It looks like you're new here. If you want to get involved, click one of these buttons!

- All Categories 2.2K
- Applied Category Theory Course 338
- Applied Category Theory Exercises 149
- Applied Category Theory Discussion Groups 48
- Applied Category Theory Formula Examples 15
- Chat 470
- Azimuth Code Project 107
- News and Information 145
- Azimuth Blog 148
- Azimuth Forum 29
- Azimuth Project 190
- - Strategy 109
- - Conventions and Policies 21
- - Questions 43
- Azimuth Wiki 707
- - Latest Changes 699
- - - Action 14
- - - Biodiversity 8
- - - Books 2
- - - Carbon 9
- - - Computational methods 38
- - - Climate 53
- - - Earth science 23
- - - Ecology 43
- - - Energy 29
- - - Experiments 30
- - - Geoengineering 0
- - - Mathematical methods 69
- - - Meta 9
- - - Methodology 16
- - - Natural resources 7
- - - Oceans 4
- - - Organizations 34
- - - People 6
- - - Publishing 4
- - - Reports 3
- - - Software 20
- - - Statistical methods 2
- - - Sustainability 4
- - - Things to do 2
- - - Visualisation 1
- General 38

Options

The general approach that we seem to agree on for better predictions of ENSO is to try to find patterns or correlations amongst various data sets.

This discussion topic is meant to collect information and analyses for the known correlation between the Quasi-Biennial Oscillation (QBO) of stratospheric wind speeds and ENSO, see [1,2,3,4,5] refs at the bottom of this post.

What I find intriguing about this correlation is that the QBO periodicity is much stronger than the erratic ENSO periodicity. One would think this would have implications for predictability of ENSO -- since a stronger period is more predictable than a weaker, and since QBO is thought to drive ENSO in some way, we may be able to isolate a component of the time series. And if this forcing is strongly periodic, it may be used to project into the future.

The reason QBO is thought to be a driver is that the QBO winds downwell over the Pacific Ocean as they cycle. One can see this in the speed vs altitude plots, where a higher atmospheric pressure corresponds to a lower altitude.

This tends to push on the Pacific Ocean surface with the same cycle, causing water to pile up in the windward direction periodically.

The QBO has an average period of about 28 months since data collection started in 1952 (data link), which explains the quasi-biennial aspect. However the cycles show a measurable amount of jitter, which is a fluctuation in a given cycle's period. One question to consider is whether this jitter is random or shows an extra periodicity, which may be due to tidal beating (?) see http://contextearth.com/2014/06/17/the-qbom/ for some evidence that I collected.

The recent claim in [5] is that the ENSO and QBO time series have almost aligned over a recent 5-year interval. See the figure below.

That near-correlation could be just happenstance as in general the waveforms don't align, with the ENSO being much more erratic and the QBO showing a stricter periodicity. The other characteristic of QBO is that the peaks are generally broader and more flat-topped than the valleys. But now that I look at what others have plotted, the asymmetry is not quite as apparent (from here)

Yet, this asymmetry is not generally seen in ENSO.

I have been analyzing using the fundamental frequency of QBO as a driver to a nonlinear DiffEq equation model of ENSO and the results are intriguing enough that I have pursued this over several ENSO data sets such as SOI and ENSO proxies. Start here and follow the links backwards. The general finding is that periods close to the QBO period appear to be strong candidates for a forcing function, but the non-linear function causes a transformation that the response is much more erratic.

That's the general idea and I will post more analyses related to QBO below this entry.

Paul Pukite

[1] N. Calvo, M. A. Giorgetta, R. Garcia‐Herrera, and E. Manzini, “Nonlinearity of the combined warm ENSO and QBO effects on the Northern Hemisphere polar vortex in MAECHAM5 simulations,” Journal of Geophysical Research: Atmospheres (1984–2012), vol. 114, no. D13, 2009.

[2] M. Geller and W. Yuan, “QBO-ENSO Connections and Influence on the Tropical Cold Point Tropopause,” presented at the AGU Fall Meeting Abstracts, 2011, vol. 1, p. 03.

[3] S. Liess and M. A. Geller, “On the relationship between QBO and distribution of tropical deep convection,” Journal of Geophysical Research: Atmospheres (1984–2012), vol. 117, no. D3, 2012.

[4] W. M. Gray, J. D. Sheaffer, and J. A. Knaff, “Inﬂuence of the stratospheric QBO on ENSO variability,” J. Meteor: Soc. Japan, vol. 70, pp. 975–995, 1992.

[5] J. L. Neu, T. Flury, G. L. Manney, M. L. Santee, N. J. Livesey, and J. Worden, “Tropospheric ozone variations governed by changes in stratospheric circulation,” Nature Geoscience, vol. 7, no. 5, pp. 340–344, 2014.

P.S. I almost finished writing this a few days ago but I inadvertently hit the back button. I had to wait a few days to get back my motivation.

## Comments

Hello Paul

No matter what the cause for periodicity you might model that by cos and sin. However I like to bring to your attention that this is a better form

Exp[d((x-a)^2+(x-b)^2)]

Cos[e(x-a)]Cos[e(x-b)]the more variable you add, the more the terms in the norm in the exp and more Cos terms.

The above gives flexible modelling with more control.

`Hello Paul No matter what the cause for periodicity you might model that by cos and sin. However I like to bring to your attention that this is a better form Exp[d((x-a)^2+(x-b)^2)]*Cos[e(x-a)]*Cos[e(x-b)] the more variable you add, the more the terms in the norm in the exp and more Cos terms. The above gives flexible modelling with more control.`

Yet cos(A)cos(B) = 1/2*(cos(A-B) - cos(A+B))

So this expands to a linear combination, but constrained by a common scaling and then damped similar to a wavelet?

`Yet cos(A)cos(B) = 1/2*(cos(A-B) - cos(A+B)) So this expands to a linear combination, but constrained by a common scaling and then damped similar to a wavelet?`

Is the ENSO/QBO index from literature reference [5]? Usually I have no access to nature articles. The two look indeed correlated. Is there a longer time scale? The QBO periodicity looks to me biannual with some irregularities, like a signal thats sometimes "out of sync" with a biannual forcing. Where a rather likely forcing would be temperature (via radiation). (If we assume that the temperature data is not yet rotten enough for showing complete bogus). Moreover it looks to me as if there is also an annual oscillation contained in the signal, like there seems to be always a little "bump" on the downslide between peaks. This bump is sometimes rather high ("a small peak", an "extrasystole") and seems to lead in that case to a lagging behind of the biannual signal peak. This happens slightly before the year 63 (almost nonvisible) , before 65 the "extrasystole" is comparibly big leading to the lag between 65 and 67 the small peak behind 67 would have been the original 67 peak if there wouldn't have been the lagging, in 69 the signal is again in sync, before 77 an extrasystole is again leading to a lagging, again before 79, leading to a lagging, the small peak at 81 would again be the peak at 81 if not for the lagging, here the signal has now a 90 degrees phaseshift, i.e. the beat is quite out of sync, it catches though up again in 85, in 87 small extrasystole with small lagging, in 89 big extrasystole, signal again out quite out of sync, catches up in 97, extrasystole before 01, 03 and 05, out of sync until it "catches up again" in 2013. If there is no extrasystole in 2014 I conjecture that the QBO index will peak again in summer 2015.

`Is the ENSO/QBO index from literature reference [5]? Usually I have no access to nature articles. The two look indeed correlated. Is there a longer time scale? The QBO periodicity looks to me biannual with some irregularities, like a signal thats sometimes "out of sync" with a biannual forcing. Where a rather likely forcing would be <a href="http://forum.azimuthproject.org/discussion/1375/quasibiennial-oscillation/?Focus=11084#Comment_11084">temperature</a> (via radiation). (If we assume that the temperature data is <a href="http://www.randform.org/blog/?p=5676">not yet rotten enough for showing complete bogus</a>). Moreover it looks to me as if there is also an annual oscillation contained in the signal, like there seems to be always a little "bump" on the downslide between peaks. This bump is sometimes rather high ("a small peak", an "extrasystole") and seems to lead in that case to a lagging behind of the biannual signal peak. This happens slightly before the year 63 (almost nonvisible) , before 65 the "extrasystole" is comparibly big leading to the lag between 65 and 67 the small peak behind 67 would have been the original 67 peak if there wouldn't have been the lagging, in 69 the signal is again in sync, before 77 an extrasystole is again leading to a lagging, again before 79, leading to a lagging, the small peak at 81 would again be the peak at 81 if not for the lagging, here the signal has now a 90 degrees phaseshift, i.e. the beat is quite out of sync, it catches though up again in 85, in 87 small extrasystole with small lagging, in 89 big extrasystole, signal again out quite out of sync, catches up in 97, extrasystole before 01, 03 and 05, out of sync until it "catches up again" in 2013. If there is no extrasystole in 2014 I conjecture that the QBO index will peak again in summer 2015.`

I would say use the Wavelet technology, the math has been worked to death and there are large number of them as opposed guessing our own by looking at plots.

But that is my thinking not necessarily a guideline

D

`> but constrained by a common scaling and then damped similar to a wavelet? I would say use the Wavelet technology, the math has been worked to death and there are large number of them as opposed guessing our own by looking at plots. But that is my thinking not necessarily a guideline D`

"Is the ENSO/QBO index from literature reference [5]? Usually I have no access to nature articles."

nad, Yes that is from reference 5. I can dig this up for you when I get a chance.

I agree looking at the waveform in the third chart is a lot like trying to read an EKG, where the extra little features and the exact timing is crucial, i.e long QT, etc.

The QBO details also change with altitude but that maintains the overall mean period, as it must otherwise the behavior would gradually get out of phase.

I have a link for QBO data from the official repository as given in the 6th paragraph, but I don't know what kind of filtering was involved in getting the third chart. When I plot from the source data, it ends up looking more like a noisy full-wave rectified sine wave, with the valleys sharper than the peaks, as shown below.

That's why I was intrigued by this other one. As nad observed, I was also curious about all the “extrasystole” features that emerged. The main peaks don't really line up with my QBO chart and so perhaps the one I plotted is a derivative of QBO? As long as the noise is not too great, derivatives are a great way of revealing extra features, as shown with edge detection algorithms. Or it more likely is just QBO data taken at a different altitude.

The one I plotted was from the lowest altitude, which I figured would have the greatest influence on ENSO. By using the data from the handful of altitudes, it may be possible to further discriminate these features. From what I understand, some QBO research uses the different altitudes to arrive at an empirical orthogonal function ( EOF ) representation of the data. I have to admit that I don't completely "get" EOFs other than assuming that they are similar to a Fourier series of various fixed frequency sine waves and to principle components analysis -- but with the sinusoids replaced with functions that are combinations of other data sources. I am OK with this as long as the other sources are isolated as in a multiple regression, but when they get combined I start losing intuition. But as I said, my understanding is not complete as I have yet to experiment with EOFs on my own and I use multiple regression spectral decomposition more than PCAs. I hope someone can help clarify if this is a possible route.

The other point by nad is important, and that is whether we could actually pick up a pattern and predict the extrasystole peaks in the future. If you can't do this for QBO, then it must be even harder for ENSO and El Ninos, where the fundamental frequency is even less well-defined.

Incidentally, the way I came across the Ref 5 article was thanks to a fierce climate skeptic who seemed a little confused. After he found out that I was working on an ENSO & QBO correlation, he dug up this article and acted like it was no big deal and that anybody could see the connection. And then he wrapped it up by saying that I was wasting my time pursuing this angle (?!?!)

I don't know what's up with these climate skeptical people, but all I can say is Thank You.

`"Is the ENSO/QBO index from literature reference [5]? Usually I have no access to nature articles." nad, Yes that is from reference 5. I can dig this up for you when I get a chance. I agree looking at the waveform in the third chart is a lot like trying to read an EKG, where the extra little features and the exact timing is crucial, i.e long QT, etc. The QBO details also change with altitude but that maintains the overall mean period, as it must otherwise the behavior would gradually get out of phase. I have a link for QBO data from the official repository as given in the 6th paragraph, but I don't know what kind of filtering was involved in getting the third chart. When I plot from the source data, it ends up looking more like a noisy full-wave rectified sine wave, with the valleys sharper than the peaks, as shown below. ![QBO plot](http://imageshack.com/a/img905/7240/F3Gam7.gif) That's why I was intrigued by this other one. As nad observed, I was also curious about all the “extrasystole” features that emerged. The main peaks don't really line up with my QBO chart and so perhaps the one I plotted is a derivative of QBO? As long as the noise is not too great, derivatives are a great way of revealing extra features, as shown with edge detection algorithms. Or it more likely is just QBO data taken at a different altitude. The one I plotted was from the lowest altitude, which I figured would have the greatest influence on ENSO. By using the data from the handful of altitudes, it may be possible to further discriminate these features. From what I understand, some QBO research uses the different altitudes to arrive at an empirical orthogonal function ( [EOF](http://en.wikipedia.org/wiki/Empirical_orthogonal_functions) ) representation of the data. I have to admit that I don't completely "get" EOFs other than assuming that they are similar to a Fourier series of various fixed frequency sine waves and to principle components analysis -- but with the sinusoids replaced with functions that are combinations of other data sources. I am OK with this as long as the other sources are isolated as in a multiple regression, but when they get combined I start losing intuition. But as I said, my understanding is not complete as I have yet to experiment with EOFs on my own and I use multiple regression spectral decomposition more than PCAs. I hope someone can help clarify if this is a possible route. The other point by nad is important, and that is whether we could actually pick up a pattern and predict the extrasystole peaks in the future. If you can't do this for QBO, then it must be even harder for ENSO and El Ninos, where the fundamental frequency is even less well-defined. --- Incidentally, the way I came across the Ref 5 article was thanks to a fierce climate skeptic who seemed a little confused. After he found out that I was working on an ENSO & QBO correlation, he dug up this article and acted like it was no big deal and that anybody could see the connection. And then he wrapped it up by saying that I was wasting my time pursuing this angle (?!?!) I don't know what's up with these climate skeptical people, but all I can say is Thank You.`

As I said I think it looks biannual. Moreover in other comments I suspected that since it is so regular the biannuity seems to indicate a planetary origin, which alters the radiation (and thus the temperature) in a biannual fashion. I have though no idea what this planetary motion should be. One thing one should look at in the context of biannuity though more carefully is surely that the earth in its orbit around the sun appears with its axis a bit like a particle with a spin (like around a nucleus). On the other hand if there would be some kind of biannuity in solar irradiance then this would probably be widely known.

`>The other point by nad is important, and that is whether we could actually pick up a pattern and predict the extrasystole peaks in the future. As I said I think it looks biannual. Moreover in other comments I suspected that since it is so regular the biannuity seems to indicate a planetary origin, which alters the radiation (and thus the temperature) in a biannual fashion. I have though no idea what this planetary motion should be. One thing one should look at in the context of biannuity though more carefully is surely that the earth in its orbit around the sun appears with its axis a bit like a particle with a <a href="http://en.wikipedia.org/wiki/Plate_trick">spin</a> (like around a nucleus). On the other hand if there would be some kind of biannuity in solar irradiance then this would probably be widely known.`

I said:

With that I mean the pattern. In particular I don't share the opinion about other periods. But actually you wanted a comment on the height of the extrasystole. For this one one would probably need to look at all sorts of climate phenomena. According to your correlation image from [5] interestingly ENSO precedes QBO (intuitively I could have guessed the other way around), so this could be one component

`I said: >As I said I think it looks biannual. With that I mean the pattern. In particular I don't share the opinion about other periods. But actually you wanted a comment on the height of the extrasystole. For this one one would probably need to look at all sorts of climate phenomena. According to your correlation image from [5] interestingly ENSO precedes QBO (intuitively I could have guessed the other way around), so this could be one component`

Thanks nad for the comments

One issue that I should clarify is that if some set of EOFs was created to model QBO, and these were based on some other phenomena or involved the QBO at different altitudes, it wouldn't really help much with an El Nino projection. Unless a fundamental equation is formulated or a simulation executed based on physical principles, there is no automatic way to extrapolate the fitted EOFs into the future. For example, if the EOFs are sinusoids no problem, but if the EOF is say, monthly rainfall in Wisconsin, it wouldn't help much. Perhaps that is being too pedantic on my part, but I have to occasionally remind myself of this argument to stay on the objective path.

You also mentioned the possibility of possible planetary effects. When I attempted a machine learning fit to QBO, one extended trial ended up like this

If you look at the frequency in one of the Fourier series compositions along the Pareto front, a frequency of 77.7 radians/year lines up with the lunar month synodic period of 29.5 days. And the 153 rad/yr is the half-month cycle. That could be just coincidental and why it is cool to get other people to cast skeptical eyes to the results.

`Thanks nad for the comments One issue that I should clarify is that if some set of EOFs was created to model QBO, and these were based on some other phenomena or involved the QBO at different altitudes, it wouldn't really help much with an El Nino projection. Unless a fundamental equation is formulated or a simulation executed based on physical principles, there is no automatic way to extrapolate the fitted EOFs into the future. For example, if the EOFs are sinusoids no problem, but if the EOF is say, monthly rainfall in Wisconsin, it wouldn't help much. Perhaps that is being too pedantic on my part, but I have to occasionally remind myself of this argument to stay on the objective path. You also mentioned the possibility of possible planetary effects. When I attempted a machine learning fit to QBO, one extended trial ended up like this ![qbofit](http://imageshack.com/a/img855/7435/femn.gif) If you look at the frequency in one of the Fourier series compositions along the Pareto front, a frequency of 77.7 radians/year lines up with the lunar month synodic period of 29.5 days. And the 153 rad/yr is the half-month cycle. That could be just coincidental and why it is cool to get other people to cast skeptical eyes to the results.`

???Shouldn't lead the term: t

cos(6.192 + 153t) to an increasing signal? This isn't visible in the graphics (but then I don't know what the graphics is supposed to show) and also not visible in the QBO (at least from what I have seen sofar)`>When I attempted a machine learning fit to QBO, one extended trial ended up like this ???Shouldn't lead the term: t*cos(6.192 + 153*t) to an increasing signal? This isn't visible in the graphics (but then I don't know what the graphics is supposed to show) and also not visible in the QBO (at least from what I have seen sofar)`

nad asks

That is true, but the value of t ranges from 1950 to 2000, so that this amplification is very slight, like 2 to 3% of the signal amplitude. I am not sure if this can be perceived visually amongst the fluctuations, but that is what the tool is finding.

So I have no control over the machine learning fit that Eureqa executes, which is good and bad I suppose. Good because it doesn't add any human bias, but bad in that there is no physics involved at this level. For example, Eureqa is not going to say that those numbers are related to lunar monthly cycles, but it is up to the human to figure out the physical mechanisms and decide whether something is just a coincidence.

I am sure that this has some relevance to the recent Azimuth blog post on models and machine learning.

`nad asks "Shouldn’t lead the term: tcos(6.192 + 153t) to an increasing signal?" That is true, but the value of t ranges from 1950 to 2000, so that this amplification is very slight, like 2 to 3% of the signal amplitude. I am not sure if this can be perceived visually amongst the fluctuations, but that is what the tool is finding. So I have no control over the machine learning fit that Eureqa executes, which is good and bad I suppose. Good because it doesn't add any human bias, but bad in that there is no physics involved at this level. For example, Eureqa is not going to say that those numbers are related to lunar monthly cycles, but it is up to the human to figure out the physical mechanisms and decide whether something is just a coincidence. I am sure that this has some relevance to the recent Azimuth blog post on models and machine learning.`

Aha. I see it seems the full text of the "solution" is printed again below. I was originally reading that term off the blue bar text (which doesn't show the whole term). That solution carries a term 1984 which quite dominates the solution, so 2000-1950 = 50, which is 50/1984 =roughly= 1/40 = roughly = 0.025, thats what you mean with 2-3% signal amplitude. But if the drawing on the right is supposed to be the graph of the "solution" then it seems the labeling on the y-axis is not only off by a simple factor, but eventually by some nonlinear scale. That is I wonder where that particular form of modulation should come from. Like assume maximal amplitude for all other terms 40.9+50+13.22 =roughly=110 =roughly=100 and 100/1984=roughly=0.05. I.e. the modulation of the signal would be in a linear scale in the range of about 5%, which it isn't in the drawing. I won't though exclude that I miscalculated something, since I am currently only half awake and in a hurry which is not the optimal situation for mental arithmetics and human bias prevention.

`>That is true, but the value of t ranges from 1950 to 2000, so that this amplification is very slight, like 2 to 3% of the signal amplitude. I am not sure if this can be perceived visually amongst the fluctuations, but that is what the tool is finding. Aha. I see it seems the full text of the "solution" is printed again below. I was originally reading that term off the blue bar text (which doesn't show the whole term). That solution carries a term 1984 which quite dominates the solution, so 2000-1950 = 50, which is 50/1984 =roughly= 1/40 = roughly = 0.025, thats what you mean with 2-3% signal amplitude. But if the drawing on the right is supposed to be the graph of the "solution" then it seems the labeling on the y-axis is not only off by a simple factor, but eventually by some nonlinear scale. That is I wonder where that particular form of modulation should come from. Like assume maximal amplitude for all other terms 40.9+50+13.22 =roughly=110 =roughly=100 and 100/1984=roughly=0.05. I.e. the modulation of the signal would be in a linear scale in the range of about 5%, which it isn't in the drawing. I won't though exclude that I miscalculated something, since I am currently only half awake and in a hurry which is not the optimal situation for mental arithmetics and human bias prevention.`

Wow, I determined that the first 1-D time-series chart of QBO is at a higher altitude than the one one I plotted at #6.

I was intrigued by nad's suggestion that it had some interesting structure. It also seemed to have a greater signal-to-noise ratio as the periods appear stronger and more distinct than the lower altitude results.

So I ran the higher-altitude QBO on Eureqa to find the Fourier components:

I looked for the lowest-error/minimum complexity representation on the Pareto curve, highlighted in blue on the left columns and red in the right columns.

There is a main frequency of 2.665 rads/yr with symmetric sidelobes at 2.487 rads/yr and 2.841 rads/year. The 2.665 corresponds to the mean period of the QBO = 28 months. The sidelobes are weaker.

Also a pair of high-frequency components are generated at 153 rads/yr and 154 yrs/yr. These are approximately equal in amplitude and correspond to about 1/2-month period each (a 1/2 month tidal factor ?). But since they are close in frequency, we should be able to take the difference (154-153)=1 rads/yr and use that as an envelope. Look at the Eureqa results in the following and you can see how the machine learning actually started with 1 rad/yr and then switches over to the higher frequency representation, since that must reduce the error in some incremental fashion.

This is where it gets neat, IMO.

I decided to apply the 2.665, 2.487, 2.841, and 1 rads/yr components as forcing factors in my SOM Mathieu differential equation evaluation, most recently evaluated here and specifically for the SOI set, which overlaps the QBOM time span.

Recall further up in this thread where I observed that the QBO is quite periodic in its waveform, while the ENSO is highly erratic. In the case of SOI specifically, the measure shows the same erratic waveform.

In the solution below, I left the Mathieu modulation as before and chose a restricted time interval for the SOI, yet I still backcasted 20 years prior to when the actual QBO data was collected. Note that I did modify the

amplitudesof the factors to improve the fit, so that the lower sidelobe is stronger than the main. The correlation coefficient is 0.63, which isn't extremely high, but the general agreement seems quite good to me.The weak fits occur at the start of the 1990's and 1980's and around 1964. (Incidentally, these do correspond to significant volcanic events, Pinatubo 1991, El Chicon 1982, and Agung 1963)

The idea here is that the non-linear Mathieu modulation (LHS of DiffEq) is transforming the regular QBO forcing (RHS of DiffEq) into something much more erratic. The Mathieu modulation is rationalized as a low-order effect in the sloshing dynamics of the equatorial Pacific Ocean.

The process now is to hammer on this formulation to determine the likelihood of this solution being statistically significant. So the questions to ask are (1) is inadvertent bias being introduced to guide the solution? (2) how much can the coefficients be tweaked without being accused of over-fitting? (3) is this a case of over-fitting as it is? and (4) the big question, justifying the math as plausible physics. In other words, am I fooling myself by going down this path?

As far as I know there is only one way to improve the fit, and that is to use a brute force differential evolution search as suggested by Dara.

`Wow, I determined that the first 1-D time-series chart of QBO is at a higher altitude than the one one I plotted at #6. I was intrigued by nad's suggestion that it had some interesting structure. It also seemed to have a greater signal-to-noise ratio as the periods appear stronger and more distinct than the lower altitude results. So I ran the higher-altitude QBO on Eureqa to find the Fourier components: ![QBO](http://imageshack.com/a/img912/3035/0bHY3I.gif) I looked for the lowest-error/minimum complexity representation on the Pareto curve, highlighted in blue on the left columns and red in the right columns. There is a main frequency of 2.665 rads/yr with symmetric sidelobes at 2.487 rads/yr and 2.841 rads/year. The 2.665 corresponds to the mean period of the QBO = 28 months. The sidelobes are weaker. Also a pair of high-frequency components are generated at 153 rads/yr and 154 yrs/yr. These are approximately equal in amplitude and correspond to about 1/2-month period each (a 1/2 month tidal factor ?). But since they are close in frequency, we should be able to take the difference (154-153)=1 rads/yr and use that as an envelope. Look at the Eureqa results in the following and you can see how the machine learning actually started with 1 rad/yr and then switches over to the higher frequency representation, since that must reduce the error in some incremental fashion. ![QBO alias](http://imageshack.com/a/img908/4517/TPj4Fs.png) This is where it gets neat, IMO. I decided to apply the 2.665, 2.487, 2.841, and 1 rads/yr components as forcing factors in my SOM Mathieu differential equation evaluation, most recently evaluated [here](http://forum.azimuthproject.org/discussion/1451/enso-proxy-records) and specifically for the SOI set, which overlaps the QBOM time span. Recall further up in this thread where I observed that the QBO is quite periodic in its waveform, while the ENSO is highly erratic. In the case of SOI specifically, the measure shows the same erratic waveform. In the solution below, I left the Mathieu modulation as before and chose a restricted time interval for the SOI, yet I still backcasted 20 years prior to when the actual QBO data was collected. Note that I did modify the *amplitudes* of the factors to improve the fit, so that the lower sidelobe is stronger than the main. The correlation coefficient is 0.63, which isn't extremely high, but the general agreement seems quite good to me. The weak fits occur at the start of the 1990's and 1980's and around 1964. (Incidentally, these do correspond to significant volcanic events, Pinatubo 1991, El Chicon 1982, and Agung 1963) ![SOI](http://imageshack.com/a/img661/5201/tOQJck.gif) The idea here is that the non-linear Mathieu modulation (LHS of DiffEq) is transforming the regular QBO forcing (RHS of DiffEq) into something much more erratic. The Mathieu modulation is rationalized as a low-order effect in the sloshing dynamics of the equatorial Pacific Ocean. The process now is to hammer on this formulation to determine the likelihood of this solution being statistically significant. So the questions to ask are (1) is inadvertent bias being introduced to guide the solution? (2) how much can the coefficients be tweaked without being accused of over-fitting? (3) is this a case of over-fitting as it is? and (4) the big question, justifying the math as plausible physics. In other words, am I fooling myself by going down this path? As far as I know there is only one way to improve the fit, and that is to use a brute force differential evolution search as suggested by Dara.`

Paul, sorry but I think this problem hasn't been adressed adequately.

I wrote in here:

OK this assertion sounds quite as been sitting on the crackpotty. So let me please outline the super vague "reasoning" which is behind this. The earth moves through the heliospheric current sheet. The sun wind interacts with the earth atmosphere mostly at the poles. The sun wind particle stream has though different directions depending on polarity. The particle stream influences cloud formation and could thus among others in principle change the albedo and temperature. I have no idea how big those influences are. Probably quite small. The magnet field of the sun seems rather unregular, but eventually some accumulated effect due to the rotation of the sun which shapes the current sheet together with the movement of the earth in that current sheet have a two year periodicity. It is unlikely but at the moment I can't fully exclude this possibility, and I have no other immediate counter arguments at hand so thats why I said one would probably need to look at this i.e. one would at least need to find some excluding arguments.

`Paul, sorry but I think this <a href="http://forum.azimuthproject.org/discussion/1471/qbo-and-enso/?Focus=12470#Comment_12470">problem</a> hasn't been adressed adequately. I wrote in <a href="http://forum.azimuthproject.org/discussion/1471/qbo-and-enso/?Focus=12452#Comment_12452">here:</a> >One thing one should look at in the context of biannuity though more carefully is surely that the earth in its orbit around the sun appears with its axis a bit like a particle with a spin (like around a nucleus). OK this assertion sounds quite as been sitting on the crackpotty. So let me please outline the super vague "reasoning" which is behind this. The earth moves through the <a href="http://en.wikipedia.org/wiki/Heliospheric_current_sheet#mediaviewer/File:Heliospheric-current-sheet.gif">heliospheric current sheet</a>. The sun wind interacts with the earth atmosphere mostly at the <a href="http://en.wikipedia.org/wiki/Magnetosphere#mediaviewer/File:Structure_of_the_magnetosphere_mod.svg">poles</a>. The sun wind particle stream has though different directions depending on polarity. The particle stream influences cloud formation and could thus among others in principle change the albedo and temperature. I have no idea how big those influences are. Probably quite small. The magnet field of the sun seems rather unregular, but eventually some accumulated effect due to the rotation of the sun which shapes the current sheet together with the movement of the earth in that current sheet have a two year periodicity. It is unlikely but at the moment I can't fully exclude this possibility, and I have no other immediate counter arguments at hand so thats why I said one would probably need to look at this i.e. one would at least need to find some excluding arguments.`

Good stuff nad.

Others have mentioned the heliospheric cuurent sheet, which leads to this paper

[1]A. Shapoval, J. L. Le Mouël, M. Shnirman, and V. Courtillot, “Can irregularities of solar proxies help understand quasi-biennial solar variations?,” Nonlinear Processes in Geophysics Discussions, vol. 1, no. 1, pp. 155–192, 2014.

http://www.nonlin-processes-geophys-discuss.net/1/155/2014/npgd-1-155-2014.pdf

`Good stuff nad. Others have mentioned the heliospheric cuurent sheet, which leads to this paper [1]A. Shapoval, J. L. Le Mouël, M. Shnirman, and V. Courtillot, “Can irregularities of solar proxies help understand quasi-biennial solar variations?,” Nonlinear Processes in Geophysics Discussions, vol. 1, no. 1, pp. 155–192, 2014. <http://www.nonlin-processes-geophys-discuss.net/1/155/2014/npgd-1-155-2014.pdf>`

In the paper they introduce some "irregularity indices" λ_WN and λ_aa of "daily series of sunspot number WN and geomagnetic index aa as a function of increasing smoothing from N = 162 to 648 days. "

In the summary they say:

I agree with that. That is I couldn't understand within a decent time what they are doing. As one result it seems they somehow found some rather sharp change in 1975 in sunspot activity and QBO. ???:

Furthermore from the abstract:

was there such a change in 1975? I haven't heard of a sharp change of sun behaviour and/or QBO behaviour....

furthermore the summary:

So it seems at least some people consider the sunwind as being able to have a possible impact on major climate features.

I find the information which is available in the net not really sufficient for saying much more on that topic.

This Nasa website has something on the sometimes weird shape (like a "conch shell") of the heliospheric current sheet. Moreover it seems (at least if the sheet looks more like flat disc) that the earth dips through the sheet. In fact the earth orbit plane seems to be tilted by about 7 degrees with respect to the sheet plane. But usually the earth seems more to travel through the sheet ripples. ???? (the image on the latter page seems to be from this 1999 article with data from 1994 (p. 28))

The weird "shell" current sheet (arising from two northpoles on the sun) was from the Ulysses mission. Now it seems they have new missons called STEREO and SOHO according to a projects participant at Max Planck Institute, and in particular the info about the old Ulysses mission somehow disappeared.

From the Max Planck Institute's page:

They seem to have now more fotographs from the sun. I couldnt though find anything there on the shape of the current sheet.

So concluding, sofar the pathway towards explaing a possible biennal (and in particular not-quasibiennal) forcing via the heliospheric sheet looks not too promising, despite the fact that there seem to be sunwind influences on global climate.

In particular the suns magnetic field seems to be too erratic than that it could account for a regular biennal forcing that is even the dipping through the sheet (which could result in annual forcings) seem to occur irregularily (?). Moreover I haven't found anything on a possible orbit related resonance between the earth and suns magnetic fields and in fact the suns magnetic field seems way to small (?haven't checked though) to influence the earth magnetic field in a significant way. ?

`>Others have mentioned the heliospheric cuurent sheet, which leads to this paper In the paper they introduce some "irregularity indices" λ_WN and λ_aa of "daily series of sunspot number WN and geomagnetic index aa as a function of increasing smoothing from N = 162 to 648 days. " In the summary they say: >The irregularity index method is promising but still not a fully understood tool. I agree with that. That is I couldn't understand within a decent time what they are doing. As one result it seems they somehow found some rather sharp change in 1975 in sunspot activity and QBO. ???: >λ_WN and λ_aa display Schwabe cycles with sharp peaks not only at cycle maxima but also at minima: we call the resulting 5.5 year variations “half Schwabe variations” (HSV). Furthermore from the abstract: >We propose that the HSV behavior of the irregularity index of WN may be linked to the presence of strong QBO before 1915–1930, a transition and their disappearance around 1975, corresponding to a change in regime of solar activity. was there such a change in 1975? I haven't heard of a sharp change of sun behaviour and/or QBO behaviour.... furthermore the summary: >Vecchio et al. (2012), using magnetic synoptic maps from 1976 to 2003, propose that QBO are fundamental modes associated with poleward magnetic flux migration from low to high latitudes (part of meridional circulation) during the maximum and descending phases of the solar cycle. A strong link between QBO and the solar dynamo is inferred from these and other works. Time variations of QBO might therefore provide information on changes in meridional flow. On the other hand, non-linearity of the solar dynamo itself could be the source of QBO. So it seems at least some people consider the sunwind as being able to have a possible impact on major climate features. I find the information which is available in the net not really sufficient for saying much more on that topic. This <a href="http://science.nasa.gov/science-news/science-at-nasa/2003/22apr_currentsheet/">Nasa website</a> has something on the sometimes weird shape (like a "conch shell") of the heliospheric current sheet. Moreover it seems (at least if the sheet looks more like flat disc) that the earth <a href="http://science.nasa.gov/media/medialibrary/2003/04/22/22apr_currentsheet_resources/sectorcrossing.mov">dips through the sheet</a>. In fact the earth orbit plane seems to be tilted by about 7 degrees with respect to the sheet plane. But usually the earth seems more to travel through the sheet <a href="http://pluto.space.swri.edu/IMAGE/glossary/IMF.html">ripples.</a> ???? (the image on the latter page seems to be from this <a href="http://trs-new.jpl.nasa.gov/dspace/bitstream/2014/18860/1/99-2157.pdf">1999 article</a> with data from 1994 (p. 28)) The weird "shell" current sheet (arising from two northpoles on the sun) was from the Ulysses mission. Now it seems they have new missons called <a href="http://en.wikipedia.org/wiki/STEREO">STEREO</a> and <a href="http://en.wikipedia.org/wiki/Solar_and_Heliospheric_Observatory">SOHO</a> according to a projects <a href="http://www.mps.mpg.de/1765421/Basics">participant at Max Planck Institute</a>, and in particular the info about the old <a href="http://www.mps.mpg.de/de/projekte/ulysses/">Ulysses</a> mission somehow disappeared. From the Max Planck Institute's page: >New fundamental knowledge about the Sun has been obtained with instruments (co-)developed by the Institute on board the space probes SOHO and STEREO. The measurements from the UV spectrometer SUMER on SOHO have led to the recognition of the decisive role of the magnetic field in dynamic processes, while STEREO allowed for the first time 3D observations of the Sun and the inner heliosphere They seem to have now more fotographs from the sun. I couldnt though find anything there on the shape of the current sheet. So concluding, sofar the pathway towards explaing a possible biennal (and in particular not-quasibiennal) forcing via the heliospheric sheet looks not too promising, despite the fact that there seem to be sunwind influences on global climate. In particular the suns magnetic field seems to be too erratic than that it could account for a regular biennal forcing that is even the dipping through the sheet (which could result in annual forcings) seem to occur irregularily (?). Moreover I haven't found anything on a possible orbit related resonance between the earth and suns magnetic fields and in fact the suns magnetic field seems way to small (?haven't checked though) to influence the earth magnetic field in a significant way. ?`

IPCC 2001:

See more at:

hth

`IPCC 2001: > All reconstructions indicate that the direct effect of variations in solar forcing over the 20th century was about 20 to 25% of the change in forcing due to increases in the well-mixed greenhouse gases. See more at: * [RealClimate](http://www.realclimate.org/index.php/archives/2006/09/the-trouble-with-sunspots/comment-page-2/#sthash.kno2Hydq.dpuf) hth`

thanks for the link.

About the IPCC 2001 citation: I am not sure wether the solar forcing meant here is variations in radiation only and/or particle flows that is the realclimate blog post you linked to indicates that the particle aspect seems to be rather negligible, if I correctly understand their critique:

The links to the CRF-article are broken.

Do you know wether CERN built that cloud chamber?

`thanks for the link. About the IPCC 2001 citation: I am not sure wether the solar forcing meant here is variations in radiation only and/or particle flows that is the realclimate blog post you linked to indicates that the particle aspect seems to be rather negligible, if I correctly understand their <a href="http://www.realclimate.org/index.php/archives/2005/05/on-veizers-celestial-climate-driver/">critique</a>: >I will try to show that CRF explanation for the recent global warming is easy to rule out. The links to the CRF-article are broken. Do you know wether CERN built that cloud chamber?`

But I found their link to a discussion about day and night time temperatures interesting. This may indicate that there exist more detailed temperature collections than the CRUTEM (which provides only monthly averages).

`But I found their <a href="http://www.grida.no/publications/other/ipcc_tar/?src=/climate/ipcc_tar/wg1/054.htm">link</a> to a discussion about day and night time temperatures interesting. This may indicate that there exist more detailed temperature collections than the <a href="http://www.randform.org/blog/?p=5676">CRUTEM</a> (which provides only monthly averages).`

I thought it meant total solar forcing. CERN must have built the simulation or else I couldn't have read the results (but I've no idea where). There had been a lot of puff from WUWTs et al. about some paper where, I think, particles were supposed to seed cloud formation.

Recent results on bio-seeding of clouds (widely publicised) are, imo, a very large potential contribution to filling what is one of the largest voids in current models.

`I thought it meant total solar forcing. CERN must have built the simulation or else I couldn't have read the results (but I've no idea where). There had been a lot of puff from WUWTs et al. about some paper where, I think, particles were supposed to seed cloud formation. Recent results on bio-seeding of clouds (widely publicised) are, imo, a very large potential contribution to filling what is one of the largest voids in current models.`

The other lines of geophysical evidence pertain to the Earth's Length of Day (LOD) and of the Chandler Wobble.

Besides the 28 month quasi-biennial period in the QBO, there is most definitely an odd 1 rad/year frequency component concealed in the time series. This also shows up in the ENSO SOI results, as I demonstrate in the other thread.

As Gross from JPL has pointed out from his analysis of LOD, there is a connection between the wobble (a beat period of 6 years) and the welling of the deep ocean.

[1]R. S. Gross, “The excitation of the Chandler wobble,” Geophysical Research Letters, vol. 27, no. 15, pp. 2329–2332, 2000.

In this whole area, the lines between what is of a geophysical origin and what is a climactic origin start to blur.

`The other lines of geophysical evidence pertain to the Earth's Length of Day (LOD) and of the Chandler Wobble. Besides the 28 month quasi-biennial period in the QBO, there is most definitely an odd 1 rad/year frequency component concealed in the time series. This also shows up in the ENSO SOI results, as I demonstrate in the [other thread](http://forum.azimuthproject.org/discussion/1451/enso-proxy-records/#Item_47). As Gross from JPL has pointed out from his analysis of LOD, there is a connection between the wobble (a beat period of 6 years) and the welling of the deep ocean. [1]R. S. Gross, “The excitation of the Chandler wobble,” Geophysical Research Letters, vol. 27, no. 15, pp. 2329–2332, 2000. In this whole area, the lines between what is of a geophysical origin and what is a climactic origin start to blur.`

I was searching for data and visualizations concerning magnetic indices like the aa index. I found links at NO AA ;). That is the links to the Laplace Institute on that page are broken. Moreover searching on the website like via search or looking at the page data sets didn't reveal anything. Googling revealed those data files, but no visualisations. For completeness I should mention that BGS holds also index data, but for the aa index they write

`I was searching for data and visualizations concerning magnetic indices like the aa index. I found links at <a href="http://www.ngdc.noaa.gov/IAGA/vdat/">NO AA</a> ;). That is the links to the Laplace Institute on that page are broken. Moreover searching on the website like via search or looking at the page <a href="http://www.ipsl.fr/en/Our-research/Observations/Data-Sets">data sets</a> didn't reveal anything. Googling revealed <a href="http://isgi.latmos.ipsl.fr/lesdonne.htm">those data files,</a> but no visualisations. For completeness I should mention that BGS holds also <a href="http://www.geomag.bgs.ac.uk/data_service/data/magnetic_indices/aaindex.html">index data</a>, but for the aa index they write >the aa indices available on this service are not the definitive values (see note on compilation and changes) . Definitive aa are published by the International Service for Geomagnetic Indices (ISGI). Operated by LATMOS, France, ISGI, has an advisory board that is appointed by the Executive Committee of the International Association of Geomagnetism and Aeronomy (IAGA) and operates as part of the French BCMT (Bureau Central du Magnétisme Terrestre).`

I wrote:

In this context and the context of the real climate notices I wanted to mention:

This sounds as if a damaged ozone layer could lead to colder regions in the high troposphere.

From Wikipedia:

and

`I wrote: >The particle stream influences cloud formation and could thus among others in principle change the albedo and temperature. In this context and the context of the real climate notices I wanted to mention: >The division of the atmosphere into layers mostly by reference to temperature is discussed above. Temperature decreases with altitude starting at sea level, but variations in this trend begin above 11 km, where the temperature stabilizes through a large vertical distance through the rest of the troposphere. In the stratosphere, starting above about 20 km, the temperature increases with height, due to heating within the ozone layer caused by capture of significant ultraviolet radiation from the Sun by the dioxygen and ozone gas in this region. This sounds as if a damaged ozone layer could lead to colder regions in the high troposphere. <a href="http://en.wikipedia.org/wiki/Cloud">From Wikipedia:</a> >Clouds of the high-étage form at altitudes of 3,000 to 7,600 m (10,000 to 25,000 ft) in the polar regions, 5,000 to 12,200 m (16,500 to 40,000 ft) in the temperate regions and 6,100 to 18,300 m (20,000 to 60,000 ft) in the tropical region.[42] <a href="en.wikipedia.org/wiki/Cloud#Clouds_and_weather_forecasting">and</a> >The presence of significant high-étage cloud cover indicates an organized low-pressure disturbance or an associated warm front is about 300 km away from the point of observation.`

Remark concerning the temperature collections:

It seems that at least private collections like wundermap and awekas have also rather few stations in northern siberia and central africa. Like wundermap has on Novaya Semlya only one station (Malye Karmakuly). It seems also that their data is not openly available.

`Remark concerning the temperature collections: It seems that at least private collections like <a href="http://www.wunderground.com/wundermap/">wundermap</a> and <a href="http://www.awekas.at/de/temp.php?nid=30">awekas</a> have also rather few stations in northern siberia and central africa. Like wundermap has on Novaya Semlya only one station (Malye Karmakuly). It seems also that their data is not openly available.`

Is seems NOAA actually has geomagnetic indices here and in principle you could download it and even look at a GIF animation, however I always get the error message: SPIDR cannot execute the requested action...

Potential reasons:

If you're using Web Services, please confirm you're using correct parameters and arguments Web Services Guide SPIDR may also be under heavy load, if you feel certain your request is correct, try again later.

If you believe this response to be in error, please contact SPIDR Support

Anyways I haven't even found an official definition of the aa index. The aa index starts in the 19th century, while the K and ap indices only in 1932 according to the NOAA page. It's also not explained here.

The aa.doc in http://isgi.latmos.ipsl.fr/source/indices/aa/ doesn't say at which time of the day the three-hourly measurements start.

all this is annoying.

`Is seems NOAA actually has geomagnetic indices <a href="http://spidr.ngdc.noaa.gov/spidr/query.do?group=geomInd">here</a> and in principle you could download it and even look at a GIF animation, however I always get the error message: SPIDR cannot execute the requested action... Potential reasons: If you're using Web Services, please confirm you're using correct parameters and arguments Web Services Guide SPIDR may also be under heavy load, if you feel certain your request is correct, try again later. If you believe this response to be in error, please contact SPIDR Support Anyways I haven't even found an official definition of the aa index. The aa index starts in the 19th century, while the K and ap indices only in 1932 according to the NOAA page. It's also not explained <a href="http://www.gfz-potsdam.de/en/research/organizational-units/departments/department-2/earths-magnetic-field/services/kp-index/theory/related-indices/">here.</a> The aa.doc in <a href="http://isgi.latmos.ipsl.fr/source/indices/aa/">http://isgi.latmos.ipsl.fr/source/indices/aa/</a> doesn't say at which time of the day the three-hourly measurements start. all this is annoying.`

I have to remember to search the Azimuth forum for previous discussions, such as the following pertaining to QBO http://forum.azimuthproject.org/discussion/1375/quasibiennial-oscillation/

As far as an exact biennial (2-year) period, there may be something on the recent thread on tides: http://forum.azimuthproject.org/discussion/1480/tidal-records-and-enso/?Focus=12570#Comment_12570

The issue with an

exact2-year period is that it is hard to understand which year the peak of the oscillation starts -- in other words, whether it is an odd or even year, and vice versa for the valley. This is a symmetry argument and so the choice must be meta-stable. I am thinking that the yearly seasonal period doubles at some point in the past, and it gets locked into a groove. And then some unknown event would come along and perhaps force the system to skip a half a cycle and go from an odd year to an even year or vice versa.This is similar to the magnetic polarity of the earth -- what decides the direction? And yet we know that the polarity does switch occasionally.

`I have to remember to search the Azimuth forum for previous discussions, such as the following pertaining to QBO <http://forum.azimuthproject.org/discussion/1375/quasibiennial-oscillation/> As far as an exact biennial (2-year) period, there may be something on the recent thread on tides: <http://forum.azimuthproject.org/discussion/1480/tidal-records-and-enso/?Focus=12570#Comment_12570> The issue with an *exact* 2-year period is that it is hard to understand which year the peak of the oscillation starts -- in other words, whether it is an odd or even year, and vice versa for the valley. This is a symmetry argument and so the choice must be meta-stable. I am thinking that the yearly seasonal period doubles at some point in the past, and it gets locked into a groove. And then some unknown event would come along and perhaps force the system to skip a half a cycle and go from an odd year to an even year or vice versa. This is similar to the magnetic polarity of the earth -- what decides the direction? And yet we know that the polarity does switch occasionally.`

You mean your comment with the period doubling? Yes eventually this could be some kind of a period doubling phenomena, the little peaks in between could belong to the smaller amplitude of a period doubling. I find it irritating though that the little peaks don't oscillate around a medium value (like in the logistic map) and that they seem to lead to a postponement of the higher peak, but then I haven't looked at many examples in dynamical systems which display period doubling. There may be examples which reflect this.

I didn't really understand what you where doing with the tidal, but then I got tired to check all the things there. In particular if Darwin is so close to Sydney (as someone said in the forum) then why should it take 3 months for the tide to arrive there?

Well at least in this diagram

it looks quite clearly (I find) as if the QBO index raises in odd years. This holds by the way also for the temperature (please add 58 to the year count):

`>As far as an exact biennial (2-year) period, there may be something on the recent thread on tides: http://forum.azimuthproject.org/discussion/1480/tidal-records-and-enso/?Focus=12570#Comment_12570 You mean your comment with the period doubling? Yes eventually this could be some kind of a period doubling phenomena, the little peaks in between could belong to the smaller amplitude of a period doubling. I find it irritating though that the little peaks don't oscillate around a medium value (like in the <a href="http://en.wikipedia.org/wiki/Logistic_map">logistic map</a>) and that they seem to lead to a postponement of the higher peak, but then I haven't looked at many examples in dynamical systems which display period doubling. There may be examples which reflect this. I didn't really understand what you where doing with the tidal, but then I got tired to check all the things there. In particular if Darwin is so close to Sydney (as someone said in the forum) then why should it take 3 months for the tide to arrive there? >The issue with an exact 2-year period is that it is hard to understand which year the peak of the oscillation starts – in other words, whether it is an odd or even year Well at least in <a href="http://forum.azimuthproject.org/discussion/1471/qbo-and-enso/?Focus=12437#Comment_12437">this diagram</a> it looks quite clearly (I find) as if the QBO index raises in odd years. This holds by the way also for the temperature (please add 58 to the year count): ![temp](http://www.randform.org/blog/wp-content/2014/09/2yearcycleElNino450.jpg)`

nad, The tidal experiment is very simple and a spreadsheet is all you need. Take the tidal gauge data, filter on a 12-month box window, and subtract the current value from a value referenced from 2 years ago. Repeat that backwards for all previous points. Then overlay with the SOI data.

Do you think the 3-month shift is significant? That simply gave the optimal fit -- a fit using the current month may be nearly as good.

`nad, The tidal experiment is very simple and a spreadsheet is all you need. Take the tidal gauge data, filter on a 12-month box window, and subtract the current value from a value referenced from 2 years ago. Repeat that backwards for all previous points. Then overlay with the SOI data. Do you think the 3-month shift is significant? That simply gave the optimal fit -- a fit using the current month may be nearly as good.`

the "out of sync" QBO oscillations are also rather clearly visible in the temperatures above (as to be somewhat expected):

The table from this comment was:

(63), 65,67

(77),79,81,(83)

(89),91

(03),05,07,09,(11)

in the temperature diagram this is (deduce 58):

(5),7,9

(19), 21, 23, (25)

(31),33

(45),47,49,51, (53)

`the <a href="http://forum.azimuthproject.org/discussion/1498/is-there-an-exact-biannual-global-temperature-oscillation/?Focus=13562#Comment_13562">"out of sync" QBO oscillations</a> are also rather clearly visible in the <a href="http://forum.azimuthproject.org/post/12578/">temperatures</a> above (as to be somewhat expected): The table from this <a href="http://forum.azimuthproject.org/discussion/1498/is-there-an-exact-biannual-global-temperature-oscillation/?Focus=13562#Comment_13562">comment </a> was: (63), 65,67 (77),79,81,(83) (89),91 (03),05,07,09,(11) in the <a href="http://forum.azimuthproject.org/post/12578/">temperature diagram</a> this is (deduce 58): (5),7,9 (19), 21, 23, (25) (31),33 (45),47,49,51, (53)`

I would like to add that the year, where the temperature anomalies do seem not to follow a similar biannual pattern as the QBO is (apart from the years >2011) the year 1971. In this year there was the volcano Teneguia.

`I would like to add that the year, where the temperature anomalies do seem not to follow a similar biannual pattern as the QBO is (apart from the years >2011) the year 1971. In this year there was the volcano <a href="https://en.wikipedia.org/wiki/Teneguia">Teneguia.</a>`

nad said on 9/3/2016:

A biennial cycle seems to be an underlying behavior for ENSO.

I'm satisfied with the physical model for QBO, as I can derive it mathematically from first principles starting from Laplace's tidal equations. The key to a successful derivation is that the important measure is the

accelerationof the QBO, and not the windvelocity. The final result is obtained by solving a Sturm-Liouville equation with a surprising simple analytical formulation. The acceleration can then be tied directly to a gravitational forcing with period related to the draconic lunar month of 27.212 days. When the acceleration is integrated to a velocity, the model fit to the data is impressive. There isn't anything close to this in the QBO research literature, starting way back to Lindzen. Some have hinted for evidence of atmospheric tides, but that discussion mainly concerns solar tides.Where I am at with the ENSO model is perhaps as impressive in terms of a fit, but the explanation is more tenuous. I start with a Mathieu wave equation, which is a standard formulation for modeling sloshing of liquid volumes. The thermocline within the equatorial Pacific ocean is very sensitive to angular momentum variations as the density differences between the warmer surface layer and colder layer can cause it to slosh longitudinally.

What I find is that two primary angular momentum cycles of 14 years and 6.5 years (related to the Chandler wobble) reproduce the general profile of the ENSO time series -- but only if an underlying biennial modulation is simultaneously applied. That's the tenuous explanation as a

strictly biennial cycleis a metastable phenomena; little motivates such a cycle apart from a possible period doubling of the annual cycle due to a nonlinear bifurcation. That metastability may have driven a biennial phase inversion which occurred during the period 1980-1996, timed to a Pacific ocean climate shift.Yet driving this model to second-order is a remarkable fit of the lunar tidal frequencies to the ENSO residual, which fills in the detail of the El Nino and La Nino peaks and valleys that the primary angular momentum variations missed. The nonlinear aliasing of the lunar cycles can actually mimic a biennial modulation of a few fundamental frequencies, which are clearly expressed as Fourier components within the ENSO time-series, as shown here.

Taken together, the common forcing frequencies discovered between the QBO and ENSO suggest that even though the QBO model has the best theoretical basis, the ENSO model will not be far behind. A couple of years have passed since the start of this thread and I can honestly say that the explanation is becoming more concise.

`nad said on 9/3/2016: > "I would like to add that the year, where the temperature anomalies do seem not to follow a similar biannual pattern as the QBO is (apart from the years >2011) the year 1971. In this year there was the volcano Teneguia." A biennial cycle seems to be an underlying behavior for ENSO. I'm satisfied with the physical model for QBO, as I can derive it [mathematically from first principles](http://contextearth.com/2016/08/23/qbo-model-final-stretch/) starting from Laplace's tidal equations. The key to a successful derivation is that the important measure is the *acceleration* of the QBO, and not the wind *velocity*. The final result is obtained by solving a Sturm-Liouville equation with a surprising simple analytical formulation. The acceleration can then be tied directly to a gravitational forcing with period related to the draconic lunar month of 27.212 days. When the acceleration is integrated to a velocity, the model fit to the data is impressive. There isn't anything close to this in the QBO research literature, starting way back to Lindzen. Some have hinted for evidence of atmospheric tides, but that discussion mainly concerns solar tides. Where I am at with the ENSO model is [perhaps as impressive in terms of a fit](http://contextearth.com/2016/05/12/deterministically-locked-on-the-enso-model/), but the explanation is more tenuous. I start with a Mathieu wave equation, which is a standard formulation for modeling sloshing of liquid volumes. The thermocline within the equatorial Pacific ocean is very sensitive to angular momentum variations as the density differences between the warmer surface layer and colder layer can cause it to slosh longitudinally. What I find is that two primary angular momentum cycles of 14 years and 6.5 years (related to the Chandler wobble) reproduce the general profile of the ENSO time series -- but only if an underlying biennial modulation is simultaneously applied. That's the tenuous explanation as a *strictly biennial cycle* is a metastable phenomena; little motivates such a cycle apart from a possible period doubling of the annual cycle due to a nonlinear bifurcation. That metastability may have driven a biennial phase inversion which occurred during the period 1980-1996, timed to a Pacific ocean climate shift. Yet driving this model to second-order is a remarkable fit of the lunar tidal frequencies to the ENSO residual, which fills in the detail of the El Nino and La Nino peaks and valleys that the primary angular momentum variations missed. The nonlinear aliasing of the lunar cycles can actually mimic a biennial modulation of a few fundamental frequencies, which are clearly expressed as Fourier components within the ENSO time-series, [as shown here](http://contextearth.com/2016/06/10/pukites-model-of-enso/). Taken together, the common forcing frequencies discovered between the QBO and ENSO suggest that even though the QBO model has the best theoretical basis, the ENSO model will not be far behind. A couple of years have passed since the start of this thread and I can honestly say that the explanation is becoming more concise.`

Here is a business article that explains why it's nice to be able to predict ENSO

http://www.bloomberg.com/news/articles/2016-09-14/a-market-roiling-la-nina-is-dividing-world-weather-forecasters

The comparison is what would happen if we didn't have a model for tides. Wouldn't know when or where you could moor your boat or when you could launch it. But with ENSO, the stakes are much higher, as economies and agricultures are sensitive to the phase of ENSO.

This is my latest ENSO model that uses known angular momentum variations in the earth''s rotation (14 and 6.5 years) along with the known lunar periods that give the lunar tractive forcing extremes. It's essentially a tidal model reconfigured for the sloshing of the ocean, which if it portrays the actual physical mechanism would be a breakthrough for climate prediction.

The distinction with the QBO model is that lunar terms are strong second-order factors in ENSO, but first-order with QBO. The angular momentum changes are irrelevant for QBO as the atmospheric mass has very little inertial mass.

It will be a learning experience to see how this tracks both into the past and into the future. As I mention here, Hanson,Brier,Maul have established that the strongest periods going back to 1525 are 14 and 6.75 years.

The problem with historical proxy data is that the highest resolution is essentially 1 year (for measures such as tree-rings & coral-rings etc) but for proxy measures based on sedimentation results, the resolution is much worse. It would be nice to have the extra resolution for proxy like we have for instrumental ENSO records (monthly pressure and temperature readings) but doubt that we ever will get that.

There is an

xkcdcartoon that has gone viral recently that doesn't acknowledge the smearing of the proxy data. One paleoclimate guy was pointing this out and it is covered in this NPR piece.`Here is a business article that explains why it's nice to be able to predict ENSO http://www.bloomberg.com/news/articles/2016-09-14/a-market-roiling-la-nina-is-dividing-world-weather-forecasters The comparison is what would happen if we didn't have a model for tides. Wouldn't know when or where you could moor your boat or when you could launch it. But with ENSO, the stakes are much higher, as economies and agricultures are sensitive to the phase of ENSO. This is my latest ENSO model that uses known angular momentum variations in the earth''s rotation (14 and 6.5 years) along with the known lunar periods that give the lunar tractive forcing extremes. It's essentially a tidal model reconfigured for the sloshing of the ocean, which if it portrays the actual physical mechanism would be a breakthrough for climate prediction. ![enso](http://imageshack.com/a/img921/6946/dNnYRc.png) The distinction with the QBO model is that lunar terms are strong second-order factors in ENSO, but first-order with QBO. The angular momentum changes are irrelevant for QBO as the atmospheric mass has very little inertial mass. It will be a learning experience to see how this tracks both into the past and into the future. As I mention [here](http://contextearth.com/2016/09/09/obscure-paper-on-enso-determinism/), Hanson,Brier,Maul have established that the strongest periods going back to 1525 are 14 and 6.75 years. The problem with historical proxy data is that the highest resolution is essentially 1 year (for measures such as tree-rings & coral-rings etc) but for proxy measures based on sedimentation results, the resolution is much worse. It would be nice to have the extra resolution for proxy like we have for instrumental ENSO records (monthly pressure and temperature readings) but doubt that we ever will get that. There is an _xkcd_ cartoon that has gone viral recently that doesn't acknowledge the smearing of the proxy data. One paleoclimate guy was pointing this out and it is [covered in this NPR piece](http://www.npr.org/sections/goatsandsoda/2016/09/14/493925781/epic-climate-cartoon-goes-viral-but-it-has-one-key-problem).`

Harvest moon time

This small-scale QBO experiment is very interesting

http://www.gfd-dennou.org/library/gfd_exp/exp_e/exp/bo/1/res.htm

The geometry of the setup doesn't match the real QBO at all, yet the results appear to agree with the solution to the primitive equations for a rotating cylinder with a periodic latitudinal forcing. Watch the membrane -- as it pulsates up-and-down the transverse oscillations keep pace. When the membrane amplitude gets bigger gradually the velocity covers a longer swing. That's because the inertial volume can keep up with it.

Also have a new blog post at http://contextearth.com/2016/09/17/enso-model-final-stretch-maybe/

`Harvest moon time ![moon](https://pbs.twimg.com/media/CshRiKbW8AA_VsK.jpg) This small-scale QBO experiment is very interesting http://www.gfd-dennou.org/library/gfd_exp/exp_e/exp/bo/1/res.htm The geometry of the setup doesn't match the real QBO at all, yet the results appear to agree with the solution to the primitive equations for a rotating cylinder with a periodic latitudinal forcing. Watch the membrane -- as it pulsates up-and-down the transverse oscillations keep pace. When the membrane amplitude gets bigger gradually the velocity covers a longer swing. That's because the inertial volume can keep up with it. Also have a new blog post at http://contextearth.com/2016/09/17/enso-model-final-stretch-maybe/`

Gavin Schmidt of NASA states here that the predictability level of ENSO is at 6-months:

http://fivethirtyeight.com/features/why-we-dont-know-if-it-will-be-sunny-next-month-but-we-know-itll-be-hot-all-year/

Also a paper by Dunkerton on the QBO http://onlinelibrary.wiley.com/doi/10.1002/2016GL070921/pdf

Dunkerton says the current QBO model is validated by lab experiments undertaken on a scale model, which is described in the previous post. Actually I think that setup confirms my closed form analysis, as the approximation I make along the equator turns a rotating sphere into a rotating cylinder.

and a blog posting: http://robertscribbler.com/2016/09/19/giant-gravity-waves-smashed-key-atmospheric-clock-during-winter-of-2016-possible-climate-change-link

I added the following comment but it went into moderation, so I will reproduce here:

Ultimately, the reason that QBO can be modeled is that it stays along the equator and behaves with a reduced dimensionality inside what amounts to a waveguide. Right at the equator, the Coriolis forces precisely cancel and the system of equations that govern fluid flow on a rotating sphere can be simplified and thus analyzed in closed form. These equations were originally formulated by the mathematician Pierre Laplace in the late 1700’s to try to understand the dynamics of ocean tides.

In fact, the QBO essentially is a manifestation of an atmospheric tide governed by external forces — the current consensus is that gravity waves are responsible. Yet, on close examination these gravity waves happen to be perfectly aligned with the gravitational tractive forcing of the lunar nodal cycle. The pull of the moon as it crosses the equator then controls the direction of the QBO cross-wind.

The nodal lunar cycle is 27.212 days, so how does the 28 month period of the QBO cycle come about? That’s actually quite straightforward to understand. The cycle “beats” with the solar seasonal cycle creating a stronger pulse that occurs every 2.369 years or approximately 28 months. These are pulses of acceleration, which when integrated become a velocity and turn into these almost squared-off sinusoidal oscillations that are characteristic of the QBO wind. Anyone that has done any signal processing knows that the integral of a sequence of delta spikes results in a square wave.

That’s the key observation that the AGW-denier Richard Lindzen missed when he formulated his original QBO theory over 40 years ago. He didn’t see the lunar tidal connection and so created his own half-baked explanation of what drove the oscillations. Everyone seemed to follow his lead and so we have gone down a deep rabbit hole of complexity to try to understand QBO ever since.

What happens outside of the equatorial latitudes is that the Coriolis forces start to exert themselves, which then will create the twisting vortices in the jet stream which are much more difficult to analyze. We do know that the polar vortex has shown correlation with the direction of the QBO.

But we really have to start somewhere and the best place is to work the foundational models from scratch. It will take a while to unwind from what Lindzen inflicted on us with his limited QBO theory over 40 years ago.

`Gavin Schmidt of NASA states here that the predictability level of ENSO is at 6-months: http://fivethirtyeight.com/features/why-we-dont-know-if-it-will-be-sunny-next-month-but-we-know-itll-be-hot-all-year/ > "Since the 1980s, we’ve had sufficient understanding of ENSO to be able to predict the occurrence and speed of these waves and, consequently, the variability of ocean temperatures in the Eastern Pacific about six months in advance." Also a paper by Dunkerton on the QBO http://onlinelibrary.wiley.com/doi/10.1002/2016GL070921/pdf Dunkerton says the current QBO model is validated by lab experiments undertaken on a scale model, which is described in the [previous post](15527/#Comment_15527). Actually I think that setup confirms my closed form analysis, as the approximation I make along the equator turns a rotating sphere into a rotating cylinder. and a blog posting: http://robertscribbler.com/2016/09/19/giant-gravity-waves-smashed-key-atmospheric-clock-during-winter-of-2016-possible-climate-change-link I added the following comment but it went into moderation, so I will reproduce here: Ultimately, the reason that QBO can be modeled is that it stays along the equator and behaves with a reduced dimensionality inside what amounts to a waveguide. Right at the equator, the Coriolis forces precisely cancel and the system of equations that govern fluid flow on a rotating sphere can be simplified and thus analyzed in closed form. These equations were originally formulated by the mathematician Pierre Laplace in the late 1700’s to try to understand the dynamics of ocean tides. In fact, the QBO essentially is a manifestation of an atmospheric tide governed by external forces — the current consensus is that gravity waves are responsible. Yet, on close examination these gravity waves happen to be perfectly aligned with the gravitational tractive forcing of the lunar nodal cycle. The pull of the moon as it crosses the equator then controls the direction of the QBO cross-wind. The nodal lunar cycle is 27.212 days, so how does the 28 month period of the QBO cycle come about? That’s actually quite straightforward to understand. The cycle “beats” with the solar seasonal cycle creating a stronger pulse that occurs every 2.369 years or approximately 28 months. These are pulses of acceleration, which when integrated become a velocity and turn into these almost squared-off sinusoidal oscillations that are characteristic of the QBO wind. Anyone that has done any signal processing knows that the integral of a sequence of delta spikes results in a square wave. That’s the key observation that the AGW-denier Richard Lindzen missed when he formulated his original QBO theory over 40 years ago. He didn’t see the lunar tidal connection and so created his own half-baked explanation of what drove the oscillations. Everyone seemed to follow his lead and so we have gone down a deep rabbit hole of complexity to try to understand QBO ever since. What happens outside of the equatorial latitudes is that the Coriolis forces start to exert themselves, which then will create the twisting vortices in the jet stream which are much more difficult to analyze. We do know that the polar vortex has shown correlation with the direction of the QBO. ![polar](http://imageshack.com/a/img921/4407/MjQRAr.png) But we really have to start somewhere and the best place is to work the foundational models from scratch. It will take a while to unwind from what Lindzen inflicted on us with his limited QBO theory over 40 years ago.`

I created a QBO page that is a concise derivation of the theory behind the oscillations:

http://contextEarth.com/compact-qbo-derivation/

Four key observations allow this derivation to work

Coriolis effect cancels at the equator and use a small angle (in latitude) approximation to capture any differential effect.

Identification of wind acceleration and not wind speed as the measure of QBO.

Associating a latitudinal displacement with a tidal elevation via a partial derivative expansion to eliminate an otherwise indeterminate parameter.

Applying a seasonal aliasing to the lunar tractive forces which ends up perfectly matching the observed QBO period.

These are obscure premises but all are necessary to derive the equations and match to the observations.

This model should have been derived long ago .... that's what has me stumped. Years ago I spent hours working on transport equations for semiconductor devices so have a good feel for how to handle these kinds of DiffEq's. You literally had to do this otherwise you would never develop the intuition on how a transistor or some other device works. The QBO for some reason reminds me quite a bit of solving the Hall effect. Maybe I am just using a different lens in solving these kinds of problems.

`I created a QBO page that is a concise derivation of the theory behind the oscillations: http://contextEarth.com/compact-qbo-derivation/ Four key observations allow this derivation to work 1. Coriolis effect cancels at the equator and use a small angle (in latitude) approximation to capture any differential effect. 2. Identification of wind acceleration and not wind speed as the measure of QBO. 3. Associating a latitudinal displacement with a tidal elevation via a partial derivative expansion to eliminate an otherwise indeterminate parameter. 4. Applying a seasonal aliasing to the lunar tractive forces which ends up perfectly matching the observed QBO period. These are obscure premises but all are necessary to derive the equations and match to the observations. This model should have been derived long ago .... that's what has me stumped. Years ago I spent hours working on transport equations for semiconductor devices so have a good feel for how to handle these kinds of DiffEq's. You literally had to do this otherwise you would never develop the intuition on how a transistor or some other device works. The QBO for some reason reminds me quite a bit of solving the [Hall effect](https://en.wikipedia.org/wiki/Hall_effect). Maybe I am just using a different lens in solving these kinds of problems.`

This might be a topic for the Strugges with the Continuum blog post

Consider the equation that I derived for the QBO time-series

If you look closely at the DiffEq, there is a clear singularity in the Sturm-Liouville equation. Its most obvious in the secant. When the arg reaches $\pi/2$ it hits the singularity. Yet the solution is bounded. What this means is that the solution either exactly zero-compensates for the infinity or its too narrow a delta to be important.

If you do a numerical integration of this DiffEq, it can get tricky with regards to picking the right delta interval, yet the closed-form solution is trivial.

`This might be a topic for the [Strugges with the Continuum](https://johncarlosbaez.wordpress.com/2016/09/23/struggles-with-the-continuum-part-7/) blog post Consider the equation that I derived for the [QBO time-series](http://contextearth.com/compact-qbo-derivation/) ![qbo](http://imagizer.imageshack.us/a/img923/8846/76Ijl3.png) If you look closely at the DiffEq, there is a clear singularity in the Sturm-Liouville equation. Its most obvious in the secant. When the arg reaches $\pi/2$ it hits the singularity. Yet the solution is bounded. What this means is that the solution either exactly zero-compensates for the infinity or its too narrow a delta to be important. If you do a numerical integration of this DiffEq, it can get tricky with regards to picking the right delta interval, yet the closed-form solution is trivial.`

With respect to the ENSO model I have been thinking about ways of evaluating the statistical significance of the fit to the data. If we train on one 70 year interval and then test on the following 70 year interval, we get the interesting effect of finding a higher correlation coefficient on the test interval. The training interval is just below 0.85 while the test is above 0.86.

The model fit is relatively aggressive in the number of degrees of freedom (DOF) it contains, since there appear to be multiple forcings involved, each with a unique period. This exacts a statistical price as the number of DOF allows one to also fit the ENSO proxy data to arbitrary models. For example, a red noise random walk mode can also give an often impressive correlation coefficient by using the

same setof parameters, but with varying amplitude and phase.That appears troubling in terms of discriminating between a real and a coincidental fit, but if we look closely at the result of out-of-band tests on trained fits to red noise models, they rapidly become uncorrelated. Below are the statistics for the “in” training run and the “out” test or validation. Even though a correlation coefficient above 0.7 is achieved during training, that means nothing within the test interval, as all phase coherence disappears when the random walk is invoked. Note below that the out statistics are centered over a coefficient of 0.0, which is essentially the extreme of uncorrelated behavior. But for model operating on the real data, one can see that both the training and test CC values are very high (arrows on the right), which means that the ENSO behavior is not stochastic and that whatever periodic behavior is defined in the first 70 years is also observed in the next 70 years.

For this model operating on the real data, at least some of the DiffEq fit is attributed to a forced alignment with a biennial term. This gives probably a 10% improvement of the CC (both for red noise and for the real data) over it not being applied. This is due to a common mode multiplicative factor to both the LHS and RHS of the DiffEq. Yet, the biennial factor is essential to providing a mechanism for phase inversions, such as what occurs between 1980 and 1996. And it also improves the fit as even slight variations away from 2 years will appreciably reduce the correlation.

`With respect to the ENSO model I have been thinking about ways of evaluating the statistical significance of the fit to the data. If we train on one 70 year interval and then test on the following 70 year interval, we get the interesting effect of finding a higher correlation coefficient on the test interval. The training interval is just below 0.85 while the test is above 0.86. ![fit](http://imageshack.com/a/img921/781/dvd5T0.png) The model fit is relatively aggressive in the number of degrees of freedom (DOF) it contains, since there appear to be multiple forcings involved, each with a unique period. This exacts a statistical price as the number of DOF allows one to also fit the ENSO proxy data to arbitrary models. For example, a red noise random walk mode can also give an often impressive correlation coefficient by using the *same set* of parameters, but with varying amplitude and phase. That appears troubling in terms of discriminating between a real and a coincidental fit, but if we look closely at the result of out-of-band tests on trained fits to red noise models, they rapidly become uncorrelated. Below are the statistics for the “in” training run and the “out” test or validation. Even though a correlation coefficient above 0.7 is achieved during training, that means nothing within the test interval, as all phase coherence disappears when the random walk is invoked. Note below that the out statistics are centered over a coefficient of 0.0, which is essentially the extreme of uncorrelated behavior. But for model operating on the real data, one can see that both the training and test CC values are very high (arrows on the right), which means that the ENSO behavior is not stochastic and that whatever periodic behavior is defined in the first 70 years is also observed in the next 70 years. ![test](http://imageshack.com/a/img923/9612/L376sq.png) For this model operating on the real data, at least some of the DiffEq fit is attributed to a forced alignment with a biennial term. This gives probably a 10% improvement of the CC (both for red noise and for the real data) over it not being applied. This is due to a common mode multiplicative factor to both the LHS and RHS of the DiffEq. Yet, the biennial factor is essential to providing a mechanism for phase inversions, such as what occurs between 1980 and 1996. And it also improves the fit as even slight variations away from 2 years will appreciably reduce the correlation.`

A recent paper analyzing historical ENSO records used a CGM to support some of their observations. What they do not explain is the strength of the biennial period in their simulation results.

Evolution and forcing mechanisms of El Niño over the past 21,000 years Zhengyu Liu, Zhengyao Lu, Xinyu Wen, B. L. Otto-Bliesner, A. Timmermann, K. M. Cobb, Nature 515, 550–553 (27 November 2014) doi:10.1038/nature13963

Intro section

Methods section

This is interesting in that they mischaracterize the periodicity as biannual, which also means semiannual, or twice a year, but that it is clearly biennial in the chart, which is defined as once every two years. That could be just a typo not caught during proof-reading. Yet the peak is sharply centered around a two-year fundamental period, which is the interesting aspect.

I replot that curve below to show the symmetry around the 2-year period. Drawing a Lorentzian curve around that frequency and linearizing the axis makes it symmetric.

A Lorentzian or Cauchy often results as the frequency response of a driven damped harmonic oscillator. If that is the case, what the simulation shows may be the result of forcing comprised of different stimuli frequencies collected over time and the accompanying frequency response (e.g. a Bode plot). Yet this implies that the characteristic frequency or eigenvalue would need to be 2 years. But why would a characteristic frequency just happen to align with a period so close to 2 years? The 2 year period is likely not an eigenvalue of the properties of a damped harmonic oscillator, e.g. a spring constant and a damper, but more than likely connected to a period doubling based on the annual cycle.

This is a spectrum of the biennial modulated ENSO data of the last 130 years. Pairs show up +/- around the 2 year central period. These are all close in value to either wobble cycles or aliased lunar cycles, the latter of which have a natural biennial modulation.

The strongest factors above are shown as pluses on the historical model below, these are not to scale in the vertical but positioned along the horizontal to highlight the symmetry:

So the ENSO power spectrum over the modern instrumental record is not a smooth Lorentizian but a discrete set of frequencies that are balanced +/- around the central biennial frequency. The question is whether these discrete frequencies are fixed over time. Based on what I found with ENSO proxy data it appears that it may be, at least for spans of 100's of years -- http://contextearth.com/2016/09/27/enso-proxy-revisited/

`A recent paper analyzing historical ENSO records used a CGM to support some of their observations. What they do not explain is the strength of the biennial period in their simulation results. ![TRACE](http://imageshack.com/a/img921/5258/QrlvKs.png) [Evolution and forcing mechanisms of El Niño over the past 21,000 years](http://www.nature.com/nature/journal/v515/n7528/fig_tab/nature13963_SF1.html) Zhengyu Liu, Zhengyao Lu, Xinyu Wen, B. L. Otto-Bliesner, A. Timmermann, K. M. Cobb, Nature 515, 550–553 (27 November 2014) doi:10.1038/nature13963 Intro section > “To understand ENSO’s evolution during the past 21 kyr, we analyse the baseline transient simulation (TRACE) conducted with the Community Climate System model version 3 (CCSM3). This simulation uses the complete set of realistic climate forcings: orbital, greenhouse gases, continental ice sheets and meltwater discharge (Fig. 1a, d and Methods). TRACE has been shown to replicate many key features of the global climate evolution” Methods section > “Model ENSO.ENSO simulated by the model for the present day shows many realistic features, although the ENSO period tends to be biased towards quasi-biannual, as opposed to a broader 2–7-year peak in the observation38. The ENSO mode resembles the SST mode29 and propagates westwards as in many CGCMs. In the past 21 kyr, the preferred period of model ENSO remains at quasi-biannual, with the power spectrum changing only modestly with time (Extended Data Fig. 1).” This is interesting in that they mischaracterize the periodicity as biannual, which also means semiannual, or twice a year, but that it is clearly biennial in the chart, which is defined as once every two years. That could be just a typo not caught during proof-reading. Yet the peak is sharply centered around a two-year fundamental period, which is the interesting aspect. I replot that curve below to show the symmetry around the 2-year period. Drawing a Lorentzian curve around that frequency and linearizing the axis makes it symmetric. ![symm](http://imageshack.com/a/img924/9210/zQf0ZI.png) A Lorentzian or Cauchy often results as the frequency response of a [driven damped harmonic oscillator](http://demonstrations.wolfram.com/ResonanceLineshapesOfADrivenDampedHarmonicOscillator/ ). If that is the case, what the simulation shows may be the result of forcing comprised of different stimuli frequencies collected over time and the accompanying frequency response (e.g. a Bode plot). Yet this implies that the characteristic frequency or eigenvalue would need to be 2 years. But why would a characteristic frequency just happen to align with a period so close to 2 years? The 2 year period is likely not an eigenvalue of the properties of a damped harmonic oscillator, e.g. a spring constant and a damper, but more than likely connected to a period doubling based on the annual cycle. This is a spectrum of the biennial modulated ENSO data of the last 130 years. Pairs show up +/- around the 2 year central period. These are all close in value to either wobble cycles or aliased lunar cycles, the latter of which [have a natural biennial modulation](http://contextearth.com/2016/09/17/enso-model-final-stretch-maybe/). ![folded](http://imageshack.com/a/img923/919/0RjPtp.png) The strongest factors above are shown as pluses on the historical model below, these are not to scale in the vertical but positioned along the horizontal to highlight the symmetry: ![spikes](http://imageshack.com/a/img921/2088/ndQEiY.png) So the ENSO power spectrum over the modern instrumental record is not a smooth Lorentizian but a discrete set of frequencies that are balanced +/- around the central biennial frequency. The question is whether these discrete frequencies are fixed over time. Based on what I found with ENSO proxy data it appears that it may be, at least for spans of 100's of years -- http://contextearth.com/2016/09/27/enso-proxy-revisited/`

Why does ENSO contain the same characteristic frequencies as variations in Length-of-Day (LOD)?

Blog post here: ENSO model maps to LOD cycles

`Why does ENSO contain the same characteristic frequencies as variations in Length-of-Day (LOD)? Blog post here: [ENSO model maps to LOD cycles](http://contextearth.com/2016/10/03/enso-recovers-lod/)`

Experimenting on the QBO acceleration time-series with very short training intervals, the part shaded in yellow. Interesting how much information content is stored in a short 3 to 4 year interval. The training part is duplicated almost exactly in several parts of the time series (see ~1998, e.g.) using the exact Draconic lunar period, aliased to the annual cycle.

And stating the obvious, but a massive hurricane about to hit Florida very soon.

The consensus is that Atlantic hurricanes start off the equatorial coast of Africa. There are apparently correlations of hurricane occurrences and strength with the direction of QBO so just another reason to understand QBO.

`Experimenting on the QBO acceleration time-series with very short training intervals, the part shaded in yellow. Interesting how much information content is stored in a short 3 to 4 year interval. The training part is duplicated almost exactly in several parts of the time series (see ~1998, e.g.) using the exact Draconic lunar period, aliased to the annual cycle. ![1](http://imageshack.com/a/img924/4158/aWl9za.png) ![2](http://imageshack.com/a/img924/5641/x2wila.png) --- And stating the obvious, but a massive hurricane about to hit Florida very soon. The consensus is that Atlantic hurricanes start off the equatorial coast of Africa. There are apparently correlations of hurricane occurrences and strength with the direction of QBO so just another reason to understand QBO.`

Blog post that goes into some of the details behind the previous comment

## Short Training Intervals for QBO

http://contextEarth.com/2016/10/10/short-training-intervals-for-qbo/

Also pulled out the machine learning for one last time to see how it would work on a short training interval:

`Blog post that goes into some of the details behind the previous comment #Short Training Intervals for QBO# http://contextEarth.com/2016/10/10/short-training-intervals-for-qbo/ Also pulled out the machine learning for one last time to see how it would work on a short training interval: ![e](http://imagizer.imageshack.us/a/img922/7145/G5zhhx.png)`

Interesting wave sloshing behavior experiment somebody did -- perhaps for a school project -- apparently trying to duplicate the PRL paper "Observation of star-shaped surface gravity waves" by Rajchenbach et al (2013)

Look closely and you can see that the 3-fold symmetry has a repeat cycle which is twice the period of the vertical forcing cycle.

`Interesting wave sloshing behavior experiment somebody did -- perhaps for a school project -- apparently trying to duplicate the PRL paper "Observation of star-shaped surface gravity waves" by Rajchenbach et al (2013) Look closely and you can see that the 3-fold symmetry has a repeat cycle which is twice the period of the vertical forcing cycle. https://youtu.be/iSE0ubrxwK0`

The odd symmetry of the wave sloshing in the previous comment may be classified as a tripole of sorts.

The period doubling seems to be an emergent feature of nonlinear waves, as modeled by the Mathieu equation

The nonlinearity comes from mutiplicative mixing of waves of different periods. Period doubling is somewhat reinforcing, as a harmonic that has a period of one year when mixed with a period of two years will recover the two year period.

$ sin(2\pi t)sin(\pi t) = \frac{1}{2} (cos(\pi t) - cos(3\pi t) )$

But the "somewhat" is a caveat, as what decides the alignment of the doubling on an odd versus even year is inherently arbitrary and therefore metastable.

One can almost see the potential tri-fold symmetry of the wave sloshing in this paper discussing a Pacific Ocean tripole.

My own theory is that climate shifts such as occur with the IPO (circa late 1970's) are associated with a period doubling phase inversion between even and odd years.

`The odd symmetry of the wave sloshing in the previous comment may be classified as a tripole of sorts. The period doubling seems to be an emergent feature of nonlinear waves, as modeled by the Mathieu equation > ![mathieu](http://imageshack.com/a/img923/6401/tORmHR.png) > from Rajchenbach, Jean, and Didier Clamond. "Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited." Journal of Fluid Mechanics 777 (2015): R2. The nonlinearity comes from mutiplicative mixing of waves of different periods. Period doubling is somewhat reinforcing, as a harmonic that has a period of one year when mixed with a period of two years will recover the two year period. $ sin(2\pi t)sin(\pi t) = \frac{1}{2} (cos(\pi t) - cos(3\pi t) )$ But the "somewhat" is a caveat, as what decides the alignment of the doubling on an odd versus even year is inherently arbitrary and therefore metastable. One can almost see the potential tri-fold symmetry of the wave sloshing in this paper discussing a Pacific Ocean tripole. > ![tripole](http://imageshack.com/a/img923/1705/QrHU54.png) > from Henley, Benjamin J., et al. "A tripole index for the interdecadal Pacific oscillation." Climate Dynamics 45.11-12 (2015): 3077-3090. My own theory is that climate shifts such as occur with the IPO (circa late 1970's) are associated with a period doubling phase inversion between even and odd years.`

In the previous comment I referenced the Rajchenbach article on Faraday waves.

There is a telling assertion within that article:

What they are suggesting is that too much focus has been placed on natural resonances and the dispersion relationships within a free fluid volume. Whereas the forced response is clearly as important if not more, and that the forcing will show through in the solution of the equations. If you haven't inferred by now, I have been pursuing this strategy for a while, and the Rajchenbach article is the first case that I have found made of what I always thought should be an obvious premise. That's a peer-reviewed article and the fact that the reviewers allowed the "astonishingly" adjective in the paper is what makes it telling. It's astonishing in the equivalent sense that Rajchenbach & Clamond are pointing out that a pendulum's motion will be impacted by a periodic push. IOW, astonishing in the sense that it should be obvious!

So, what I take from it is that there has been a misguided focus on unconstrained or free fluid flow, as opposed to solving the dynamics of a Newtonian fluid within an enclosing container. Read through it and see if you come up with a similar understanding.

Further, what I am finding is that in the cases of ENSO and QBO, the fundamental seasonality forcing does not jump out at you, but in fact is mixed into the stew and is not revealed by conventional methods. What you need is a good model of the physics and novel means of extracting the parameters from the dynamics, such as using the Mathieu equation for fluid dynamics -- as Rajchenbach suggests and I have been advocating for ENSO. That's why it hasn't been obvious over the years, and the continuing claim being made that ENSO and QBO are driven by natural/chaotic resonances lives on ...

`In the previous comment I referenced the [Rajchenbach article on Faraday waves](http://contextearth.com/wp-content/uploads/2016/10/rajchenbach2015.pdf ). There is a telling assertion within that article: > "For instance, to the best of our knowledge, the dispersion relation (relating angular frequency ω and wavenumber k) of parametrically forced water waves has astonishingly not been explicitly established hitherto. Indeed, this relation is often improperly identified with that of free unforced surface waves, despite experimental evidence showing significant deviations" What they are suggesting is that too much focus has been placed on natural resonances and the dispersion relationships within a free fluid volume. Whereas the forced response is clearly as important if not more, and that the forcing will show through in the solution of the equations. If you haven't inferred by now, I have been pursuing this strategy for a while, and the Rajchenbach article is the first case that I have found made of what I always thought should be an obvious premise. That's a peer-reviewed article and the fact that the reviewers allowed the "astonishingly" adjective in the paper is what makes it telling. It's astonishing in the equivalent sense that Rajchenbach & Clamond are pointing out that a pendulum's motion will be impacted by a periodic push. IOW, astonishing in the sense that it should be obvious! So, what I take from it is that there has been a misguided focus on unconstrained or free fluid flow, as opposed to solving the dynamics of a Newtonian fluid within an enclosing container. Read through it and see if you come up with a similar understanding. Further, what I am finding is that in the cases of ENSO and QBO, the fundamental seasonality forcing does not jump out at you, but in fact is mixed into the stew and is not revealed by conventional methods. What you need is a good model of the physics and novel means of extracting the parameters from the dynamics, such as using the Mathieu equation for fluid dynamics -- as Rajchenbach suggests and I have been advocating for ENSO. That's why it hasn't been obvious over the years, and the continuing claim being made that ENSO and QBO are driven by natural/chaotic resonances lives on ... ![enso23](http://imageshack.com/a/img921/6679/tlDzQG.png)`

Over at Nick Stokes moyhu blog, JCH asserts:

This is how I responded:

To be able to forecast anything, you have to first understand it. Most current weather forecasts are at the level of heuristics. A heuristic doesn't have to be based on science but only on the fact that it has worked sufficiently well in the past. The heuristic is obviously pretty bad for ENSO because the best I have seen works for only a few months in advance. IOW, it's essentially at the level of "slow train coming", to quote Dylan.

That is changing for ENSO as we are developing a much better physical model. In this blog post, I include some ideas from the latest research on wave dynamics.

The central premises to making the model work include:

A biennial mode is assumed operating based on the doubling of the annual period.

A phase inversion occurs between 1980 and 1996, justified by metastability of the biennial model with respect to even and odd starting years.

Mathieu equation formulation for sloshing behavior.

A right-hand-side (RHS) forcing due to known angular momentum and gravity terms, with additional seasonal aliasing of the tidal terms.

Given the quality of the results, it won't be long before our understanding of ENSO approaches that of oceanic tides, and we know how well tides can be forecasted. That's essentially predictable for many years in advance, with precision down to the hour. In fact, anything even close to this will be a boon. Take a look at the detail of the fits in the linked post and you can see the potential predictability.

`Over at Nick Stokes moyhu blog, JCH asserts: > "The fact is, nobody at this time is reliably forecasting ENSO." This is how I responded: To be able to forecast anything, you have to first understand it. Most current weather forecasts are at the level of heuristics. A heuristic doesn't have to be based on science but only on the fact that it has worked sufficiently well in the past. The heuristic is obviously pretty bad for ENSO because the best I have seen works for only a few months in advance. IOW, it's essentially at the level of "slow train coming", to quote Dylan. That is changing for ENSO as we are developing a much better physical model. [In this blog post](http://contextearth.com/2016/10/24/solver-vs-multiple-linear-regression-for-enso/), I include some ideas from the latest research on wave dynamics. The central premises to making the model work include: 1. A biennial mode is assumed operating based on the doubling of the annual period. 2. A phase inversion occurs between 1980 and 1996, justified by metastability of the biennial model with respect to even and odd starting years. 3. Mathieu equation formulation for sloshing behavior. 4. A right-hand-side (RHS) forcing due to known angular momentum and gravity terms, with additional seasonal aliasing of the tidal terms. Given the quality of the results, it won't be long before our understanding of ENSO approaches that of oceanic tides, and we know how well tides can be forecasted. That's essentially predictable for many years in advance, with precision down to the hour. In fact, anything even close to this will be a boon. Take a look at the detail of the fits in the linked post and you can see the potential predictability. ![enso](http://imagizer.imageshack.us/a/img924/3039/kZKD7S.png)`

This one expands in size if opened in another window. I am getting to the point where I am at with QBO, in that the focus is on the discrepancies between model and data. Are these caused by volcanos? Or are they due to other factors not considered?

`This one expands in size if opened in another window. I am getting to the point where I am at with QBO, in that the focus is on the discrepancies between model and data. Are these caused by volcanos? Or are they due to other factors not considered? ![a](http://imageshack.com/a/img924/2586/u0iQcy.png)`

This is an Excel Solver fit to ENSO without the wave equation transform applied. Only the angular momentum variations of 14 and 6.48 year periods along with the 3 aliased tidal periods were included.

The training was divided into a 70 year early interval and a 63 year later interval. They essentially capture the same underlying time-series profile, suggesting that the ENSO process is ergodic on a relatively short time scale.

Again, this is not easily detected unless we assume that a biennial Mathieu modulation is applied to the wave equation and that a biennial phase inversion between even-and-odd years applies between the years 1980 to 1996.

`This is an Excel Solver fit to ENSO without the wave equation transform applied. Only the angular momentum variations of 14 and 6.48 year periods along with the 3 aliased tidal periods were included. ![fit](http://imageshack.com/a/img923/383/C0aeGQ.png) The training was divided into a 70 year early interval and a 63 year later interval. They essentially capture the same underlying time-series profile, suggesting that the ENSO process is ergodic on a relatively short time scale. Again, this is not easily detected unless we assume that a biennial Mathieu modulation is applied to the wave equation and that a biennial phase inversion between even-and-odd years applies between the years 1980 to 1996.`

There is a discussion going on at the Moyhu blog concerning chaos in climate modeling and how well GCMs work.

Nick rightly says:

Yet the bigger question concerns why we can't easily map out the silly oscillations of ENSO like we can with tides. For heaven's sakes, ENSO is just a standing wave dipole of ocean sloshing and should have been figured out long ago, maybe not as long ago as the last Cubbies championship but certainly by now.

What makes it that hard to figure out? The key is that it isn't necessarily a chaotic system but one that is defined more by metastability. Everything about ENSO points to an underlying periodicity governed by a period doubling of the annual cycle. Yet period doubling from 1 to 2 years implies that something has to set the bifurcation parity to either an odd-year cycle or an even-year cycle. Energetically, there is nothing that obviously determines the even vs odd cycle ... except perhaps how tidal gravitational forcing interact with the seasonal cycle. In fact, as I have shown on my blog there is a distinct biennial parity for each of the three classes of lunar tides (nodal, anomalistic, and tropical), which occurs after a nonlinear mixing with the seasonal cycle.

$sin(\pi t) \cdot sin(\omega_m t) = \frac{1}{2} ( cos(\pi t - \omega_m t) - cos(\pi t + \omega_m t) ) $

If you expand all the terms with this biennial factor, then you can reconstruct a strongly aliased lunar tide cycle.

The metastability revolves around how easily this balance is tipped. From what I have been able to discern this metastability has only flipped once, and that was during the 1980 to 1996 time interval. This can explain why standard signal processing techniques have not uncovered the metastability but the one described recently by Astudillo has detected the disturbance at 1980 as well:

H. Astudillo, R. Abarca-del-Río, and F. Borotto, “Long-term potential nonlinear predictability of El Niño–La Niña events,” Climate Dynamics, pp. 1–11, 2016.

Cubs finally win, and perhaps we are nearing an understanding of ENSO

Here is the latest analysis: http://contextearth.com/2016/11/03/short-training-intervals-for-enso/

`There is a discussion going on at the Moyhu blog concerning chaos in climate modeling and how well GCMs work. [Nick rightly says](https://moyhu.blogspot.com/2016/10/chaos-cfd-and-gcms.html#comment-form): > "The key word there is work. They do." Yet the bigger question concerns why we can't easily map out the silly oscillations of ENSO like we can with tides. For heaven's sakes, ENSO is just a standing wave dipole of ocean sloshing and should have been figured out long ago, maybe not as long ago as the last Cubbies championship but certainly by now. What makes it that hard to figure out? The key is that it isn't necessarily a chaotic system but one that is defined more by metastability. Everything about ENSO points to an underlying periodicity governed by a period doubling of the annual cycle. Yet period doubling from 1 to 2 years implies that something has to set the bifurcation parity to either an odd-year cycle or an even-year cycle. Energetically, there is nothing that obviously determines the even vs odd cycle ... except perhaps how tidal gravitational forcing interact with the seasonal cycle. In fact, as I have [shown on my blog](http://contextearth.com/2016/04/21/biennial-mode-of-sst-and-enso/) there is a distinct biennial parity for each of the three classes of lunar tides (nodal, anomalistic, and tropical), which occurs after a nonlinear mixing with the seasonal cycle. $sin(\pi t) \cdot sin(\omega_m t) = \frac{1}{2} ( cos(\pi t - \omega_m t) - cos(\pi t + \omega_m t) ) $ If you [expand all the terms with this biennial factor](http://contextearth.com/2015/11/17/the-math-of-seasonal-aliasing/), then you can reconstruct a strongly aliased lunar tide cycle. The metastability revolves around how easily this balance is tipped. From what I have been able to discern this metastability has only flipped once, and that was during the 1980 to 1996 time interval. This can explain why standard signal processing techniques have not uncovered the metastability but the one described recently by Astudillo has detected the disturbance at 1980 as well: H. Astudillo, R. Abarca-del-Río, and F. Borotto, “Long-term potential nonlinear predictability of El Niño–La Niña events,” Climate Dynamics, pp. 1–11, 2016. Cubs finally win, and perhaps we are nearing an understanding of ENSO Here is the latest analysis: http://contextearth.com/2016/11/03/short-training-intervals-for-enso/`

Here is an example of a tight-fitting training interval for ENSO, with a gap interval for validation.

Even though we may be over-fitting to the noise within the dual training intervals (with a correlation coefficient nearing 0.95), the validation interval is still at 0.75 and obviously has the same general time-series profile. However the fit to the earlier data prior to 1920 is lower at 0.48.

The timing is perfect for putting the finishing touches on the ENSO and QBO models as I will be presenting at the AGU meeting next month.

Program link:

OS11B-04: Analytical Formulation of Equatorial Standing Wave Phenomena: Application to QBO and ENSO

Everything in this abstract still holds and so I continue to be confident in presenting the results. I will either sink or swim on relying on such simple models but I think the findings are rock solid.

`Here is an example of a tight-fitting training interval for ENSO, with a gap interval for validation. ![1](http://imageshack.com/a/img923/4999/dDKcbW.png) Even though we may be over-fitting to the noise within the dual training intervals (with a correlation coefficient nearing 0.95), the validation interval is still at 0.75 and obviously has the same general time-series profile. However the fit to the earlier data prior to 1920 is lower at 0.48. The timing is perfect for putting the finishing touches on the ENSO and QBO models as I will be presenting at the AGU meeting next month. --- Program link: [OS11B-04: Analytical Formulation of Equatorial Standing Wave Phenomena: Application to QBO and ENSO ](https://agu.confex.com/agu/fm16/meetingapp.cgi/Paper/120573) ![paper](http://imageshack.com/a/img921/588/3j75T9.png) Everything in this abstract still holds and so I continue to be confident in presenting the results. I will either sink or swim on relying on such simple models but I think the findings are rock solid.`

The following science cartoon is making the rounds this morning, courtesy of this twitter person :

Yet the reality is the following for much of physics:

because they did not consider the entire truth table of complexity vs correctness.

The complex & wrong model of QBO by Richard Lindzen and the complex & wrong model of ENSO by Anastasios Tsonis are essentially worthless junk that we need to do a U-turn on.

I may actually use the lower graphic as part of my AGU presentation.

`The following science cartoon is making the rounds this morning, courtesy of [this twitter person](https://twitter.com/Science_Hooker) : ![2](http://imageshack.com/a/img922/1505/DzAVCY.png) Yet the reality is the following for much of physics: ![1](http://imageshack.com/a/img923/7091/Z13xvT.png) because they did not consider the entire truth table of complexity vs correctness. The complex & wrong model of QBO by Richard Lindzen and the complex & wrong model of ENSO by Anastasios Tsonis are essentially worthless junk that we need to do a U-turn on. I may actually use the lower graphic as part of my AGU presentation.`

This is good. Using the known tidal and Chandler wobble terms, and only training data from 1910 to 1950, here is a validated Mathieu wave equation fit.

Notice the almost excessive overfitting applied to the training interval, yet the validated fit beyond 1950 shows a still very reasonable agreement with the data.

The intriguing aspect to this is that if the Chandler wobble terms were known as of 1950, the El Nino of 1998 could have been predicted 48 years in advance. Apparently the Southern Oscillation was discovered in the 1920's

`This is good. Using the known tidal and Chandler wobble terms, and only training data from 1910 to 1950, here is a validated Mathieu wave equation fit. ![1](http://imageshack.com/a/img923/5286/jH72x5.png) Notice the almost excessive overfitting applied to the training interval, yet the validated fit beyond 1950 shows a still very reasonable agreement with the data. The intriguing aspect to this is that if the Chandler wobble terms were known as of 1950, the El Nino of 1998 could have been predicted 48 years in advance. Apparently the Southern Oscillation was [discovered in the 1920's](https://ams.confex.com/ams/annual2003/techprogram/paper_58909.htm)`