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# QBO and ENSO

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

## Refs

[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.

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edited September 2014

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.

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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?

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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.

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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

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"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.

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edited September 2014

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 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.

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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

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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.

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edited September 2014

When I attempted a machine learning fit to QBO, one extended trial ended up like this

???Shouldn't lead the term: tcos(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)

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 "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.

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edited September 2014

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.

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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 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)

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.

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edited September 2014

I wrote in here:

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 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.

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edited September 2014

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

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edited September 2014

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 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:

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. ?

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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:

hth

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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:

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?

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edited September 2014

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).

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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.

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edited September 2014

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.

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edited September 2014

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

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).

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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.

From Wikipedia:

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]

and

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.

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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.

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edited September 2014

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.

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.

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I have to remember to search the Azimuth forum for previous discussions, such as the following pertaining to QBO http://azimuth.mathforge.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://azimuth.mathforge.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.

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As far as an exact biennial (2-year) period, there may be something on the recent thread on tides: http://azimuth.mathforge.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 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?

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 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):

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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.

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New paper, Paul, in JAMES : https://doi.org/10.1029/2021MS002530

I paid attention to it because Michael Ghil is a co-author: "Extratropical low-frequency variability with ENSO forcing: A reduced-order coupled model study."