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## Comments

Eli Rabett, who knows the material and the players far better than me, said a couple of interesting things about Chris Gray, commenting on his passing:

http://rabett.blogspot.com/2016/04/bill-gray-has-passed-appreciation-and.html

and linked http://www.washingtonpost.com/wp-dyn/content/article/2006/05/23/AR2006052301305.html.

`<a href="http://rabett.blogspot.com/">Eli Rabett</a>, who knows the material and the players far better than me, said a couple of interesting things about Chris Gray, commenting on his passing: <a href="http://rabett.blogspot.com/2016/04/bill-gray-has-passed-appreciation-and.html">http://rabett.blogspot.com/2016/04/bill-gray-has-passed-appreciation-and.html</a> and linked <a href="http://www.washingtonpost.com/wp-dyn/content/article/2006/05/23/AR2006052301305.html">http://www.washingtonpost.com/wp-dyn/content/article/2006/05/23/AR2006052301305.html</a>.`

Interesting post by Eli on Gray:

That's the definition of a heuristic, which can model a behavior but with no physical explanation. Some people consider heuristics to be wrong by definition, while others say they are correct until they stop working (then you find a different heuristic).

`Interesting post by Eli on Gray: > "but by careful observation developed a set of ad hoc models, which turned out to be way wrong but extremely useful for prediction. " That's the definition of a heuristic, which can model a behavior but with no physical explanation. Some people consider heuristics to be wrong by definition, while others say they are correct until they stop working (then you find a different heuristic).`

The theory of QBO is largely based on the work of Richard Lindzen. Yet Lindzen does not have the greatest track record when it comes to research findings

http://www.theguardian.com/environment/climate-consensus-97-per-cent/2014/jan/06/climate-change-climate-change-scepticism

One example of where Lindzen was wildly wrong was in projecting AGW trends:

That's part of the reason why we have a chance of developing a valid model for QBO. Like many research areas, it is easiest to make progress where the effort has been historically weak. And from what I have read on Lindzen, he has somehow made a name for himself while simultaneously getting debunked on his various theories.

On the other hand, what we have in this thread is an elegant and plausible model for QBO:

Somehow, someway Lindzen did not deduce this lunisolar correlation to QBO, even though he had cited the possibility.

Acceptance of a new model is partly dependent on the weaknesses of prior models. Lindzen has been on shaky standing for his contrarian views on climate science topics, and I think this is another example of that.

`The theory of QBO is largely based on the work of Richard Lindzen. Yet Lindzen does not have the greatest track record when it comes to research findings http://www.theguardian.com/environment/climate-consensus-97-per-cent/2014/jan/06/climate-change-climate-change-scepticism >" Richard Lindzen is one of the approximately 3 percent of climate scientists who believe the human influence on global warming is relatively small (though Lindzen is now retired, no longer doing scientific research). More importantly, he's been wrong about nearly every major climate argument he's made over the past two decades. Lindzen is arguably the climate scientist who's been the wrongest, longest. " One example of where Lindzen was wildly wrong was in projecting AGW trends: ![graph](http://i.guim.co.uk/img/static/sys-images/Guardian/Pix/pictures/2014/1/4/1388804857237/Hansen_vs_Lindzen_450.jpg?w=620&q=55&auto=format&usm=12&fit=max&s=a71a3a9bd623b7c4016d1b8140d1d547) That's part of the reason why we have a chance of developing a valid model for QBO. Like many research areas, it is easiest to make progress where the effort has been historically weak. And from what I have read on Lindzen, he has somehow made a name for himself while simultaneously getting debunked on his various theories. On the other hand, what we have in this thread is an elegant and plausible model for QBO: ![model](http://imageshack.com/a/img907/5050/CMo0fK.png) Somehow, someway Lindzen did not deduce this lunisolar correlation to QBO, even though he had cited the possibility. Acceptance of a new model is partly dependent on the weaknesses of prior models. Lindzen has been on shaky standing for his contrarian views on climate science topics, and I think this is another example of that.`

Another interesting quote about Richard Lindzen and his climate science theories:

This is the video of the lecture

Note how emphatic Pierrehumbert is in saying that Lindzen has been wrong:

and this one

Reminds me of college where some professors write big X's across student's exam results if they get it wrong.

I may have to do something similar with Lindzen's QBO theory.

`Another interesting quote about Richard Lindzen and his climate science theories: > The Weekly Standard devotes the first page of its piece to establishing how smart Lindzen is – and he certainly is a smart man, but as climate scientist Ray Pierrehumbert put it, >> "It's okay to be wrong, and [Lindzen] is a smart person, but most people don't really understand that one way of using your intelligence is to spin ever more clever ways of deceiving yourself, ever more clever ways of being wrong. And that's okay because if you are wrong in an interesting way that advances the science, I think it's great to be wrong, and he has made a career of being wrong in interesting ways about climate science." > Make no mistake about it; Lindzen has made a career of being wrong about climate science. [This is the video of the lecture](https://youtu.be/RICBu_P8JWI?t=2080) Note how emphatic Pierrehumbert is in saying that Lindzen has been wrong: ![Lindzen1](http://imageshack.com/a/img923/5943/K911ES.png) and this one ![Lindzen1](http://imageshack.com/a/img922/5918/H6zdZT.png) Reminds me of college where some professors write big X's across student's exam results if they get it wrong. I may have to do something similar with Lindzen's QBO theory.`

I looked at old data for length-of-day (LOD) measurements of the earth to get an idea of angular momentum changes in the rotation. I did not use data from the official IERS website because it looks like it gets filtered too aggressively.

The results are here

Not hard to believe because it is predictable based on what I inferred with the QBO and ENSO model, but sure enough, the primary angular momentum variations corresponds to a seasonally aliased 27.21 day Draconic lunar cycle.

Via dumb machine learning of over 160 years worth of data, the first experiment resulted in an unaliased value of 27.2121 days, to be compared to the known value of 27.21222 days. The precise value is important as any error of the period will induce a gradual phase change over a long duration, thus destroying any coherence in the effect.

The explanation is basic Newtonian physics in that variations of an orbiting satellite like the moon are going to induce changes in the earth's rotation -- and that this will transitively provide a forcing to induce a sloshing mode in the ocean ala ENSO or in atmospheric winds such as QBO. The short period -- diurnal and semidurnal -- effects have been detected in the LOD measurements before but not these seasonally aliased periodic cycles of 2 to 3 years. Nothing in science is provable but these results certainly substantiate the QBO and ENSO models. In other words, if I didn't find a 27.212 day period and it came out to (for example) 26.98 days instead, the QBO and ENSO models would likely be invalidated, as the seasonally-aliased forcing period would not correlate with the fundamental periods found in the QBO and ENSO data.

`I looked at old data for length-of-day (LOD) measurements of the earth to get an idea of angular momentum changes in the rotation. I did not use data from the official IERS website because it looks like it gets filtered too aggressively. The results are [here](http://contextearth.com/2016/06/02/seasonal-aliasing-of-long-period-tides-found-in-length-of-day-data/) Not hard to believe because it is predictable based on what I inferred with the QBO and ENSO model, but sure enough, the primary angular momentum variations corresponds to a seasonally aliased 27.21 day Draconic lunar cycle. Via dumb machine learning of over 160 years worth of data, the first experiment resulted in an unaliased value of 27.2121 days, to be compared to the known value of 27.21222 days. The precise value is important as any error of the period will induce a gradual phase change over a long duration, thus destroying any coherence in the effect. The explanation is basic Newtonian physics in that variations of an orbiting satellite like the moon are going to induce changes in the earth's rotation -- and that this will transitively provide a forcing to induce a sloshing mode in the ocean ala ENSO or in atmospheric winds such as QBO. The short period -- diurnal and semidurnal -- effects have been detected in the LOD measurements before but not these seasonally aliased periodic cycles of 2 to 3 years. Nothing in science is provable but these results certainly substantiate the QBO and ENSO models. In other words, if I didn't find a 27.212 day period and it came out to (for example) 26.98 days instead, the QBO and ENSO models would likely be invalidated, as the seasonally-aliased forcing period would not correlate with the fundamental periods found in the QBO and ENSO data.`

Thanks for the Tamino stats video links - I'm sure I'll learn a lot if I get round to watching them; notwithstanding your point about him smoothing out what he thought was red noise. BTW Over on the blog you've got a rogue "principle" :).

`Thanks for the Tamino stats video links - I'm sure I'll learn a lot if I get round to watching them; notwithstanding your point about him smoothing out what he thought was red noise. BTW Over on the blog you've got a rogue "principle" :).`

The gist of his statistical time series analysis is that some modeled behavior fits well due to chance alone. What the red noise analysis does is provide a gauge to compare a non-noisy model against.

I gave up on watching the Tamino videos. I started and realized that I shouldn't waste my time with material that's doesn't add much value to what I am doing and that in any case could be skimmed from reading other sources.

`The gist of his statistical time series analysis is that some modeled behavior fits well due to chance alone. What the red noise analysis does is provide a gauge to compare a non-noisy model against. I gave up on watching the Tamino videos. I started and realized that I shouldn't waste my time with material that's doesn't add much value to what I am doing and that in any case could be skimmed from reading other sources.`

What other sources are those?

`What other sources are those?`

This one is available online for free

An Introduction to Statistical Signal Processing Robert M. Gray and Lee D. Davisson

When I was working more on fossil fuel depletion modeling several years ago, all I really did was stochastic modeling and applying elements of statistical signal processing. That's really because the models are of large ensembles of events and processes.

In contrast, the big climate models like ENSO and QBO are singular processes likely driven by cyclic forcings. There is actually little that is statistical about this and so regular signal processing and deterministic models are more applicable.

On one of the skeptic sites, I chuckle over the fact that they ascribe it all to noise

No significant periodic components? Awfully narrow view imo.

`This one is available online for free An Introduction to Statistical Signal Processing Robert M. Gray and Lee D. Davisson When I was working more on fossil fuel depletion modeling several years ago, all I really did was stochastic modeling and applying elements of statistical signal processing. That's really because the models are of large ensembles of events and processes. In contrast, the big climate models like ENSO and QBO are singular processes likely driven by cyclic forcings. There is actually little that is statistical about this and so regular signal processing and deterministic models are more applicable. On one of the skeptic sites, I chuckle over the fact that they ascribe it all to noise > "Without sufficient individual model runs to compare to the single observed realization, I have found that using Singlular Spectrum Analysis allows for non linear trends and decomposition of the temperature series into trend, quasi periodical/cyclical and red/white noise. In these analyses I find no evidence for significant periodic components in either the modeled or observed series and thus I can model the residual noise with ARMA. I have found that confidence intervals determined from those models having multiple runs agrees well with those determined from Monte Carlo simulations using ARMA models." No significant periodic components? Awfully narrow view imo.`

Thanks for the reference, that's great.

`Thanks for the reference, that's great.`

All these phenomena are related via forcing stemming from lunar gravitational effects

Look at intervals that show excursions marked by red X, like around 1983:

The LOD is the most direct measure of a sporadic angular momentum change in the earth's rotation. Perhaps this is related to the El Chichon volcanic eruption in 1982. For QBO and ENSO, the model also showed transient deviations from the data.

`All these phenomena are related via forcing stemming from lunar gravitational effects Look at intervals that show excursions marked by red X, like around 1983: ![all](http://imageshack.com/a/img923/9113/dZWJCe.png) The LOD is the most direct measure of a sporadic angular momentum change in the earth's rotation. Perhaps this is related to the El Chichon volcanic eruption in 1982. For QBO and ENSO, the model also showed transient deviations from the data.`

Gavin Schmidt of NASA tweeted a cite to a recent paper on QBO by a group of 12 scientists

Geller, M. A., Zhou, T., Shindell, D., Ruedy, R., Aleinov, I., Nazarenko, L., Tausnev, N.L., Kelley, M., Sun, S., Cheng, Y., Field, R.D. and Faluvegi, G. (2016), Modeling the QBO – Improvements Resulting from Higher Model Vertical Resolution. J. Adv. Model. Earth Syst.. doi:10.1002/2016MS000699

They claim that the base frequency of QBO relates inversely proportional to the pressure:

I disagree with this interpretation. I know that this might sound like I am using the argument of personal astonishment, but I can't see how a physical model could operate this way. If anything, pressure may modify the amplitude of the oscillations, i.e. the peak speed, but asserting that the pressure sets the oscillation period is likely the result of a model that no one understands. If it was indeed caused by the pressure, they should be able to come up with a simplified formula, like deriving the resonant frequency of a Rijke tube, instead of suggesting that it is an emergent property of a complicated model.

The 28 month period is ubiquitous in a number of geophysical oscillations. It shows up in (1) the seasonally aliased Draconic lunar cycle, (2) the strongest factor in the model of ENSO, (3) exactly half that period in the Chandler Wobble, (4) its defined as the mean annual flood recurrence interval by the USGS (first reported in 1960), and (5) the fundamental QBO period.

No one wants to admit that seeing this number show up in all these measures is most likely a forced response from the first lunar cycle factor, as opposed to a natural resonance. Problem is that a forced response is not as "sexy" as finding a natural frequency eigenvalue.

But of course there is no way to verify either way, since a controlled experiment is not available to test against.

`Gavin Schmidt of NASA tweeted a cite to a recent paper on QBO by a group of 12 scientists Geller, M. A., Zhou, T., Shindell, D., Ruedy, R., Aleinov, I., Nazarenko, L., Tausnev, N.L., Kelley, M., Sun, S., Cheng, Y., Field, R.D. and Faluvegi, G. (2016), Modeling the QBO – Improvements Resulting from Higher Model Vertical Resolution. [J. Adv. Model. Earth Syst.](http://onlinelibrary.wiley.com/doi/10.1002/2016MS000699/epdf). doi:10.1002/2016MS000699 They claim that the base frequency of QBO relates inversely proportional to the pressure: > "The mean zonal winds from the ERA-Interim reanalysis are shown between 4.5 and 4.5 N between heights of 100 and 1 hPa for the 20 years 1991-2010. Note that ERA-Interim shows that approximately 8 1/2 QBO cycles occurred during this period, implying a QBO period averaging about 28 months. The model results in figure 1b show that no coherent QBO resembling observations exists for the gravity wave momentum flux forcing of 1.5 mPa, which is consistent with the steady jets at low wave forcing demonstrated by Yoden and Holton (1988). It also shows that the QBO-like oscillation for a gravity wave momentum flux forcing of 2.0 mPa has a period of about 8 years, while a forcing of 2.5 mPa gives a period of about 37 months, and a forcing of 3.0 mPa gives a period of about 25 months, and a forcing of 3.5 mPa gives a period of about 21 months. In fact, we find that the best fit to observed QBO periods is for a gravity wave momentum flux forcing of 2.9 mPa, as will be shown in the next section. " I disagree with this interpretation. I know that this might sound like I am using the argument of personal astonishment, but I can't see how a physical model could operate this way. If anything, pressure may modify the amplitude of the oscillations, i.e. the peak speed, but asserting that the pressure sets the oscillation period is likely the result of a model that no one understands. If it was indeed caused by the pressure, they should be able to come up with a simplified formula, like deriving the [resonant frequency of a Rijke tube](http://hyperphysics.phy-astr.gsu.edu/hbase/waves/rijke.html), instead of suggesting that it is an emergent property of a complicated model. The 28 month period is ubiquitous in a number of geophysical oscillations. It shows up in (1) the seasonally aliased Draconic lunar cycle, (2) the strongest factor in the model of ENSO, (3) exactly half that period in the Chandler Wobble, (4) its defined as the mean annual flood recurrence interval by the USGS ([first reported in 1960](http://pubs.usgs.gov/wsp/1543a/report.pdf)), and (5) the fundamental QBO period. No one wants to admit that seeing this number show up in all these measures is most likely a forced response from the first lunar cycle factor, as opposed to a natural resonance. Problem is that a forced response is not as "sexy" as finding a natural frequency eigenvalue. But of course there is no way to verify either way, since a controlled experiment is not available to test against.`

Wow, this is what I think is a huge breakthrough in the QBO model. I started from first principles and applied some what I consider rather obvious simplifications.

`Wow, this is what I think is a [huge breakthrough in the QBO model](http://contextearth.com/2016/06/29/simplifying-laplaces-tidal-equations-for-qbo/). I started from first principles and applied some what I consider rather obvious simplifications.`

(following the last post) For posterity, the idea is to simplify the Laplace Tidal Equation along the equator, where the QBO resides.

Many of the terms evaluated along the equator are second-order .

With the substitution

$ \frac{d\Theta}{d\mu} = \frac{d\Theta}{dt} \frac{dt}{d\mu} $

and for $\sigma \gg \mu$ and $\mu = sin \phi \cong\phi $ the equation reduces to

$ \frac{1}{\sigma^2} \left( \frac{d}{dt}(\frac{d\Theta}{dt} \frac{dt}{d\mu}) \frac{dt}{d\mu} + \left[\frac{s}{\sigma} + s^2\right] \Theta \right) + \gamma \Theta = 0 $

With an educated guess that $\frac{d\mu}{dt}$ is a sinusoid plus a constant, the equation solves as a Sturm-Liouville variant, where $\Theta(t)$ is replaced by $f(t)$ (to keep it consistent with the generic DiffEq I have been using).

The solution is a sinusoid of a sinusoid, which has interesting properties that map to the QBO time-series.

One of these properties is a gradually clipping for large sinusoidal excursions, which matches to what is observed with QBO (choosing B=0). Note the squaring of the waveform.

The generic solution's inner modulation : $ f(t) = k \cdot sin( \sqrt{A} \cdot sin(\omega t + \psi_0) + \phi_0) $

can also be replaced by a set of sinusoids of different frequency. That's where the decomposition of the lunar forcing into a seasonally-aliased set of factors is applied. The sinusoidal clipping also suppresses the beat modulation caused by adding cycles of differing frequency.

It really does collapse almost perfectly into a concise formulation. That's the quasi-elevator-pitch, more here.

`(following the last post) For posterity, the idea is to simplify the Laplace Tidal Equation along the equator, where the QBO resides. ![laplace](http://contextearth.com/wp-content/uploads/2016/06/laplaceTidalOperator.gif) Many of the terms evaluated along the equator are second-order . With the substitution $ \frac{d\Theta}{d\mu} = \frac{d\Theta}{dt} \frac{dt}{d\mu} $ and for $\sigma \gg \mu$ and $\mu = sin \phi \cong\phi $ the equation reduces to $ \frac{1}{\sigma^2} \left( \frac{d}{dt}(\frac{d\Theta}{dt} \frac{dt}{d\mu}) \frac{dt}{d\mu} + \left[\frac{s}{\sigma} + s^2\right] \Theta \right) + \gamma \Theta = 0 $ With an educated guess that $\frac{d\mu}{dt}$ is a sinusoid plus a constant, the equation solves as a Sturm-Liouville variant, where $\Theta(t)$ is replaced by $f(t)$ (to keep it consistent with the generic DiffEq I have been using). ![sl](http://imageshack.com/a/img923/302/q857zj.png) The solution is a sinusoid of a sinusoid, which has interesting properties that map to the QBO time-series. One of these properties is a gradually clipping for large sinusoidal excursions, which matches to what is observed with QBO (choosing B=0). Note the squaring of the waveform. ![fit](http://imageshack.com/a/img924/4915/S7OmbX.png) The generic solution's inner modulation : $ f(t) = k \cdot sin( \sqrt{A} \cdot sin(\omega t + \psi_0) + \phi_0) $ can also be replaced by a set of sinusoids of different frequency. That's where the decomposition of the lunar forcing into a seasonally-aliased set of factors is applied. The sinusoidal clipping also suppresses the beat modulation caused by adding cycles of differing frequency. It really does collapse almost perfectly into a concise formulation. That's the quasi-elevator-pitch, [more here](http://contextearth.com/2016/06/10/pukites-model-of-enso/).`

Here is an alternate derivation of reducing Laplace's tidal equations along the equator.

For a fluid sheet of average thickness $D$, the vertical tidal elevation $\zeta$, as well as the horizontal velocity components $u$ and $v$ (in the latitude $\varphi$ and longitude $\lambda$ directions).

This is the set of Laplace's tidal equations (Wikipedia). Along the equator, for $\varphi$ at zero we can reduce this.

$ {\begin{aligned}{\frac {\partial \zeta }{\partial t}}&+{\frac {1}{a\cos(\varphi )}}\left[{\frac {\partial }{\partial \lambda }}(uD)+{\frac {\partial }{\partial \varphi }}\left(vD\cos(\varphi )\right)\right]=0,\\[2ex]{\frac {\partial u}{\partial t}}&-v\left(2\Omega \sin(\varphi )\right)+{\frac {1}{a\cos(\varphi )}}{\frac {\partial }{\partial \lambda }}\left(g\zeta +U\right)=0\qquad {\text{and}}\\[2ex]{\frac {\partial v}{\partial t}}&+u\left(2\Omega \sin(\varphi )\right)+{\frac {1}{a}}{\frac {\partial }{\partial \varphi }}\left(g\zeta +U\right)=0,\end{aligned}} $

where $\Omega$ is the angular frequency of the planet's rotation, $g$ is the planet's gravitational acceleration at the mean ocean surface, $a$ is the planetary radius, and $U$ is the external gravitational tidal-forcing potential.

The main candidates for removal due to the small-angle approximation are the second terms in the second and third equation. The plan is to then substitute the isolated $u$ and $v$ terms into the first equation, after taking of that equation with respect to $t$.

$ {\begin{aligned}{\frac {\partial \zeta }{\partial t}}&+{\frac {1}{a}}\left[{\frac {\partial }{\partial \lambda }}(uD)+{\frac {\partial }{\partial \varphi }}\left(vD\right)\right]=0,\\[2ex]{\frac {\partial u}{\partial t}}&+{\frac {1}{a}}{\frac {\partial }{\partial \lambda }}\left(g\zeta +U\right)=0\qquad {\text{and}}\\[2ex]{\frac {\partial v}{\partial t}}&+{\frac {1}{a}}{\frac {\partial }{\partial \varphi }}\left(g\zeta +U\right)=0,\end{aligned}} $

Taking another derivative of the first equation:

$ {\begin{aligned}{a\frac {\partial^2 \zeta }{\partial t^2}}&+ \frac {\partial }{\partial t} \left[{\frac {\partial }{\partial \lambda }}(uD)+{\frac {\partial }{\partial \varphi }}\left(vD\right)\right]=0,\end{aligned}} $

Next, on the bracketed pair we invert the order of derivatives (and pull out

D, which may not be right for ENSO where the thermocline depth varies!)$ {\begin{aligned}{a\frac {\partial^2 \zeta }{\partial t^2}}&+ D \left[{\frac {\partial }{\partial \lambda }}( \frac {\partial u }{\partial t} )+{\frac {\partial }{\partial \varphi }}(\frac {\partial v }{\partial t} )\right]=0,\end{aligned}} $

Notice now that the bracketed terms can be replaced by the 2nd and 3rd of Laplace's equations

${\begin{aligned}{\frac {\partial u}{\partial t}}&=-{\frac {1}{a}}{\frac {\partial }{\partial \lambda }}\left(g\zeta +U\right)\end{aligned}}$

${\begin{aligned}{\frac {\partial v}{\partial t}}&=-{\frac {1}{a}}{\frac {\partial }{\partial \varphi }}\left(g\zeta +U\right)\end{aligned}}$

as

$ {\begin{aligned}{a^2\frac {\partial^2 \zeta }{\partial t^2}}&- D \left[{\frac {\partial }{\partial \lambda }}( {{\frac {\partial }{\partial \lambda }}\left(g\zeta +U\right)} )+{\frac {\partial }{\partial \varphi }}({{\frac {\partial }{\partial \varphi }}\left(g\zeta +U\right)})\right]=0\end{aligned}} $

The $\lambda$ terms are in longitude so that we can use a separation of variables approach and create a spatial standing wave for QBO, $SW(s)$ where $s$ is a wavenumber.

$ {\begin{aligned}{\frac {\partial^2 \zeta }{\partial t^2}}&- D \left[( SW(s) \zeta )+{\frac {\partial }{\partial \varphi }}({{\frac {\partial }{\partial \varphi }}\left(g\zeta +U\right)})\right]=0\end{aligned}} $

The next bit is the connection between a change in latitudinal forcing with a temporal change:

$ \begin{aligned} {\frac {\partial \zeta }{\partial \varphi } = \frac {\partial \zeta }{\partial t} \frac {\partial t }{\partial \varphi } } \end{aligned}$

so if

$ \begin{aligned} \frac {\partial \varphi }{\partial t } = \sum_{i=1}^{i=N} k_i \omega_i \cos(\omega_i t) \end{aligned}$

to describe the external gravitational forcing terms, then the solution is the following:

$ \begin{aligned} \zeta(t) = \sin( \sqrt{A} \sum_{i=1}^{i=N} k_i \sin(\omega_i t) ) \end{aligned} $

where $A$ is an aggregate of the constants of the diffEq.

So we can eventually get to a fit that looks like the actual QBO:

The QBO may actually be the $v(t)$ term - the horizontal longitudinal velocity of the fluid, the wind in other words - which can be derived from the above by applying the solution to Laplace's third tidal equation in simplified form above.

Consider if instead of $U$ being a constant, it varies in a similar sinusoidal fashion, then that term will go to the RHS as a forcing, and the above impulse response function will need to be convolved with that forcing . The same goes for the $D$ factor which I foreshadowed earlier. Having these vary with time changes the character of the solution. So instead of having a relatively constant amplitude envelope characteristic of QBO, the waveform amplitude will become more erratic, and more similar to the dynamics of ENSO, where the peak excursions vary a considerable amount.

Modifying the $D$ factor in fact makes the solution closer to a Mathieu equation formulation. The stratosphere is constant in depth (i.e. stratified) so that's why we set that to a constant. Yet remember that with ENSO, it the

thermoclinedepth that is sloshing wildly. The density differences of water above and below the thermocline is what is sensitive to the lunar gravitational forcing. That's why with a strict biennial modulation applied to the $D$ term allows an ENSO model that shows a behavior that matches the data so well:This is a fit for ENSO from 1880 to 1950, with the validation to the right Note how well the amplitudes vary in prediction

In contrast for QBO, a training run from 1953 to 1983 shows this:

The validated model to the right of 1983 shows much less variation with amplitude, in keeping with the excursion-limiting behavior of the sin-of-a-sin formulation (i.e. sin(sin(t))).

`Here is an alternate derivation of reducing Laplace's tidal equations along the equator. For a fluid sheet of average thickness $D$, the vertical tidal elevation $\zeta$, as well as the horizontal velocity components $u$ and $v$ (in the latitude $\varphi$ and longitude $\lambda$ directions). This is the set of Laplace's tidal equations ([Wikipedia](https://en.wikipedia.org/wiki/Theory_of_tides)). Along the equator, for $\varphi$ at zero we can reduce this. $ {\begin{aligned}{\frac {\partial \zeta }{\partial t}}&+{\frac {1}{a\cos(\varphi )}}\left[{\frac {\partial }{\partial \lambda }}(uD)+{\frac {\partial }{\partial \varphi }}\left(vD\cos(\varphi )\right)\right]=0,\\[2ex]{\frac {\partial u}{\partial t}}&-v\left(2\Omega \sin(\varphi )\right)+{\frac {1}{a\cos(\varphi )}}{\frac {\partial }{\partial \lambda }}\left(g\zeta +U\right)=0\qquad {\text{and}}\\[2ex]{\frac {\partial v}{\partial t}}&+u\left(2\Omega \sin(\varphi )\right)+{\frac {1}{a}}{\frac {\partial }{\partial \varphi }}\left(g\zeta +U\right)=0,\end{aligned}} $ where $\Omega$ is the angular frequency of the planet's rotation, $g$ is the planet's gravitational acceleration at the mean ocean surface, $a$ is the planetary radius, and $U$ is the external gravitational tidal-forcing potential. The main candidates for removal due to the small-angle approximation are the second terms in the second and third equation. The plan is to then substitute the isolated $u$ and $v$ terms into the first equation, after taking of that equation with respect to $t$. $ {\begin{aligned}{\frac {\partial \zeta }{\partial t}}&+{\frac {1}{a}}\left[{\frac {\partial }{\partial \lambda }}(uD)+{\frac {\partial }{\partial \varphi }}\left(vD\right)\right]=0,\\[2ex]{\frac {\partial u}{\partial t}}&+{\frac {1}{a}}{\frac {\partial }{\partial \lambda }}\left(g\zeta +U\right)=0\qquad {\text{and}}\\[2ex]{\frac {\partial v}{\partial t}}&+{\frac {1}{a}}{\frac {\partial }{\partial \varphi }}\left(g\zeta +U\right)=0,\end{aligned}} $ Taking another derivative of the first equation: $ {\begin{aligned}{a\frac {\partial^2 \zeta }{\partial t^2}}&+ \frac {\partial }{\partial t} \left[{\frac {\partial }{\partial \lambda }}(uD)+{\frac {\partial }{\partial \varphi }}\left(vD\right)\right]=0,\end{aligned}} $ Next, on the bracketed pair we invert the order of derivatives (and pull out *D*, which may not be right for ENSO where the thermocline depth varies!) $ {\begin{aligned}{a\frac {\partial^2 \zeta }{\partial t^2}}&+ D \left[{\frac {\partial }{\partial \lambda }}( \frac {\partial u }{\partial t} )+{\frac {\partial }{\partial \varphi }}(\frac {\partial v }{\partial t} )\right]=0,\end{aligned}} $ Notice now that the bracketed terms can be replaced by the 2nd and 3rd of Laplace's equations ${\begin{aligned}{\frac {\partial u}{\partial t}}&=-{\frac {1}{a}}{\frac {\partial }{\partial \lambda }}\left(g\zeta +U\right)\end{aligned}}$ ${\begin{aligned}{\frac {\partial v}{\partial t}}&=-{\frac {1}{a}}{\frac {\partial }{\partial \varphi }}\left(g\zeta +U\right)\end{aligned}}$ as $ {\begin{aligned}{a^2\frac {\partial^2 \zeta }{\partial t^2}}&- D \left[{\frac {\partial }{\partial \lambda }}( {{\frac {\partial }{\partial \lambda }}\left(g\zeta +U\right)} )+{\frac {\partial }{\partial \varphi }}({{\frac {\partial }{\partial \varphi }}\left(g\zeta +U\right)})\right]=0\end{aligned}} $ The $\lambda$ terms are in longitude so that we can use a separation of variables approach and create a spatial standing wave for QBO, $SW(s)$ where $s$ is a wavenumber. $ {\begin{aligned}{\frac {\partial^2 \zeta }{\partial t^2}}&- D \left[( SW(s) \zeta )+{\frac {\partial }{\partial \varphi }}({{\frac {\partial }{\partial \varphi }}\left(g\zeta +U\right)})\right]=0\end{aligned}} $ The next bit is the connection between a change in latitudinal forcing with a temporal change: $ \begin{aligned} {\frac {\partial \zeta }{\partial \varphi } = \frac {\partial \zeta }{\partial t} \frac {\partial t }{\partial \varphi } } \end{aligned}$ so if $ \begin{aligned} \frac {\partial \varphi }{\partial t } = \sum_{i=1}^{i=N} k_i \omega_i \cos(\omega_i t) \end{aligned}$ to describe the external gravitational forcing terms, then the solution is the following: $ \begin{aligned} \zeta(t) = \sin( \sqrt{A} \sum_{i=1}^{i=N} k_i \sin(\omega_i t) ) \end{aligned} $ where $A$ is an aggregate of the constants of the diffEq. So we can eventually get to a fit that looks like the actual QBO: ![qbo](http://imageshack.com/a/img924/4915/S7OmbX.png) The QBO may actually be the $v(t)$ term - the horizontal longitudinal velocity of the fluid, the wind in other words - which can be derived from the above by applying the solution to Laplace's third tidal equation in simplified form above. <hr/> Consider if instead of $U$ being a constant, it varies in a similar sinusoidal fashion, then that term will go to the RHS as a forcing, and the above impulse response function will need to be convolved with that forcing . The same goes for the $D$ factor which I foreshadowed earlier. Having these vary with time changes the character of the solution. So instead of having a relatively constant amplitude envelope characteristic of QBO, the waveform amplitude will become more erratic, and more similar to the dynamics of ENSO, where the peak excursions vary a considerable amount. Modifying the $D$ factor in fact makes the solution closer to a Mathieu equation formulation. The stratosphere is constant in depth (i.e. stratified) so that's why we set that to a constant. Yet remember that with ENSO, it the *thermocline* depth that is sloshing wildly. The density differences of water above and below the thermocline is what is sensitive to the lunar gravitational forcing. That's why with a strict biennial modulation applied to the $D$ term allows an ENSO model that shows a behavior that matches the data so well: This is a fit for ENSO from 1880 to 1950, with the validation to the right ![enso](http://imageshack.com/a/img921/1870/SDw7kW.png) Note how well the amplitudes vary in prediction In contrast for QBO, a training run from 1953 to 1983 shows this: ![qbo](http://imageshack.com/a/img922/9528/urDjx6.png) The validated model to the right of 1983 shows much less variation with amplitude, in keeping with the excursion-limiting behavior of the sin-of-a-sin formulation (i.e. sin(sin(t))).`

This is a caveat to the Laplace's tidal equation fit to the QBO data. Although the seasonally aliased Draconic period works very well, a shifted value from 27.212 days to 27.209 provides an arguably better fit. Although not noticeable to the eye, the correlation coefficient does increase by almost 0.01 with the slightly shorter period. Compare upper vs lower curves below:

This is all about keeping a coherent phase across over 60 years of QBIO data. After 54 years or 650 Draconic-monthly periods, the difference between a seasonally aliased 27.212 and 27.209 will become apparent as a gradual phase shift, as shown below.

The phase buildup is about a tenth of a period. The issue is whether this is due to fitting slop as the multiple linear regression tries to account for possible jitter noise or whether this is something real in the physical process. The latter may occur if some other forcing alignments in sync with the lower period value exist. The model is likely within some uncertainty margins, but I don't have a good handle on how to quantify the margin itself.

No matter how well a model works, there is always room for creeping doubt. If you don't have this in your own mind, someone else will certainly express it.

`This is a caveat to the Laplace's tidal equation fit to the QBO data. Although the seasonally aliased Draconic period works very well, a shifted value from 27.212 days to 27.209 provides an arguably better fit. Although not noticeable to the eye, the correlation coefficient does increase by almost 0.01 with the slightly shorter period. Compare upper vs lower curves below: ![2QBOs](http://imageshack.com/a/img923/9910/APVmQ3.png) This is all about keeping a coherent phase across over 60 years of QBIO data. After 54 years or 650 Draconic-monthly periods, the difference between a seasonally aliased 27.212 and 27.209 will become apparent as a gradual phase shift, as shown below. ![qbo](http://imageshack.com/a/img922/7535/uamalS.png) The phase buildup is about a tenth of a period. The issue is whether this is due to fitting slop as the multiple linear regression tries to account for possible [jitter](https://en.wikipedia.org/wiki/Jitter) noise or whether this is something real in the physical process. The latter may occur if some other forcing alignments in sync with the lower period value exist. The model is likely within some uncertainty margins, but I don't have a good handle on how to quantify the margin itself. No matter how well a model works, there is always room for creeping doubt. If you don't have this in your own mind, someone else will certainly express it.`

Here is another quote from Richard Lindzen, originator of the consensus QBO model. This is on picking out tidal periods from the data:

PLANETARY WAVES ON BETA-PLANES , 1967

The lunar nodal tidal period matches the QBO period exactly when seasonally aliased. If Lindzen believed what he wrote, he would have to say that the nature of the QBO forcing is now clear.

`Here is another quote from Richard Lindzen, originator of the consensus QBO model. This is on picking out tidal periods from the data: ![tide](https://pbs.twimg.com/media/Cn_RTvrWYAAuv9D.jpg) [PLANETARY WAVES ON BETA-PLANES](http://eaps.mit.edu/faculty/lindzen/18_bet~1.pdf) , 1967 The lunar nodal tidal period matches the QBO period exactly when seasonally aliased. If Lindzen believed what he wrote, he would have to say that the nature of the QBO forcing is now clear.`

I don't think I showed this piece of evidence on this forum yet. It shows how the three important draconic aliased periods for QBO are discovered via machine learning.

Fig 1: Eureqa results when applied to a QBO time-series 2nd derivative. The lowest error solution (shown in reverse highlighted blue) feature as strongest sinusoidal factors the three seasonally aliased Draconic harmonics.

Fig. 2 : Alignment of the three strongest sinusoidal periods found via machine learning with those predicted via seasonally reinforced Draconic lunar tidal periods.

More details here: http://contextearth.com/2016/02/13/qbo-model-validation/

`I don't think I showed this piece of evidence on this forum yet. It shows how the three important draconic aliased periods for QBO are discovered via machine learning. ![eu](http://imagizer.imageshack.us/a/img912/1410/pX5ZeD.jpg) Fig 1: Eureqa results when applied to a QBO time-series 2nd derivative. The lowest error solution (shown in reverse highlighted blue) feature as strongest sinusoidal factors the three seasonally aliased Draconic harmonics. ![line](http://imagizer.imageshack.us/a/img921/6223/oi6jJE.png) Fig. 2 : Alignment of the three strongest sinusoidal periods found via machine learning with those predicted via seasonally reinforced Draconic lunar tidal periods. More details here: http://contextearth.com/2016/02/13/qbo-model-validation/`

This is a very troublesome article that just came out

"Evidence of a tidal effect on the Polar Jet Stream" http://CliveBest.com/blog/?p=7278

Based on my experience, would I think twice of trying to convince someone that a cc of 0.024 was significant? Looking at the time series, I could generate countless red or white noise trials that would work as well.

Clive Best is an AGW skeptic, but one of the smarter guys blogging on science topics so am of mixed mind on this particular research.

`This is a very troublesome article that just came out "Evidence of a tidal effect on the Polar Jet Stream" http://CliveBest.com/blog/?p=7278 ![cc](http://clivebest.com/blog/wp-content/uploads/2016/08/Figure4.png) > "The Spearman correlation coefficient calculated between all values of DA/DT and T(45) evaluates to 2.4%, which although small still deviates from zero (no correlation) by 3.7 $\sigma$ equating to a statistical significance of >99.7%." Based on my experience, would I think twice of trying to convince someone that a cc of 0.024 was significant? Looking at the time series, I could generate countless red or white noise trials that would work as well. Clive Best is an AGW skeptic, but one of the smarter guys blogging on science topics so am of mixed mind on this particular research.`

There is already a connection between the QBO and the polar climate, as this plot shows the relationship between the temperature of the North Pole stratosphere and the direction of the QBO

With the explanation of QBO via seasonal aliasing of tides, this is a much better correlation to polar stratospheric effects.

`There is already a connection between the QBO and the polar climate, as this plot shows the relationship between the temperature of the North Pole stratosphere and the direction of the QBO ![polar](https://imagizer.imageshack.us/v2/449x673q90/921/MjQRAr.png) With the explanation of QBO via seasonal aliasing of tides, this is a much better correlation to polar stratospheric effects.`

Deriving the QBO from Laplace's tidal equations, it becomes apparent that the

accelerationof wind and not the windspeedis the fundamental measure to characterize. With that premise, things seem to fall into place much more naturally.First, multiply the nodal/draconic frequency by a sharply peaked yearly modulation and we get the following as a fit for QBO acceleration:

The fitting region is 1970 to 1982, and the rest is extrapolated. The correlation coefficient is not extremely high but note the amount of fine detail that gets exposed by the model.

To get back the wind speed QBO, the curves are integrated

Again the model fitting is only conducted on the interval from 1970 to 1982. Outside of that interval the model doesn't track every peak but enough of the fine detail is captured that it's obvious that the model has predictive power.

The supporting experiment involves running a symbolic regression machine learning trial on this same interval.

Amazingly, the ML finds all the same harmonics caused by multiplying the sharply peaked yearly signal (freq ~ 2π ) with the Draconic tide (aliased ~2.7 year). No other periods are detected.

`Deriving the QBO from Laplace's tidal equations, it becomes apparent that the *acceleration* of wind and not the wind *speed* is the fundamental measure to characterize. With that premise, things seem to fall into place much more naturally. First, multiply the nodal/draconic frequency by a sharply peaked yearly modulation and we get the following as a fit for QBO acceleration: ![qboPrime](http://imageshack.com/a/img922/5399/n5gGWi.png) The fitting region is 1970 to 1982, and the rest is extrapolated. The correlation coefficient is not extremely high but note the amount of fine detail that gets exposed by the model. To get back the wind speed QBO, the curves are integrated ![qbo](http://imageshack.com/a/img924/741/KJVfpb.png) Again the model fitting is only conducted on the interval from 1970 to 1982. Outside of that interval the model doesn't track every peak but enough of the fine detail is captured that it's obvious that the model has predictive power. The supporting experiment involves running a symbolic regression machine learning trial on this same interval. ![eureqa](http://imageshack.com/a/img922/5362/m1ooXj.png) Amazingly, the ML finds all the same harmonics caused by multiplying the sharply peaked yearly signal (freq ~ 2π ) with the Draconic tide (aliased ~2.7 year). No other periods are detected. ![Eu2](http://imageshack.com/a/img924/2507/QrnMZH.png)`

This is a table describing geophysical periods impacting the Earth's nutation (from [1] below)

The calculation appears to use the Delaunay arguments, which is essentially a sum of frequency components. My problem is the first entry. That has one component which appears to be close to the Draconic lunar month of 21.212 days. And I think that is what it should be according to other sources; yet it says that it is 27.20986 days.

Is this a typo or am I missing something fundamental in how the Earth responds to the lunar cycle? This frequency should be synched solidly to the observed lunar frequency, otherwise it will gradually go out of phase. The difference between the two gives a time of over 400 years before the two numbers phases cancel, so it is a slight difference but significant over long intervals.

So its not much of a gap, and in fact the 27.20986 number is closer to what I am seeing as an optimized aliased frequency component in the QBO. In other words, the optimal fit occurs for around this value.

According to the paper

Fis the difference between the mean longitude of the Moon and the mean longitude of the node of the Moon. That sounds like the same definition for the Draconic month, if the reference is the Earth.This is an older paper, but still, these geophysicists work on the numbers to make them more precise, and being sloppy about it completely defeats that purpose. So I have to treat this with number with due respect.

Anybody have any ideas on the discrepancy?

[1] C. Bizouard, M. Folgueira, and J. Souchay, “Comparison of the short period rigid Earth nutation series,” presented at the IAU Colloq. 178: Polar Motion: Historical and Scientific Problems, 2000, vol. 208, p. 613. Online

`This is a table describing geophysical periods impacting the Earth's nutation (from [1] below) ![nut](http://imageshack.com/a/img923/5129/9UOz0v.png) The calculation appears to use the Delaunay arguments, which is essentially a sum of frequency components. My problem is the first entry. That has one component which appears to be close to the Draconic lunar month of 21.212 days. And I think that is what it should be according to other sources; yet it says that it is 27.20986 days. Is this a typo or am I missing something fundamental in how the Earth responds to the lunar cycle? This frequency should be synched solidly to the observed lunar frequency, otherwise it will gradually go out of phase. The difference between the two gives a time of over 400 years before the two numbers phases cancel, so it is a slight difference but significant over long intervals. So its not much of a gap, and in fact the 27.20986 number is closer to what I am seeing as an optimized aliased frequency component in the QBO. In other words, the optimal fit occurs for around this value. According to the paper *F* is the difference between the mean longitude of the Moon and the mean longitude of the node of the Moon. That sounds like the same definition for the Draconic month, if the reference is the Earth. ![del](http://imageshack.com/a/img921/2008/PxInTf.png) This is an older paper, but still, these geophysicists work on the numbers to make them more precise, and being sloppy about it completely defeats that purpose. So I have to treat this with number with due respect. Anybody have any ideas on the discrepancy? [1] C. Bizouard, M. Folgueira, and J. Souchay, “Comparison of the short period rigid Earth nutation series,” presented at the IAU Colloq. 178: Polar Motion: Historical and Scientific Problems, 2000, vol. 208, p. 613. [Online](http://adsabs.harvard.edu/full/2000ASPC..208..613B)`

The period in space is shown as 1.03521, which appears to have enough significant digits but the leading value of 1 is thrown away and left with the Delaunay value of F=1/(1/1.03521-1-1/365.242) =-27.2106. Then the value is closer to what is in the paper. Perhaps one more significant digit is needed after 1.03521_ to resolve this issue. To get to 27.212, the value needs to be 1.035207, which indicates that the value

could have beenrounded up. This is frustrating because the final result has 7 digits, which is more precision than the "period in space" shown.The other possibility is in what is considered a year. There is the tropical year which is essentially the calendar year 365.242 days. Or the sidereal year which is 365.256 days and the anomalistic year which is 365.2596 days. If one of these other definitions is necessary for the calculation -- for example they do mention sidereal time which is used for a space-relative reference frame -- that would be enough slop to close up much of the difference.

Ft = 27.2106*365.242/365.256 = 27.20956 tropical days

This is closer to the 27.20986 days they give.

This book Space-Time Reference Systems may give some insight.

Sorry for this thread getting too pedantic but that always seems to happen when the result is homing in on a potentially solid match.

This is the difference between using 27.209 days 27.212 days in a QBO fit. The value of 27.209 days results in a correlation improvement, which is marginally noticeable by eye.

`The period in space is shown as 1.03521, which appears to have enough significant digits but the leading value of 1 is thrown away and left with the Delaunay value of F=1/(1/1.03521-1-1/365.242) =-27.2106. Then the value is closer to what is in the paper. Perhaps one more significant digit is needed after 1.03521_ to resolve this issue. To get to 27.212, the value needs to be 1.035207, which indicates that the value *could have been* rounded up. This is frustrating because the final result has 7 digits, which is more precision than the "period in space" shown. The other possibility is in what is considered a year. There is the tropical year which is essentially the calendar year 365.242 days. Or the sidereal year which is 365.256 days and the anomalistic year which is 365.2596 days. If one of these other definitions is necessary for the calculation -- for example they do mention sidereal time which is used for a [space-relative reference frame](https://en.wikipedia.org/wiki/Year#Sidereal.2C_tropical.2C_and_anomalistic_years) -- that would be enough slop to close up much of the difference. Ft = 27.2106*365.242/365.256 = 27.20956 tropical days This is closer to the 27.20986 days they give. This book [Space-Time Reference Systems](https://books.google.com/books?hl=en&lr=&id=FhKfr3ZPX6kC&oi=fnd&pg=PR5&ots=F45F0Af2YJ&sig=53Gk5ijEK95YnEaq7SDf61_DSXg#v=onepage&q&f=false) may give some insight. ![time](http://imageshack.com/a/img923/9909/abi0EQ.png) Sorry for this thread getting too pedantic but that always seems to happen when the result is homing in on a potentially solid match. This is the difference between using 27.209 days 27.212 days in a QBO fit. The value of 27.209 days results in a correlation improvement, which is marginally noticeable by eye. ![fit](http://imageshack.com/a/img922/3632/vM3ZBx.png)`

Related to the discussion about planetary periods in relation to the QBO periodicity and your comment in theENSO thread at 202. I now finally looked at the cycles. You claimed that 2.34 is an important moon period. How does this come about? Like if I calculate 365.242 *2.34 and divide this by the lunar month 27.321 I get 31.284. Not that a good match. Or is the 2.34 rounded?

By the way did you look also at other cycles in the terrestial planet system? Like Jupiter seems to have quite an impact on the sun (on a first glance it looks this cycle is related to the solar activity) so other planets could also play a role.

`Related to the discussion about planetary periods in relation to the QBO periodicity and your comment in the<a href="https://forum.azimuthproject.org/discussion/comment/15479/#Comment_15479">ENSO thread at 202.</a> I now finally looked at the cycles. You claimed that 2.34 is an important moon period. How does this come about? Like if I calculate 365.242 *2.34 and divide this by the lunar month 27.321 I get 31.284. Not that a good match. Or is the 2.34 rounded? By the way did you look also at other cycles in the terrestial planet system? Like Jupiter seems to have quite an impact on the sun (on a first glance it looks this cycle is related to the solar activity) so other planets could also play a role.`

nad asked:

The important periods are interferences of 27.212 days (the Draconic lunar month) with a strongly peaked

seasonalsignal.so in terms of frequencies where $f = 365.242/27.212 y^{-1}$, the list is

$ 1/f, 1/f-1, 1/f-2, ... , 1/f-13, 1/f -14, ... $

The frequency that is closest to $1 y^{-1}$ is $1/f-13$ = 0.422. This has a period of the reciprocal of this or 2.369 years. More in-depth math here

This is sometimes referred to as "nonlinear aliasing" or "natural aliasing" as it comes about from the nonlinear product of more than 1 frequency, leading to a spread of selected harmonics.

There is a good explanation of the effect in this meteorology book : "Mesoscale Dynamics", Yuh-Lang Lin, Cambridge University Press, 2007

That book references the AGW skeptic Roger Peilke, who wrote about the effect more recently here, "Mesoscale Meteorological Modeling", Roger A. Pielke Sr. Elsevier, 2013

It can also lead to noisy, spiky behavior in periodograms: Assessing statistical significance of periodogram peaks

So according to textbooks on meteorology, this effect can occur. And to top that off, this behavior is described in textbooks on

mesoscalephenomena, of which the QBO of stratospheric winds is a prime example at the extreme upper end of the scale.The moon and the sun have first-order effects on the earth's tides, and the other planets are second-order. So if the QBO is a tidal effect, which is what I am proposing, the other planets should probably be considered but only after the first-order effects are verified.

Thanks for the questions, as it provoked me to do a Google search on nonlinear aliasing. I knew about this from engineering experience but didn't realize how well known it is in climate modeling. Why is it not surprising that an AGW skeptic such as Pielke, or other AGW-denying QBO experts such Lindzen or Salby, would not pick up on this when it was right under their noses? Isn't that rich?

`nad asked: > "You claimed that 2.34 is an important moon period. How does this come about?" The important periods are interferences of 27.212 days (the Draconic lunar month) with a strongly peaked *seasonal* signal. so in terms of frequencies where $f = 365.242/27.212 y^{-1}$, the list is $ 1/f, 1/f-1, 1/f-2, ... , 1/f-13, 1/f -14, ... $ The frequency that is closest to $1 y^{-1}$ is $1/f-13$ = 0.422. This has a period of the reciprocal of this or 2.369 years. More in-depth math [here](http://contextearth.com/2015/11/17/the-math-of-seasonal-aliasing/) This is sometimes referred to as "nonlinear aliasing" or "natural aliasing" as it comes about from the nonlinear product of more than 1 frequency, leading to a spread of selected harmonics. There is a good explanation of the effect in this meteorology book : "Mesoscale Dynamics", Yuh-Lang Lin, Cambridge University Press, 2007 ![nonlinear](http://imageshack.com/a/img921/4153/t8MMRg.png) That book references the AGW skeptic Roger Peilke, who wrote about the effect more recently here, "Mesoscale Meteorological Modeling", Roger A. Pielke Sr. Elsevier, 2013 ![nonlinear2](http://imageshack.com/a/img923/8332/Kga45H.png) It can also lead to noisy, spiky behavior in periodograms: [Assessing statistical significance of periodogram peaks](http://arxiv.org/pdf/0711.0330.pdf) So according to textbooks on meteorology, this effect can occur. And to top that off, this behavior is described in textbooks on *mesoscale* phenomena, of which the QBO of stratospheric winds is a prime example at the extreme upper end of the scale. > "By the way did you look also at other cycles in the terrestial planet system?" The moon and the sun have first-order effects on the earth's tides, and the other planets are second-order. So if the QBO is a tidal effect, which is what I am proposing, the other planets should probably be considered but only after the first-order effects are verified. Thanks for the questions, as it provoked me to do a Google search on nonlinear aliasing. I knew about this from engineering experience but didn't realize how well known it is in climate modeling. Why is it not surprising that an AGW skeptic such as Pielke, or other AGW-denying QBO experts such Lindzen or Salby, would not pick up on this when it was right under their noses? Isn't that rich?`

Sorry. This makes no sense to me and your math explanation neither. What is y here?

`>The frequency that is closest to $1y^{−1}$ is 1/f−13 = 0.422. Sorry. This makes no sense to me and your math explanation neither. What is y here?`

.

`.`

Jim said:

Frequency is per year, so units are reciprocal of a year or 1 over y.

Why does the observed date of Easter wander from year to year? How often will Easter coincide closely with the first day of spring?

`Jim said: > "Sorry. This makes no sense to me and your math explanation neither. What is y here?" Frequency is per year, so units are reciprocal of a year or 1 over y. Why does the observed date of Easter wander from year to year? How often will Easter coincide closely with the first day of spring?`

Hint:

`Hint: ![easter](http://imageshack.com/a/img923/8150/zPV56S.png)`

Aha finally an image! I think I start to understand what you could mean. If easter would be on 1st Jan and full moon on day 27 this would be roughly the same as in this mathics.org code (Warning I just hacked that in fastly might be full of mistakes):

`Clear[modf,amod,ilist]; modf[n_]:=N[(n`

27.321582/365.241891-Floor[n27.321582/365.241891])365.241891]; a=0; amod={}; ilist={}; For[i=1,i<150,i=i+1,If[modf[i]<365.241891/12,amod=Append[amod,N[(i27.321582/365.241891-Floor[i27.321582/365.241891])365.241891]],ilist=Append[ilist,i]]];amod {27.321582,17.260257,7.19893199999999954,24.4591890000000031,14.3978640000000029,4.33653900000000231,21.5967960000000024,11.5354710000000022,1.47414600000000162,28.7957280000000058,18.7344030000000052,8.67307799999999006,25.9333350000000228} Sorry for the strange coding but I couldnt find non-integer Mod on mathics and I keep having trouble with the Mathematica notation of assignments (that is I was only able to assign values to amod here by appending...) So for the moon months 1, 14, 27, 41, 54, 67, 81, 94, 107, (108), 121, 134, 148 which seem interestingly all apart from 108 either 13 or 14 months distance away, you end up in the first solar month of the year where the integer part of the list amod says on which day and apart from some exceptions (here 108) this is "the" day which would be "easter". Moreover the day distance from Jan. 1st is sort of oscillating. Funnily before you had tried to explain the what you call "antialiasing" I had tried to do this type of calculation myself on a calculator, but then gave up and largely postponed the calculation. Your "hint" made me look at it again. So for the first year one has 27 days delay then the second 17 then 7, then 24, 14,4,22,12,1,(29),19,9,26.

Yes this looks indeed interesting!

I havent checked yet on this Jupiter-Sunactivity thing, but as I have the suspicion that sunwind may play a role Jupiter might be at least something which should be kept in mind.

`Aha finally an image! I think I start to understand what you could mean. If easter would be on 1st Jan and full moon on day 27 this would be roughly the same as in this mathics.org code (Warning I just hacked that in fastly might be full of mistakes): <code> Clear[modf,amod,ilist]; modf[n_]:=N[(n*27.321582/365.241891-Floor[n*27.321582/365.241891])*365.241891]; a=0; amod={}; ilist={}; For[i=1,i<150,i=i+1,If[modf[i]<365.241891/12,amod=Append[amod,N[(i*27.321582/365.241891-Floor[i*27.321582/365.241891])*365.241891]],ilist=Append[ilist,i]]]; amod {27.321582,17.260257,7.19893199999999954,24.4591890000000031,14.3978640000000029,4.33653900000000231,21.5967960000000024,11.5354710000000022,1.47414600000000162,28.7957280000000058,18.7344030000000052,8.67307799999999006,25.9333350000000228} </code> Sorry for the strange coding but I couldnt find non-integer Mod on mathics and I keep having trouble with the Mathematica notation of assignments (that is I was only able to assign values to amod here by appending...) So for the moon months 1, 14, 27, 41, 54, 67, 81, 94, 107, (108), 121, 134, 148 which seem interestingly all apart from 108 either 13 or 14 months distance away, you end up in the first solar month of the year where the integer part of the list amod says on which day and apart from some exceptions (here 108) this is "the" day which would be "easter". Moreover the day distance from Jan. 1st is sort of oscillating. Funnily before you had tried to explain the what you call "antialiasing" I had tried to do this type of calculation myself on a calculator, but then gave up and largely postponed the calculation. Your "hint" made me look at it again. So for the first year one has 27 days delay then the second 17 then 7, then 24, 14,4,22,12,1,(29),19,9,26. Yes this looks indeed interesting! >The moon and the sun have first-order effects on the earth's tides, and the other planets are second-order. I havent checked yet on this Jupiter-Sunactivity thing, but as I have the suspicion that sunwind may play a role Jupiter might be at least something which should be kept in mind.`

You are definitely right and a clever trick to use the

floor()function to strip off the integer part.Besides Easter Sunday, another example is calculating the oscillation of the Harvest Moon, which is the full moon nearest the first day of fall. There is a physical justification for this date, as it is a day that will provide the longest duration of light -- so that farmers can collect their harvest well in to the night.

And the connection to QBO is that lunisolar tides are strong at this time

The next step is to understand why the QBO period requires the Draconic lunar month (27.2122 days) and not the Tropical lunar month (27.3215 days).

Hint: symmetry

`You are definitely right and a clever trick to use the *floor()* function to strip off the integer part. Besides Easter Sunday, another example is calculating the oscillation of the Harvest Moon, which is the full moon nearest the first day of fall. There is a physical justification for this date, as it is a day that will provide the longest duration of light -- so that farmers can collect their harvest well in to the night. And the connection to QBO is that lunisolar tides are [strong at this time](http://earthsky.org/space/harvest-moon-2) > "Here’s what you can notice, if you live on a coastline. Watch for this full moon to bring along wide-ranging spring tides along ocean coastlines for several days following full moon. That is, high tides will climb extra high and the low tides will fall exceptionally low." The next step is to understand why the QBO period requires the Draconic lunar month (27.2122 days) and not the Tropical lunar month (27.3215 days). Hint: symmetry`

A tangentially related question is : How often is "Once in a Blue Moon"?

There is actually a statistically exact value for this!

`A tangentially related question is : How often is "Once in a Blue Moon"? https://youtu.be/3_TL6f5Fmtc There is actually a statistically exact value for this!`

Paul. In #178 you wrote:

'Jim said: "Sorry. This makes no sense to me and your math explanation neither. What is y here?"'.

Twern't me guv. Probable nad.

`Paul. In #178 you wrote: 'Jim said: "Sorry. This makes no sense to me and your math explanation neither. What is y here?"'. Twern't me guv. Probable nad.`

Sorry Jim,

optical illusion :-B fooled me with that empty comment

`Sorry Jim, optical illusion :-B fooled me with that empty comment`

Once in a blue moon chart.

Two definitions for once in a blue moon, but they apparently give the same resultant period.

pdf

`Once in a blue moon chart. ![bluemoon](http://imageshack.com/a/img921/6357/646UzN.png) Two definitions for once in a blue moon, but they apparently give the same resultant period. [pdf](https://digital.library.txstate.edu/bitstream/handle/10877/4033/fulltext.pdf)`

Sorry I dont see this. I took the tropical month because thats (if I havent misunderstood some explanations) the lunar period, as perceived from earth.

Concerning the above behaviour. Since the draconic month is rather close to the tropical month you'll also get a similar behaviour, just that the "hickups" are now at different places. The interesting point would be to see how long one stays in that pattern (were are the hickups), but today I couldnt do the computations even until 150 as in the above comment but only to 120 before running into a

`General::timeout: Timeout reached. $Aborted`

. May be I should say at this place: the computation time at mathics was sponsored by Angus Griffith, a young australian mathematician.Here you see the same for the draconic month:

`Clear[modf,aomod,iolist]; modof[n_]:=N[(n`

27.21222/365.241891-Floor[n27.21222/365.241891])365.241891]; aomod={}; iolist={}; For[i=1,i<120,i=i+1,If[modof[i]<365.241891/12,aomod=Append[aomod,N[(i27.21222/365.241891-Floor[i27.21222/365.241891])365.241891]],iolist=Append[iolist,i]]];aomod {27.21222,15.7291890000000012,4.24615800000000037,19.9753470000000016,8.49231600000000077,24.221505000000002,12.7384740000000011,1.25544299999999999,28.4676630000000024,16.9846320000000015}

and the months are:

1,14,27,41,54,68,81,94,(95),108,

If I understand correctly the blue moon is in a "hickupmonth" like for the tropical month the 108 and for the draconic the 95. What is a statistically exact value?

`>The next step is to understand why the QBO period requires the Draconic lunar month (27.2122 days) and not the Tropical lunar month (27.3215 days). Hint: symmetry Sorry I dont see this. I took the tropical month because thats (if I havent misunderstood some explanations) the lunar period, as perceived from earth. Concerning the above behaviour. Since the draconic month is rather close to the tropical month you'll also get a similar behaviour, just that the "hickups" are now at different places. The interesting point would be to see how long one stays in that pattern (were are the hickups), but today I couldnt do the computations even until 150 as in the above comment but only to 120 before running into a <code>General::timeout: Timeout reached. $Aborted</code>. May be I should say at this place: the computation time at mathics was sponsored by <a href="https://www.angusgriffith.com/">Angus Griffith</a>, a young australian mathematician. Here you see the same for the draconic month: <code>Clear[modf,aomod,iolist]; modof[n_]:=N[(n*27.21222/365.241891-Floor[n*27.21222/365.241891])*365.241891]; aomod={}; iolist={}; For[i=1,i<120,i=i+1,If[modof[i]<365.241891/12,aomod=Append[aomod,N[(i*27.21222/365.241891-Floor[i*27.21222/365.241891])*365.241891]],iolist=Append[iolist,i]]]; aomod {27.21222,15.7291890000000012,4.24615800000000037,19.9753470000000016,8.49231600000000077,24.221505000000002,12.7384740000000011,1.25544299999999999,28.4676630000000024,16.9846320000000015} </code> and the months are: 1,14,27,41,54,68,81,94,(95),108, >There is actually a statistically exact value for this! If I understand correctly the blue moon is in a "hickupmonth" like for the tropical month the 108 and for the draconic the 95. What is a statistically exact value?`

The reason one uses the Draconic month and not the Tropical month for explaining QBO is that the Tropical month will only give the return period of the phase of the moon for a

particular longitudinal location. But the QBO is a longitudinally invariant behavior, as it completely encircles the equator with a largely in-phase wind at any instant of time. So the Draconic month is the cycle that gives the maximum excursions between latitudunal nodes of the moon -- independent of longitude -- which is the necessary cyclic forcing for stimulating a cross-wind at the equator (the curl term following Laplace's tidal equations described here ).The QBO model is a perfect blend of intuitive physical reasoning and a precise mathematical formulation, just like the theory of ocean tides. I don't understand why this has been missed by Lindzen and his followers over the years. All I know is that it will be difficult to unwind a "just-so" explanation offered up by Lindzen for QBO that has been reinforced into a questionable consensus over the course of time.

Once in a blue moon statistically occurs every 2.715 years, with the Tropical month beating with a yearly cycle. This uses the Tropical month because it is a

human observableevent, meaning that it occurs at the same longitude over time. Same reason that Easter uses the Tropical month as it is an observation of the Full Moon in the Middle East region, not at an arbitrary longitude.`The reason one uses the Draconic month and not the Tropical month for explaining QBO is that the Tropical month will only give the return period of the phase of the moon for a *particular longitudinal location*. But the QBO is a longitudinally invariant behavior, as it completely encircles the equator with a largely in-phase wind at any instant of time. So the Draconic month is the cycle that gives the maximum excursions between latitudunal nodes of the moon -- independent of longitude -- which is the necessary cyclic forcing for stimulating a cross-wind at the equator (the curl term following Laplace's tidal equations [described here](http://contextearth.com/2016/07/04/alternate-simplification-of-qbo-from-laplaces-tidal-equations/) ). The QBO model is a perfect blend of intuitive physical reasoning and a precise mathematical formulation, just like the [theory of ocean tides](https://en.wikipedia.org/wiki/Theory_of_tides). I don't understand why this has been missed by Lindzen and his followers over the years. All I know is that it will be difficult to unwind a "just-so" explanation offered up by Lindzen for QBO that has been reinforced into a questionable consensus over the course of time. > "What is a statistically exact value?" Once in a blue moon statistically occurs every 2.715 years, with the Tropical month beating with a yearly cycle. This uses the Tropical month because it is a *human observable* event, meaning that it occurs at the same longitude over time. Same reason that Easter uses the Tropical month as it is an observation of the Full Moon in the Middle East region, not at an arbitrary longitude.`

There has been a recent push to understand the lunar-climate connection by none other than NASA JPL.

A former researcher at JPL, Claire Perigaud wrote a proposal a few years ago that apparently never got funded: ftp://ftp.cerfacs.fr/pub/globc/exchanges/cassou/GOASIS/Fermat_2009.pdf

In that research proposal, there is a clear indication that they understood the lunar origin of the equatorial climate forcing:

The 13.606 number is half the period of the Draconic cycle (27.212 days) and one can infer that this is the same lunar acceleration vector applied to Laplace's tidal equations in the gravity-forced QBO model.

Perigaud is maintaining a website up called http://www.MoonClimate.org/ -- which doesn't look very active. I think this was started because they were committed to following up on the ideas that JPL would not fund

They apparently submitted an article to Nature Climate Change called "Earth-Moon-Sun alignments influencing tropical climate events" in 2011 that was rejected. No sign of any article with that title or those co-authors when I googled.

`There has been a recent push to understand the lunar-climate connection by none other than NASA JPL. A former researcher at JPL, Claire Perigaud wrote a proposal a few years ago that apparently never got funded: ftp://ftp.cerfacs.fr/pub/globc/exchanges/cassou/GOASIS/Fermat_2009.pdf In that research proposal, there is a clear indication that they understood the lunar origin of the equatorial climate forcing: > ![drac](http://imageshack.com/a/img923/934/iMWso1.png) The 13.606 number is half the period of the Draconic cycle (27.212 days) and one can infer that this is the same lunar acceleration vector applied to Laplace's tidal equations in the [gravity-forced QBO model](http://contextearth.com/2016/08/23/qbo-model-final-stretch/). Perigaud is maintaining a website up called http://www.MoonClimate.org/ -- which doesn't look very active. I think this was started because they were committed to following up on the ideas that JPL would not fund > ![proposal](http://imageshack.com/a/img922/8351/N9by9B.png) They apparently [submitted an article](http://www.moonclimate.org/docs/2011_NCC_CoverLetter.pdf) to Nature Climate Change called "Earth-Moon-Sun alignments influencing tropical climate events" in 2011 that was rejected. No sign of any article with that title or those co-authors when I googled.`

New paper in

Journal of Climateavailable at http://journals.ametsoc.org/doi/full/10.1175/JCLI-D-16-0122.1:H. M. Christensen, J. Berner, D. R. B. Coleman, T. N. Palmer, ``Stochastic Parameterization and El Niño–Southern Oscillation''

`New paper in <em>Journal of Climate</em> available at http://journals.ametsoc.org/doi/full/10.1175/JCLI-D-16-0122.1: H. M. Christensen, J. Berner, D. R. B. Coleman, T. N. Palmer, ``Stochastic Parameterization and El Niño–Southern Oscillation''`

My general question is what's with the wind as a driving force?

Could not the wind be the

resultof the pressure differential caused by the ENSO dipole? Wind is always the result of a pressure differential from what I understand.But if the wind is the forcing mechanism, what ultimately forces the wind? They will say it is random fluctuations apparently. And that is the premise of that paper.

QBO is also a wind, yet I have shown that is likely due to lunisolar forcing. Yet I believe most think that QBO is driven by ENSO or by gravity waves in the lower troposphere. That is a circular reasoning logic flaw if you ask me. Wind => ENSO => QBO Wind ????

So a much more plausible and parsimonious theory is to assume that ENSO and QBO are both driven by lunisolar forcing. But because ENSO is a longitudinally constrained effect whereas QBO is world-wide, the forcing factors are not precisely in phase. Only when the lunar periods match up transiently do you see some synchronization.

Those are the questions and ideas that I have that no one can answer.

They probably can't answer this odd finding either which I added to another forum thread this morning:

https://forum.azimuthproject.org/discussion/comment/15688/#Comment_15688

Interesting how that the tidal gauge SLH readings can anticipate ENSO by two years?

One thing I noticed attending the AGU is a lack of genuine intellectual curiosity compared to other scientific disciplines that I have been involved in. Like this paper, they generate way too many wordy narratives that are just too much of a snooze. Much more interesting to do apply fresh types of analysis to see what you can find. Just my opinion.

`My general question is what's with the wind as a driving force? Could not the wind be the *result* of the pressure differential caused by the ENSO dipole? Wind is always the result of a pressure differential from what I understand. But if the wind is the forcing mechanism, what ultimately forces the wind? They will say it is random fluctuations apparently. And that is the premise of that paper. QBO is also a wind, yet I have shown that is likely due to lunisolar forcing. Yet I believe most think that QBO is driven by ENSO or by gravity waves in the lower troposphere. That is a circular reasoning logic flaw if you ask me. Wind => ENSO => QBO Wind ???? So a much more plausible and parsimonious theory is to assume that ENSO and QBO are both driven by lunisolar forcing. But because ENSO is a longitudinally constrained effect whereas QBO is world-wide, the forcing factors are not precisely in phase. Only when the lunar periods match up transiently do you see some synchronization. Those are the questions and ideas that I have that no one can answer. They probably can't answer this odd finding either which I added to another forum thread this morning: https://forum.azimuthproject.org/discussion/comment/15688/#Comment_15688 Interesting how that the tidal gauge SLH readings can anticipate ENSO by two years? ![](http://imageshack.com/a/img922/5033/PFmH57.png) One thing I noticed attending the AGU is a lack of genuine intellectual curiosity compared to other scientific disciplines that I have been involved in. Like this paper, they generate way too many wordy narratives that are just too much of a snooze. Much more interesting to do apply fresh types of analysis to see what you can find. Just my opinion.`