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
@theresphysics @nevaudit Lindzen & Houghton's tour de force QBO paper will apparently be overthrown by @WHUT's physics-free curve-fitting
— mtobis (@mtobis) November 20, 2015
//platform.twitter.com/widgets.js
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.
GeoEnergy Math 
Top panel is a fit of the 2nd derivative of QBO, while the lower panel is a backcast.
Highlighted in yellow, one can see that one part of the cycle is positioned correctly but modeled with less strength than is observed. This may be another non-linear effect, or perhaps higher order derivatives such as the third derivative.
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WHT – I think there is a point to the idea that this is to this point just curve fitting; there are no physical equations. The lunar force is gravity – yet the sun also exerts a gravitational force on the Earth. It’s slightly smaller – about 44% of the moon’s force – but a physical explanation would have to include both lunar and solar gravitational forces.
Kevin, I have the solar terms in there (1) a yearly term, (2) a semi-annual term, and a 1/3 year term
Obviously at times these two forces will oppose each other and at times reinforce each other, but both forces are always present everywhere on earth. If gravity is the forcing agent, then a more complete formula that includes changes in solar gravitational effects needs to be included.
Including solar gravity shouldn’t change your results much since it will be spread out over a year, but it’s a step towards a full description of the actual physics.
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Kevin, Machine learning found the correlation to lunar tides. I will have to ask the vendor if what they are doing classifies as curve fitting.
The rest is just a tidal model, which is solving a system of equations where the forcing amplitudes are first estimated as order-of-magnitude and then refined. No one questions that set of physics because the details are dependent on local conditions.
I have tried to come up with an example of comparing theory to experiment that doesn’t include curve fitting. I have yet to come up with a good one.
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I should also point out that the tidal potentials I gave you include the solar potentials as well as the lunar.
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Yes, good of you to point that out Keith.
There are essentially two approaches to take.
1. The direct gravitational forcing approach that Keith has been pioneering and I am trying to adapt to.
2. The indirect regression fit using the tidal Doodson parameters.
Which is better in the long run is open to discussion, with tidal predictors going with #2 because they like to calibrate against measured data, and thus tweak the fit to the spatial coordinates of the measurement.
My first couple of posts on QBO in the latest installment was essentially a mash of the two, since I used Ray’s forcing time-series but didn’t have the data behind it, so I tried to compare against the limited Fourier series that was generated.
http://contextearth.com/2015/10/09/qbo-is-a-lunar-solar-forced-system/
http://contextearth.com/2015/10/22/pukites-model-of-the-quasi-biennial-oscillation/
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WHT – I must have missed something then because I didn’t remember any solar terms in your “Lunar Tidal Potential” letter nor in any of the posts you’ve written here.
A quick look back finds this in your post Pukite’s Model of the Quasi-Biennial Oscillation: “This empirical fit uses only three factors — the lunar cycles corresponding to the Draconic month, the Anomalistic month, and the Tropical month.”
Nor do I recall any discussion of the relative magnitudes of the two tidal forces and their effect.
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Kevin,
The relevant article is http://contextearth.com/2015/11/14/whuts-up-with-posting-all-these-qbo-fits/ and the following chart, showing the relative strengths of the factors
The empirical fit you reference was based off of the early machine learning results. That is the extent of the complexity in the symbolic regression that the ML tool could stand before going into combinatorial overload.
What I did next was apply the secondary factors from the lunisolar tidal tables, and that was went into the above chart. Those were an eye-opener as they did have significance. The annual, semi-annual, and 1/3-year do show up, especially with respect to the 2nd-derivative time series profiles.
BTW, machine learning would flip out if it saw the complexity in the tidal tables. This is Richard Ray’s table, and I highlighted the strongest factors that I applied:
What is amazing is that all of these factors can be accounted for and that is why tidal prediction programs work so well.
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Regarding year length, the canonical value of 365.2425 is a long-term average, while individual years will vary from the long-term average by a few thousandths of a day due to lunar gravitational effects. So the fits you’ve got are fine.
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It actually bounces up and down about the 365.24 days depending on how many tidal coefficients I add. The last few I added moved it to the 365.23. It went as high as 365.34 days with just two factors.
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That’s a craacerjkck answer to an interesting question
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The idea that I’m a gatekeeper is, sadly, ludicrous. The only think I’m gatekeeping is my own time.
I don’t know why I am banned from commenting on the And Then There’s Physics blog if it isn’t a gatekeeping issue
If you can’t explain what you are doing in such a way that I or perhaps someone smarter than me can follow it and find it interesting, never mind replicate it, you have only made a claim, not done publishable science.
I gave a link in the above article on how to do the seasonal aliasing. Here it is repeated http://contextearth.com/2015/11/17/the-math-of-seasonal-aliasing/. Easy to replicate.
You can still win this if your claim is true, absent any believable theory. If your method is so precise, give us a prediction of the next few QBO cycles, publish them, and find something else to do meanwhile while they verify.
No thanks. Doing an extra few cycles won’t convince anyone of anything.
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“believable theory”
Belief systems do not play into it. What you and I need to do is be able debunk the theory and show how it is wrong. It’s very easy to do, all you have to show is that the aliased periods from the lunar and solar cycles don’t match with the data.
Recall that this is what Lindzen said in 1974:
So if they match and nothing else can explain that match, there is only one possibility to consider — that aliased lunar periods are producing the behavior. As Lindzen said, likely nothing else can replicate the periodicity. So Lindzen had to fabricate some other magical explanation.
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If I had an amazing electrical circuit that was perfect in all respects, but when tested, I detected a 60 Hz hum in the output, I am not going to make up some fantasy story about how it is an emergent property of the circuit. No way. I am going to say that the 60 Hz signal is forced through the circuit via the line voltage. Not very exciting but true.
That’s all there is to this analysis. It i inconceivable that the top 6 tidal factors all show up when aliased against the seasonal signal simply as a result of coincidence of some emergent property of the system.
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WHT – as I understand it, GCMs are still relying on parameterization to various degrees to create the QBO (and some still can’t create a QBO even with this parameterization). One avenue to pursue may be to work with a modeler of GCMs to turn your results either into better parametrization or direct physical equations.
Imagine a CMIP5 model using your QBO forcing compared to one without – that should be enough proof to catch anyone’s attention – especially if you used one of the CMIP5 models that couldn’t generate a QBO on its own.
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Good idea. One interesting aspect that I cannot catch is the asymmetry of the wind speed depending on east-west direction. This would only emerge I assume on other physical properties of the aerodynamics.
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I wanted to repeat a citation to what I think was important prior research to some of these findings.
Li, GuoQing, and HaiFeng Zong. “27.3-day and 13.6-day atmospheric tide.” Science in China Series D: Earth Sciences 50.9 (2007): 1380-1395.
https://www.researchgate.net/publication/238361432_273-day_and_136-day_atmospheric_tide
I cited this first here: http://contextearth.com/2014/08/15/change-of-tide-in-thought/
Again, they were among the first to question Richard Lindzen’s assertion that lunar-forced atmospheric tides were too weak to be observed:
The cites 19 & 21 are to Lindzen papers.
So when I discovered these 27-day periods in the QBO via Eureqa machine learning experiments here , the Chinese papers were the first to hit my radar.
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