regression fit. The upper-right inset is a magnified region about 365 days, and the yellow highlighting points to the most highly correlated regions.


The best fit is 365.23 and is close enough to 365.242 in my book. Both are also right on the plateau of maximum correlation, indicating that this may be within the sampling error.

After some prodding on my part, I got this tweet from a climate science gatekeeper

It's straightforward to argue that Richard Lindzen's model [1] is wrong by an exclusionary principle. If Lindzen neglected to include the gravitational forcing due to the moon's orbit, yet an absolutely trivial model shows a nearly unity correlation with a precise alignment, then Lindzen must be over-fitting. So by excluding lunar forcing, what Lindzen came up with is not the real physics but more likely a biased fit, based on the unlimited number of adjustable factors he could draw from. I would suggest that his "tour de force" paper will need to be completely reconsidered from scratch.

So the most plausible and parsimonious explanation is a model not much different than that used to predict the ocean's tides.

From the NASA site:

"The Ocean Motion website provides resources developed for inquiring minds both in and outside the classroom, for reading level grades 9-12 (Flesch-Kincaid)."
"Observing the changing water levels caused by astronomical tides is relatively simple and has long been important for major ports."

Although simple, this is real science -- no free parameters and nearly perfect agreement with the empirical data.

[1]R. S. Lindzen and J. R. Holton, “A theory of the quasi-biennial oscillation,” Journal of the Atmospheric Sciences, vol. 25, no. 6, pp. 1095–1107, 1968.

Project Loon and QBO

Missed this recent development related to QBO. Google is planning on deploying balloons to the stratosphere which carry transceivers capable of improving global Internet capability.

This is the stratosphere, so remember that the Quasi-Biennial Oscillation (QBO) of winds will likely play an important role in how the balloons get carried aloft around the earth. Google scientists and engineers might want to consider in more detail how the winds alternate in east-west directions, as their image below shows.

Based on a detailed modeling of previous wind data collected by similar high-altitude deployed balloons, the origin of the QBO is becoming less mysterious. From the data collected from these so-called radiosonde measurements, the origin of this alternation is not according to that proposed by the AGW-denier scientist Richard Lindzen, but via the periodic gravitational pull of the moon, aliased by a seasonal modulation. This is described in a short two-page paper worked out both on this site, and at the collaborative Azimuth Project forum.

The model's ability in capturing the dynamics of QBO is amazing:

Fig 1: Modeling the second derivative of QBO to lunar tidal periods allows a highly detailed predictive capability.  Training the data on a known set of lunar tidal periods (red interval), forecasts the "out-of-bad" data remarkably well !

This modeling is actually quite trivial and as straightforward as modeling the cyclic behavior of ocean tides, but one has to understand how aliasing works.  That is what  Lindzen apparently missed during his 50 years worth of hapless research efforts.

Besides the importance for this Google project, the QBO has lots of relevance for climate, including predicting storm activity and occurrence of El Ninos.

More on Project Loon

Project Loon balloon
Loon for All – Project Loon

How Loon Works – Project Loon

Project Loon - Wikipedia


Why is the QBO important?

After making a breakthrough on modeling the Quasi-Biennial Oscillation of atmospheric winds (QBO) and simultaneously debunking the fearsome AGW skeptic Richard Lindzen's original theory for QBO, it might be wise to take a step back and note the potential significance of having a highly predictive model.

From this QBO site

Why the QBO is important?

  1. The phase of the QBO affects hurricanes in the Atlantic and is widely used as a prognostic in hurricane forecasts.
    Increased hurricane activity occurs for westerly (or positive) zonal wind anomalies; reduced hurricane activity for easterly or negative zonal wind anomalies.
  2. The QBO along with sea surface temperatures and El Niño Southern Oscillation are thought to affect the monsoon.
  3. Tropical cyclone frequency in the northwest Pacific increases during the westerly phase of the QBO. Activity in the southwest Indian basin, however, increases with the easterly phase of the QBO.
  4. Major winter stratospheric warmings preferentially occur during the easterly phase of the QBO, Holton and Tan (1980).
  5. Predictions of ENSO use the expected wind anomalies at 30mb and 50mb to forecast the strength and timing of the event.
  6. The QBO is thought to affect the Sahel rainfall pattern and is used in forecasts for the region.
  7. The decay of aerosol loading following volcanic eruptions such as El Chichon and Pinatubo depends on the phase of the QBO.

And that site does not even list the connection between QBO and El Nino that has been observed [1], and definitely by my own eyes as well, and not to forget Eureqa's machine learning eyes.

That last bit is intriguing, a I am still making progress on ENSO modeling, with this QBO effort providing a foundation for lots of ideas on how to go forward. The connection between QBO, lunar forcing, and ENSO is much too compelling to think otherwise. As far as AGW, I will continue to report on what I can discover from models of the data. This is important when you realize that Lindzen had over 50 years to contribute to the field, and essentially left with the sad fact of contributing nothing, and perhaps even stalling progress in the field of atmospheric sciences for a generation.

[1] Liess, Stefan, and Marvin A. Geller. "On the relationship between QBO and distribution of tropical deep convection." Journal of Geophysical Research: Atmospheres (1984–2012) 117.D3 (2012).