Atmosphere

Models and Simulations of the atmosphere

Inferring forced response from QBO wave equation

/ Paul Pukite (@whut) / 2 Comments

[mathjax]If the wave equation is to be effective in modeling a forced behavior such as QBO then certain invariant properties would need to be observed.

f''(t) + omega_0^2 f(t) = F(t)

So if we split this into two parts

f''(t) approx k_1 F(t)

and

omega_0^2 f(t) approx k_2 F(t)

I have been applying multiple linear regression to fit the wave equation via either of the above two forms

So if we combine these two and see how well the Fourier series coefficients align — keeping care to compensate for the $$omega^2$$ scaling factor after taking the second derivative of a sinusoid — the linear result should be:

frac{f''(t)}{f(t)} = omega_0^2 k_1/k_2

and indeed ! (see Figure 1 below)

Fig 1: Correlation between direct f(t) and second-derivative f”(t) coefficient strengths. It should be linear with an intercept at the origin. The strongest term in the upper right is the aliased Draconic tidal period.  The rest are other lunar periods, solar periods, and aliased lunar harmonics.

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Forced versus Natural Response, not a secret

/ Paul Pukite (@whut) / 4 Comments

This is a concept that is beaten into the head of electrical engineers.

Engineers tend to assume the forced response first, as being pessimists they always assume the worst case.  For example: Why is that annoying 60 Hz hum coming out of that circuit?  hmmm .. probably because the AC power somehow leaked into the input and the hum resulted from the forced response of the internal transfer function?  Issue solved.

Physicists tend to look for a natural response first. Being optimists, a research physicist first and foremost wants to make a discovery. A natural response would tell us something interesting about the behavior … and perhaps a new physical law!  Look and See! The eigenvalues!   Often to them, the forced response is a nuisance that needs to be removed to observe the natural response.

But what happens when the forced response is actually the interesting physical behavior?

A prototypical example of a forced response is the phenomenon of cyclic ocean tides. The periods of tides, though complex in breadth, are merely a reflection of the orbital rates of the moon and sun.

Pictures of (left) high tide (right) low tide, linked from Science Buddies

Obviously the ocean has a natural response to a forcing stimulus, obeying some variation of the wave equation, but the overriding output directly links to the observed orbits. Even the second-order effects such as the 9-day tides are simply beats arising from nonlinear interactions of the fortnightly and monthly long period tides.  Everything from the semi-diurnal periods to the long-period monthly periods are forced responses to the gravitational tug of the lunisolar orbits.  Any resonant frequencies in the ocean’s natural response are over-ridden by the forced response.

And so it likely goes for the quasi-biennial oscillations (QBO) of the equatorial stratospheric winds, as these most definitely are a forced response to the inputs … in spite of what the current literature infers.

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QBO Model Validation

/ Paul Pukite (@whut) / 4 Comments

[mathjax]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?

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Common Origins of Climate Behaviors

/ Paul Pukite (@whut) / 1 Comment

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 …

/ Paul Pukite (@whut) / 1 Comment

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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Biennial Connection from QBO to ENSO

/ Paul Pukite (@whut) / 22 Comments

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

/ Paul Pukite (@whut) / 18 Comments

[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.

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.

Eureqa!

/ Paul Pukite (@whut) / 3 Comments

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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Project Loon and QBO

/ Paul Pukite (@whut) / Leave a comment

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