#### Howdy, Stranger!

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

Options

# QBO and ENSO

• Options
51.

There is no need to respiel ie. mention emeritus nonsense to neophytes; it will divert their attention. I say stick to what you've done that's new!

Comment Source:There is no need to respiel ie. mention emeritus nonsense to neophytes; it will divert their attention. I say stick to what *you*'ve done that's new!
• Options
52.

Jim said:

"There is no need to respiel ie. mention emeritus nonsense to neophytes; it will divert their attention. I say stick to what you've done that's new!"

Thanks. The significance is that Lindzen is the developer of the foundational model for QBO. This was done well before he became emeritus. Not that I can't defend the new model on its own terms, but I doubt anyone will stick up for Lindzen at the conference now that he has become a pariah of AGW researchers.

There is this concept of "normalizing viewpoints" that is quite common in political journalism, whereby someone like Trump is given equal weighting to a much more qualified Clinton. That's actually counterproductive to convincing others of the objective truth and I think it applies to science as well. The last thing I want to do is normalize the views of Lindzen, but instead intend to completely marginalize whatever work that he has done -- by asserting that he went down a completely wrong path. But I only have 15 minutes for the entire presentation so I have to dismiss his prior work quickly. I do it in the abstract in a single sentence so will probably go that route instead of using the cartoon above.

Comment Source:Jim said: > "There is no need to respiel ie. mention emeritus nonsense to neophytes; it will divert their attention. I say stick to what you've done that's new!" Thanks. The significance is that Lindzen is the developer of the foundational model for QBO. This was done well before he became emeritus. Not that I can't defend the new model on its own terms, but I doubt anyone will stick up for Lindzen at the conference now that he has become a pariah of AGW researchers. There is this concept of "normalizing viewpoints" that is quite common in political journalism, whereby someone like Trump is given equal weighting to a much more qualified Clinton. That's actually counterproductive to convincing others of the objective truth and I think it applies to science as well. The last thing I want to do is normalize the views of Lindzen, but instead intend to completely marginalize whatever work that he has done -- by asserting that he went down a completely wrong path. But I only have 15 minutes for the entire presentation so I have to dismiss his prior work quickly. I do it in the abstract in a single sentence so will probably go that route instead of using the cartoon above. 
• Options
53.

A single sentence sounds good to me :)

Comment Source:A single sentence sounds good to me :) 
• Options
54.

Interesting take on Richard Lindzen here

"... Lindzen’s theories that were wrong. Unfortunately, bad scientists don’t acknowledge their mistakes; Lindzen keeps inventing ever more arcane theories to avoid admitting he was wrong."

I tend to agree with this view. The point with the QBO model is that no one has actually said that Lindzen's theory was incorrect. I only say it is wrong because the Laplace formulation with the nodal gravitational forcing works so well, and that Lindzen never considered that. So perhaps it is not so much that it is wrong but that it has been superceded by a simple and rather obvious mechanism.

Comment Source:Interesting take on Richard Lindzen [here](http://www.easterbrook.ca/steve/2012/12/successful-predictions-agu-2012-tyndall-lecture/) > "... Lindzen’s theories that were wrong. Unfortunately, bad scientists don’t acknowledge their mistakes; Lindzen keeps inventing ever more arcane theories to avoid admitting he was wrong." I tend to agree with this view. The point with the QBO model is that no one has actually said that Lindzen's theory was incorrect. I only say it is wrong because the Laplace formulation with the nodal gravitational forcing works so well, and that Lindzen never considered that. So perhaps it is not so much that it is wrong but that it has been superceded by a simple and rather obvious mechanism. 
• Options
55.

Congratulations on getting a slot at the AGU meeting. Will you be updating your arXiv paper for it? If it's appropriate I think I remember John (Baez) suggesting that you might do some ENSO forecasts for, say, the next 15 or 20 years, and escrow them on the Azimuth wiki. I think you can then demonstrate your level of confidence at the AGU, letting them know you're prepared to put your money where your mouth is, as they say. Best wishes for your presentation.

Comment Source:Congratulations on getting a slot at the AGU meeting. Will you be updating your arXiv paper for it? If it's appropriate I think I remember John (Baez) suggesting that you might do some ENSO forecasts for, say, the next 15 or 20 years, and escrow them on the Azimuth wiki. I think you can then demonstrate your level of confidence at the AGU, letting them know you're prepared to put your money where your mouth is, as they say. Best wishes for your presentation. 
• Options
56.

thanks, I do have an update of the paper

Because of the uncertainty of the biennial modulation, I won't make any predictions. It has switched between even and odd alignments at least twice in the last 130+ years and who knows when it will switch again.

Comment Source:thanks, I do have an update of the paper Because of the uncertainty of the biennial modulation, I won't make any predictions. It has switched between even and odd alignments at least twice in the last 130+ years and who knows when it will switch again. 
• Options
57.

Aha, I did consider that an even-odd switch would invalidate any forecast (until we can understand and then predict those) but thought that that could be qualified. I'd still like to see your model run into the future, even just here. Indeed this might provide a measure against which any deviation might be a detector in advance of a 3rd switch?

Comment Source:Aha, I did consider that an even-odd switch would invalidate any forecast (until we can understand and then predict those) but thought that that could be qualified. I'd still like to see your model run into the future, even just here. Indeed this might provide a measure against which any deviation might be a detector in advance of a 3rd switch? 
• Options
58.
edited November 2016

Jim, I have only used data for training up to the year 2013, so I do have blind predictions for the current ENSO data.

For example, see the graph I posted a few days ago here

https://forum.azimuthproject.org/discussion/comment/15568/#Comment_15568

This extended the model from 2013 up to 2020.

Comment Source:Jim, I have only used data for training up to the year 2013, so I do have blind predictions for the current ENSO data. For example, see the graph I posted a few days ago [here](15568/#Comment_15568) https://forum.azimuthproject.org/discussion/comment/15568/#Comment_15568 This extended the model from 2013 up to 2020. 
• Options
59.

Great! I missed that. Off topic, you might like to know that people have been asking about The Oil Conundrum on G+.

Comment Source:Great! I missed that. Off topic, you might like to know that people have been asking about The Oil Conundrum on G+.
• Options
60.

That's cool. Making predictions for fossil fuel depletion is a less harrowing task than for climate. First of all, depletion is purely statistical and probabilistic so there is less a binary right or wrong answer to the result. Secondly, there is more wiggle room since humans are involved and they don't always follow the laws of physics. Third, the data for fossil fuels is so bad that the models are vitally important to make sense of what's happening. The results have to be judged on that.

Contrast that to a deterministic climate model, where you will be hung out to dry if it doesn't predict the next cycle. You really have to walk on egg shells if its all geophysics you are talking about. Think about a predictive tidal model. If you get that wrong and the owner of some boat finds it dry-docked on the rocks, he will go somewhere else for predictions.

That's why I am really concentrating on getting the physics correct.

Comment Source:That's cool. Making predictions for fossil fuel depletion is a less harrowing task than for climate. First of all, depletion is purely statistical and probabilistic so there is less a binary right or wrong answer to the result. Secondly, there is more wiggle room since humans are involved and they don't always follow the laws of physics. Third, the data for fossil fuels is so bad that the models are vitally important to make sense of what's happening. The results have to be judged on that. Contrast that to a deterministic climate model, where you will be hung out to dry if it doesn't predict the next cycle. You really have to walk on egg shells if its all geophysics you are talking about. Think about a predictive tidal model. If you get that wrong and the owner of some boat finds it dry-docked on the rocks, he will go somewhere else for predictions. That's why I am really concentrating on getting the physics correct. 
• Options
61.

That's an interesting comparison. The chap who mentioned the oil book will almost certainly read it.

Comment Source:That's an interesting comparison. The chap who mentioned the oil book will almost certainly read it. 
• Options
62.

With the AGU meeting coming up, I am rationalizing my confidence in the QBO and ENSO models. Earlier I was much more confident in the QBO model, as the results were so clean. But now my confidence in the ENSO model has risen to the same level as for QBO.

The confidence test is the following. I take a pair of non-overlapping and non-contiguous intervals in the ENSO SOI time-series. Then I use the solver to extract the underling components in each of the intervals and essentially compare the two. If the behavior of ENSO were either highly chaotic or red-noise Markovian, the fits would be markedly different as each would follow a different trajectory. However, if the two are composed of the same periodic factors, then the odds are that ENSO is a deterministic cycle.

Here is the lower fit with the given training interval described by the dotted green line. I split the fit into two intervals to capture the longer periods accurately.

Here is the high interval fit, which as you can see does not overlap with the lower interval fit.

As I have described before, I flip the phase in the years 1980-1996 to capture the known climate shift, and use the wave-equation transform of the data instead of solving the DiffEq directly.

These are the periodic component comparisons between the lower and higher intervals

These all align very tightly, with the discrepancies indicating that it isn't some artifact of a flawed fitting process (i.e. the x=x problem).

The following shows the comparison between the annual harmonics (1/2, 1/3, etc periods with the annual period effectively filtered out in the original SOI time series) for the low and high regions and also the Mathieu modulation comparison

The years prior to 1910 were not used in the fit because the data appears much more noisy than the data post that date. Incidentally, no filtering was used during the model fitting process.

Here is the model fit over the interval from 1895 to 2013. The Mathieu modulation comes stronger as the correlation of the longer interval reinforces the fit/

Also note that only data to 2013 was used and that the extrapolated fit predicts that the 2016 SOI spike was almost as strong as the 1998 event and next to that the strongest in the last 100 years. This is a graph focused only on the last 50 years, so you can see the predictive extrapolation more closely.

These are all based on the known Earth wobble and lunar tidal periods and really confirms that ENSO is a nearly pure deterministic stationary process driven by known geophysical forcings. And like the tidal models that this ENSO model emulates, the longer the period to extract from and the more lunar periods that are included, the better the fit becomes.

It only deviates from stationary determinism in terms of the odd vs even parity of the biennial modulation. The biennial modulation flips from an even-year parity before 1980 to an odd-year parity between the years 1980-1996. I only lack confidence in how to predict these flips, much like I lack the confidence to predict volcanic events which seems to impact the QBO sporadically.

Comment Source:With the AGU meeting coming up, I am rationalizing my confidence in the QBO and ENSO models. Earlier I was much more confident in the QBO model, as the results were so clean. But now my confidence in the ENSO model has risen to the same level as for QBO. The confidence test is the following. I take a pair of non-overlapping and non-contiguous intervals in the ENSO SOI time-series. Then I use the solver to extract the underling components in each of the intervals and essentially compare the two. If the behavior of ENSO were either highly chaotic or red-noise Markovian, the fits would be markedly different as each would follow a different trajectory. However, if the two are composed of the same periodic factors, then the odds are that ENSO is a deterministic cycle. Here is the lower fit with the given training interval described by the dotted green line. I split the fit into two intervals to capture the longer periods accurately. ![](http://imageshack.com/a/img924/3416/3PDDmj.png) Here is the high interval fit, which as you can see does not overlap with the lower interval fit. ![](http://imageshack.com/a/img923/2949/Q9BUkM.png) As I have described before, I flip the phase in the years 1980-1996 to capture the known climate shift, and use the wave-equation transform of the data instead of solving the DiffEq directly. These are the periodic component comparisons between the lower and higher intervals ![http://imageshack.com/a/img921/3135/veqzU0.png](http://imageshack.com/a/img921/3135/veqzU0.png) These all align very tightly, with the discrepancies indicating that it isn't some artifact of a flawed fitting process (i.e. the x=x problem). The following shows the comparison between the annual harmonics (1/2, 1/3, etc periods with the annual period effectively filtered out in the original SOI time series) for the low and high regions and also the Mathieu modulation comparison ![http://imageshack.com/a/img924/6391/NphZ5M.png](http://imageshack.com/a/img924/6391/NphZ5M.png) The years prior to 1910 were not used in the fit because the data appears much more noisy than the data post that date. Incidentally, no filtering was used during the model fitting process. Here is the model fit over the interval from 1895 to 2013. The Mathieu modulation comes stronger as the correlation of the longer interval reinforces the fit/ ![all](http://imageshack.com/a/img924/575/s0kJuJ.png) Also note that only data to 2013 was used and that the extrapolated fit predicts that the 2016 SOI spike was almost as strong as the 1998 event and next to that the strongest in the last 100 years. This is a graph focused only on the last 50 years, so you can see the predictive extrapolation more closely. ![](http://imageshack.com/a/img924/9074/5VVywf.png) These are all based on the known Earth wobble and lunar tidal periods and really confirms that ENSO is a nearly pure deterministic stationary process driven by known geophysical forcings. And like the tidal models that this ENSO model emulates, the longer the period to extract from and the more lunar periods that are included, the better the fit becomes. It only deviates from stationary determinism in terms of the odd vs even parity of the biennial modulation. The biennial modulation flips from an even-year parity before 1980 to an odd-year parity between the years 1980-1996. I only lack confidence in how to predict these flips, much like I lack the confidence to predict volcanic events which seems to impact the QBO sporadically. 
• Options
63.
edited November 2016

Here is another clincher for long-term deterministic stationary properties of ENSO. Some time ago I was running machine-learning on the Universal ENSO Proxy (UEP) records. This goes back to the year 1650 and is an annual record. Interestingly, the primary component that the symbolic reasoning finds is the aliased anomalistic tide!

If I overlay the part of the UEP fit that overlaps the modern day SOI records, we get this.

Note that this is trivial to do as the symbolic reasoner provides a sinusoidal function.

The aliased frequency of 7.821 rads/year that the UEP fit maps to is precisely the aliased anomalistic frequency factor of 4.085 rads/year shifted by 2$\pi$, which is equivalent under yearly sampling.

$2 \pi (1+1/4.085) = 7.8213$

!!!!

Comment Source:Here is another clincher for long-term deterministic stationary properties of ENSO. Some time ago I was running machine-learning on the Universal ENSO Proxy (UEP) records. This goes back to the year 1650 and is an annual record. Interestingly, the primary component that the symbolic reasoning finds is the aliased anomalistic tide! ![](http://imagizer.imageshack.us/a/img539/73/13y2CN.gif) If I overlay the part of the UEP fit that overlaps the modern day SOI records, we get this. ![](http://imageshack.com/a/img924/8135/ql0AAO.png) Note that this is trivial to do as the symbolic reasoner provides a sinusoidal function. ![](http://imageshack.com/a/img922/262/zdusWF.png) The aliased frequency of 7.821 rads/year that the UEP fit maps to is precisely the aliased anomalistic frequency factor of 4.085 rads/year shifted by 2$\pi$, which is equivalent under yearly sampling. $2 \pi (1+1/4.085) = 7.8213$ !!!! 
• Options
64.

Great stuff. I've found these threads a fun and fascinating ML, stats and geophys. tutorial. I don't know which ML and regression graphs you're going to present but, apart from the 1980-1996 biennial flip, I think I'd find a bullet point list of the different time periods for different judgements would obviate any cherry picking doubts. Was it the Astrudillo et al. paper which first posulated the flip? Causes of the flip seems like a great sign off challenge for an audience.

Comment Source:Great stuff. I've found these threads a fun and fascinating ML, stats and geophys. tutorial. I don't know which ML and regression graphs you're going to present but, apart from the 1980-1996 biennial flip, I think I'd find a bullet point list of the different time periods for different judgements would obviate any cherry picking doubts. Was it the Astrudillo et al. paper which first posulated the flip? Causes of the flip seems like a great sign off challenge for an audience.
• Options
65.

Jim, The Astudillo paper asserted that ENSO was deterministic apart from the disturbance right after 1980. They did not mention a phase flip specifically.

The phase reversal idea came from consideration of this disturbance with respect to a biennial mode, with supplemental references here:

http://contextearth.com/2016/05/13/enso-phase-reversal/

And this wavelet graph by Roundy shows how standing modes of the SST got inverted after 1980 but came back in sync around 2000:

Roundy, P.E. "On the interpretation of EOF analysis of ENSO, atmospheric Kelvin waves, and the MJO." Journal of Climate 28.3 (2015): 1148-1165.

Comment Source:Jim, The Astudillo paper asserted that ENSO was deterministic apart from the disturbance right after 1980. They did not mention a phase flip specifically. The phase reversal idea came from consideration of this disturbance with respect to a biennial mode, with supplemental references here: http://contextearth.com/2016/05/13/enso-phase-reversal/ And this wavelet graph by Roundy shows how standing modes of the SST got inverted after 1980 but came back in sync around 2000: ![](http://imageshack.com/a/img661/9797/gOJ1aM.gif) Roundy, P.E. "On the interpretation of EOF analysis of ENSO, atmospheric Kelvin waves, and the MJO." Journal of Climate 28.3 (2015): 1148-1165. 
• Options
66.

I think the phase flip and biennial mode go hand-in-hand. I have noticed a recent spate of papers on the biennial mode of ENSO. Here is a very recent one by NASA Godard scientists:

Achuthavarier, Deepthi, Siegfried D. Schubert, and Yury V. Vikhliaev. "North Pacific decadal variability: insights from a biennial ENSO environment." Climate Dynamics (2016): 1-19.

The above excerpt is the typical explanation via a wordy rationale of how a succeeding year is prevented by the previous year from cycling, thus creating a biennial period.

Which differs from the purely mathematical explanation of a Mathieu sloshing formulation showing an inherent period doubling (or frequency halving), such as described here by a group at CNRS in France:

Rajchenbach, Jean, and Didier Clamond. "Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited." Journal of Fluid Mechanics 777 (2015): R2.

What I have been doing is noting the empirical observations from the climate scientists and then tying that into the math that the physicists and engineers have been developing for other hydrodynamics applications. What's interesting is that these research efforts are concurrently advancing and the timing is perfect to tie the hydrodynamics concepts to the biennial ENSO concept.

Comment Source:I think the phase flip and biennial mode go hand-in-hand. I have noticed a recent spate of papers on the biennial mode of ENSO. Here is a very recent one by NASA Godard scientists: > Achuthavarier, Deepthi, Siegfried D. Schubert, and Yury V. Vikhliaev. "North Pacific decadal variability: insights from a biennial ENSO environment." Climate Dynamics (2016): 1-19. > ![](http://imageshack.com/a/img922/1935/3BNPw5.png) The above excerpt is the typical explanation via a wordy rationale of how a succeeding year is prevented by the previous year from cycling, thus creating a biennial period. Which differs from the purely mathematical explanation of a Mathieu sloshing formulation showing an inherent period doubling (or frequency halving), such as described here by a group at CNRS in France: > Rajchenbach, Jean, and Didier Clamond. "Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited." Journal of Fluid Mechanics 777 (2015): R2. > ![](http://imagizer.imageshack.us/a/img923/6401/tORmHR.png) What I have been doing is noting the empirical observations from the climate scientists and then tying that into the math that the physicists and engineers have been developing for other hydrodynamics applications. What's interesting is that these research efforts are concurrently advancing and the timing is perfect to tie the hydrodynamics concepts to the biennial ENSO concept.
• Options
67.

Thanks for the links. If I can see straight your scenario graph to 2020 looks like large El Ninos are followed by large La Ninas and that's what your graph shows round about 2018.

Comment Source:Thanks for the links. If I can see straight your scenario graph to 2020 looks like large El Ninos are followed by large La Ninas and that's what your graph shows round about 2018. 
• Options
68.

Jim, I am reading the bigger La Nina closer to 2020, but as I said before I want to get the physics solidified before.

Should also revisit the sea-level height readings as I found some biennial characteristics in the long-term Sydney Harbor readings.

I also have an idea on how to solve for the best fit model by finding the roots to an extended polynomial equation. I am sure readers of this forum can guess the approach as it is one of those clever mathy things.

Comment Source:Jim, I am reading the bigger La Nina closer to 2020, but as I said before I want to get the physics solidified before. Should also revisit the sea-level height readings as I found some biennial characteristics in the long-term Sydney Harbor readings. I also have an idea on how to solve for the best fit model by finding the roots to an extended polynomial equation. I am sure readers of this forum can guess the approach as it is one of those clever mathy things. 
• Options
69.

Another model training fit comparison, where the data included goes back to 1895. The earlier data looked noisy but I wanted to see how robust the fitting process is.

Lower

Higher

Periodic factor comparisons, they all match apart from the average magnitude, which varies during the solver process but is in arbitrary units anyways.

Comment Source:Another model training fit comparison, where the data included goes back to 1895. The earlier data looked noisy but I wanted to see how robust the fitting process is. Lower ![](http://imageshack.com/a/img923/2863/AnurTW.png) Higher ![](http://imageshack.com/a/img924/7893/Rr33cH.png) Periodic factor comparisons, they all match apart from the average magnitude, which varies during the solver process but is in arbitrary units anyways. ![](http://imageshack.com/a/img923/2852/3oA0Py.png) 
• Options
70.

There are two key technical approaches that I am using in doing these DiffEq fits.

One is to use the NINO34 data to smooth out the SOI data for calculating the 2nd-derivative only. The SOI data is so noisy that the NINO34 is smooth enough that it removes the significant amount of noise that two successive differentiations will introduce.

Second is to apply a hybrid goodness-of-fit metric. The hybrid nature of the metric is that it combines a correlation coefficient (which emphasizes shape of the time series) and absolute error minimization (which reduces the relative error).

The hybrid is essentially $\frac{CC}{\Delta Err}$

So my process is to apply the hybrid metric first to the Solver. Once that converges to get the scale right, I run a correlation coefficient goal to emphasize the shape. And when that converges, I run the hybrid again to reduce the relative error, which is really an energy minimization -- in that a good correlation needs to be balanced by minimum energy.

With the Excel Solver, the process takes a few hours total and I let it run in the background while I do other stuff. The linear multiple regression solver only takes a second but it can't handle the nonlinear Mathieu modulation, so that is tweaked manually to get a good fit. But the Solver approach is great in that I can start with a completely blank slate and find a largely reproducible solution in just a few hours.

So this is the starting point, with a zeroed model:

and this is the finishing point:

What the biennial Mathieu modulation does is exaggerate the peaks and valleys until the forcing RHS matches the DiffEq LHS. The Solver iterates on the two sides until it converges to achieve an optimal metric.

And always the caveat in that there is a biennial phase inversion of ENSO between the years 1980-1996. Without that premise, the fit would not work, and which is again the likely reason that the underlying model has escaped notice all these years. Try doing any kind of fit on recent data of the last 50 years without inverting the phase over that 16-year interval and all you will find is anti-correlations and will soon give up. But as Ronald Coates said in the early 1960's "if you torture the data enough, nature will confess".

Comment Source:There are two key technical approaches that I am using in doing these DiffEq fits. One is to use the NINO34 data to smooth out the SOI data for calculating the 2nd-derivative *only*. The SOI data is so noisy that the NINO34 is smooth enough that it removes the significant amount of noise that two successive differentiations will introduce. Second is to apply a hybrid goodness-of-fit metric. The hybrid nature of the metric is that it combines a correlation coefficient (which emphasizes shape of the time series) and absolute error minimization (which reduces the relative error). The hybrid is essentially $\frac{CC}{\Delta Err}$ So my process is to apply the hybrid metric first to the Solver. Once that converges to get the scale right, I run a correlation coefficient goal to emphasize the shape. And when that converges, I run the hybrid again to reduce the relative error, which is really an energy minimization -- in that a good correlation needs to be balanced by minimum energy. With the Excel Solver, the process takes a few hours total and I let it run in the background while I do other stuff. The linear multiple regression solver only takes a second but it can't handle the nonlinear Mathieu modulation, so that is tweaked manually to get a good fit. But the Solver approach is great in that I can start with a completely blank slate and find a largely reproducible solution in just a few hours. So this is the starting point, with a zeroed model: ![](http://imageshack.com/a/img924/8898/grhQfE.png) and this is the finishing point: ![](http://imageshack.com/a/img922/677/l0If8t.png) What the biennial Mathieu modulation does is exaggerate the peaks and valleys until the forcing RHS matches the DiffEq LHS. The Solver iterates on the two sides until it converges to achieve an optimal metric. And always the caveat in that there is a biennial phase inversion of ENSO between the years 1980-1996. Without that premise, the fit would not work, and which is again the likely reason that the underlying model has escaped notice all these years. Try doing any kind of fit on recent data of the last 50 years without inverting the phase over that 16-year interval and all you will find is anti-correlations and will soon give up. But as Ronald Coates said in the early 1960's *"if you torture the data enough, nature will confess"*. 
• Options
71.
edited November 2016

With these comments, I am trying to put the nail in the coffin (edit: that sounds bad, I mean stick a pin in it) on the ENSO model

One result I noticed from the independent low and high model fits, is that a slight temporal shift exists between the two profiles. This is best observed as a ~1.5 month difference in the Mathieu modulations (i.e. the LHS modulation of the DiffEq):

This translates to the same shift in the forcing factors (i.e. the RHS of the DiffEq). So if I translate each of the factor curves by either 1 or 2 months, this is the alignment I get:

Note that the RED curve is the high interval and the BLUE is the low interval and so the stronger the blue, the greater degree the two agree, since the red will hide behind the blue. The correlation coefficients are shown as well.

So consider that this is the result of about two hours of the Excel Solver grinding away on an optimal solution for each interval. There was no overlap between the two intervals, yet we still get this stunning a result. The 1.5 month discrepancy may essentially be the uncertainty in the collection of the ENSO data. I don't think it's significant as this could be simply a local minimum that the Solver converged to due to inherent noise in the data.

Because it takes a while to run these trials, I haven't tried too many other sets of input parameters. Yet if I take some arbitrary values to substitute for the ( 6.41, 14.6, 4.085, 18.6, 9.3 ) set, the resulting fit is horrible and there is absolutely no correlation in the Low vs High output factors. There may in fact be another set that works as well, but the periods would have to be physically significant to make any sense in terms of the periodic geophysical forcing mechanisms at work.

The remaining two factors that I can add are derived second-order tidal factors corresponding to 5.643 and 3.447 years which are associated with the fortnightly long period tides stemming from nonlinear multiplicative interactions of the nodal, anomalistic, and tropical monthly periods. These are critical for achieving highly precise tidal predictions and so would think they would be relevant here as well. But I have to be careful of overfitting at this stage. These would require twice as long an interval for convergence and so I would lose the low vs high training validation.

Comment Source:With these comments, I am trying to put the nail in the coffin (edit: that sounds bad, I mean stick a pin in it) on the ENSO model One result I noticed from the independent low and high model fits, is that a slight temporal shift exists between the two profiles. This is best observed as a ~1.5 month difference in the Mathieu modulations (i.e. the LHS modulation of the DiffEq): ![](http://imageshack.com/a/img923/201/dzJSy8.png) This translates to the same shift in the forcing factors (i.e. the RHS of the DiffEq). So if I translate each of the factor curves by either 1 or 2 months, this is the alignment I get: ![](http://imageshack.com/a/img922/1315/DAs5W0.png) Note that the <font color=red>RED</font> curve is the high interval and the <font color=blue>BLUE</font> is the low interval and so the stronger the blue, the greater degree the two agree, since the red will hide behind the blue. The correlation coefficients are shown as well. So consider that this is the result of about two hours of the Excel Solver grinding away on an optimal solution for each interval. There was no overlap between the two intervals, yet we still get this stunning a result. The 1.5 month discrepancy may essentially be the uncertainty in the collection of the ENSO data. I don't think it's significant as this could be simply a local minimum that the Solver converged to due to inherent noise in the data. Because it takes a while to run these trials, I haven't tried too many other sets of input parameters. Yet if I take some arbitrary values to substitute for the ( 6.41, 14.6, 4.085, 18.6, 9.3 ) set, the resulting fit is horrible and there is absolutely no correlation in the Low vs High output factors. There may in fact be another set that works as well, but the periods would have to be physically significant to make any sense in terms of the periodic geophysical forcing mechanisms at work. The remaining two factors that I can add are derived second-order tidal factors corresponding to 5.643 and 3.447 years which are associated with the fortnightly long period tides stemming from nonlinear multiplicative interactions of the nodal, anomalistic, and tropical monthly periods. These are critical for achieving highly precise tidal predictions and so would think they would be relevant here as well. But I have to be careful of overfitting at this stage. These would require twice as long an interval for convergence and so I would lose the low vs high training validation. 
• Options
72.

In the last comment, I mentioned that the 5.643 and 3.447 year aliased tidal factors may contribute to the ENSO model fit. I needed to increase the interval width at the expense of overlapping the Low and High intervals. The reason for this is that 5.643 is relatively close to 6.41 and so the discrimination region needs to be wider.

This factor does markedly improve the overall fit, as you can see in both the Low and High fits how well the 2016 El Nino peak was captured.

Shown next are the correlations for the individual factors. The 5.643 period correlation Low vs High is at the bottom. The CC values are all ~0.8 or greater.

In contrast as shown next, the 3.447 period correlation Low vs High is not strong and the contribution to the fit is relatively weak

This does not imply the factor is not there, only that more work is needed to reveal it. The Mathieu modulation converges as well, but this is at least partly due to the overlapping interval fit:

Comment Source:In the last comment, I mentioned that the 5.643 and 3.447 year aliased tidal factors may contribute to the ENSO model fit. I needed to increase the interval width at the expense of overlapping the Low and High intervals. The reason for this is that 5.643 is relatively close to 6.41 and so the discrimination region needs to be wider. ![](http://imageshack.com/a/img921/7470/8Nlccw.png) ![](http://imageshack.com/a/img921/5482/C7NNzz.png) This factor does markedly improve the overall fit, as you can see in both the Low and High fits how well the 2016 El Nino peak was captured. Shown next are the correlations for the individual factors. The 5.643 period correlation Low vs High is at the bottom. The CC values are all ~0.8 or greater. ![](http://imageshack.com/a/img923/4323/2ouGbx.png) In contrast as shown next, the 3.447 period correlation Low vs High is not strong and the contribution to the fit is relatively weak ![](http://imageshack.com/a/img921/4845/hK8cKD.png) This does not imply the factor is not there, only that more work is needed to reveal it. The Mathieu modulation converges as well, but this is at least partly due to the overlapping interval fit: ![](http://imageshack.com/a/img924/8319/ZVt6P0.png) 
• Options
73.

Found the coolest paper on ARXIV. This is tangentially related to ENSO in terms of understanding how water will reach an equilibrium shape on a rotating planet.

The dawning of the theory of equilibrium figures: a brief historical account from the 17th through the 20th century -- Giuseppe Iurato

What's really interesting is that a Who's Who of scientists and mathematicians have worked on this problem over the centuries: Newton, Maclauren, Jacobi, Liouville, Dirichlet, Dedekind, Riemann, Roche, Lichtenstein, Poincare, Cartan, Chandresekhar, Lagrange, Legendre, Gauss, Laplace, Lord Kelvin, Poisson, Tchebychev, Lyapunov, Markov, the other Darwin, and others.

Amazing the amount of intellectual effort that has been applied to this topic.

Comment Source:Found the coolest paper on ARXIV. This is tangentially related to ENSO in terms of understanding how water will reach an equilibrium shape on a rotating planet. [The dawning of the theory of equilibrium figures: a brief historical account from the 17th through the 20th century -- Giuseppe Iurato](https://arxiv.org/abs/1409.3858) What's really interesting is that a Who's Who of scientists and mathematicians have worked on this problem over the centuries: Newton, Maclauren, Jacobi, Liouville, Dirichlet, Dedekind, Riemann, Roche, Lichtenstein, Poincare, Cartan, Chandresekhar, Lagrange, Legendre, Gauss, Laplace, Lord Kelvin, Poisson, Tchebychev, Lyapunov, Markov, the other Darwin, and others. Amazing the amount of intellectual effort that has been applied to this topic. 
• Options
74.

This is really a metrology exercise in terms of trying to deconvolve the forcing signal from a combined signal, given limited or noisy measurement data. Like if you take a pendulum and infer the force of gravity from the measured period. To make an analogy, a Mathieu formulation can model an inverted pendulum that is is positioned on a moving platform. If the underlying platform motion is set to the right range of periods, then the inverted pendulum can undergo a stable yet complex oscillatory pattern as the pair of focings interact with the non-linear natural response. That's similar to what this ENSO sloshing model describes.

As far as extracting the periodic signal, the technical challenge is similar to this common test: Consider that a 60 Hz noise source is generated and you are trying to detect it with as short an interval as possible.

Using the Solver technique, input a model paremeter set consisting of a 60 Hz sinusoid with a varying phase and amplitude, along with harmonics as coonstraints. This represents the unknown 60 Hz "Hum" noise. With a sample interval just a bit stronger and longer then the rectified period, this can reveal the underlying full-wave modulated signal .

Using a hybrid $\frac{CC}{\Delta Err}$ objective to get the best fit

The full-wave recified signal is considered a good test case for checking the precision of the solver. A rectified signal is rich in harmonics due to the sharp reversal around the zero-crossing, but if these aren't handled properly they can end up producing artifacts due to overfitting. In this case the pure correlation coefficient target metric shows more artifacts than the hybrid metric.

If the training interval is twice as long, it captures the shape more precisely. This fit includes the fundamental 60Hz plus 7 harmonics.

So for fitting ENSO, I am doing something akin to this. Only considering known tidal and wobble periods, along with the seasonal harmonics $\omega_o \pm n2\pi$ (which differ from the traditional harmonics, where $n\omega_0$ ), we can get a very good fit and one that validates on the intervals that are outside the training intervals.

Comment Source:This is really a metrology exercise in terms of trying to deconvolve the forcing signal from a combined signal, given limited or noisy measurement data. Like if you take a pendulum and infer the force of gravity from the measured period. To make an analogy, a Mathieu formulation can model an inverted pendulum that is is positioned on a moving platform. If the underlying platform motion is set to the right range of periods, then the inverted pendulum can undergo a stable yet complex oscillatory pattern as the pair of focings interact with the non-linear natural response. That's similar to what this ENSO sloshing model describes. As far as extracting the periodic signal, the technical challenge is similar to this common test: Consider that a 60 Hz noise source is generated and you are trying to detect it with as short an interval as possible. Using the Solver technique, input a model paremeter set consisting of a 60 Hz sinusoid with a varying phase and amplitude, along with harmonics as coonstraints. This represents the unknown 60 Hz "Hum" noise. With a sample interval just a bit stronger and longer then the rectified period, this can reveal the underlying full-wave modulated signal . Using a hybrid $\frac{CC}{\Delta Err}$ objective to get the best fit ![](http://imageshack.com/a/img921/1637/3N37MX.png) Using just a correlation cooefficient to get the best fit ![](http://imageshack.com/a/img924/9840/RuhzsA.png ) The full-wave recified signal is considered a good test case for checking the precision of the solver. A rectified signal is rich in harmonics due to the sharp reversal around the zero-crossing, but if these aren't handled properly they can end up producing artifacts due to overfitting. In this case the pure correlation coefficient target metric shows more artifacts than the hybrid metric. If the training interval is twice as long, it captures the shape more precisely. This fit includes the fundamental 60Hz plus 7 harmonics. ![](http://imageshack.com/a/img921/3421/W5I6BK.png) So for fitting ENSO, I am doing something akin to this. Only considering known tidal and wobble periods, along with the seasonal harmonics $\omega_o \pm n2\pi$ (which differ from the traditional harmonics, where $n\omega_0$ ), we can get a very good fit and one that validates on the intervals that are outside the training intervals. 
• Options
75.

"With a sample interval just a bit stronger and longer then the rectified period, this can reveal the underlying full-wave modulated signal ."

What is a stronger sampling interval? Can you give me a reference for how this method works? I only know the Nyquist sampling theorem?

Comment Source:"With a sample interval just a bit stronger and longer then the rectified period, this can reveal the underlying full-wave modulated signal ." What is a stronger sampling interval? Can you give me a reference for how this method works? I only know the Nyquist sampling theorem?
• Options
76.

Stronger as in noise-free, which is more important for a real-world signal such as ENSO. There is no noise in this artificial fullwave rectified example, so all that matters is the duration.

Nyquist sampling is important but so is the number of sinusoidal factors that can contribute. Also important is the number of periods that are quite close in value. This takes a longer interval to disambiguate the two.

How it works is just a nonlinear gradient search to find a set of values that maximizes an objective criteria such as the correlation coefficient. From scratch, it takes about 30 minutes to set up in an Excel spreadsheet.

Keep questions coming, if I don't explain lucidly. I am certain to get these equestions again.

There was a paper on how they used it for tidal prediction, which I get back to later.

Comment Source:Stronger as in noise-free, which is more important for a real-world signal such as ENSO. There is no noise in this artificial fullwave rectified example, so all that matters is the duration. Nyquist sampling is important but so is the number of sinusoidal factors that can contribute. Also important is the number of periods that are quite close in value. This takes a longer interval to disambiguate the two. How it works is just a nonlinear gradient search to find a set of values that maximizes an objective criteria such as the correlation coefficient. From scratch, it takes about 30 minutes to set up in an Excel spreadsheet. Keep questions coming, if I don't explain lucidly. I am certain to get these equestions again. There was a paper on how they used it for tidal prediction, which I get back to later.
• Options
77.

Thanks for the explanation.

Comment Source:Thanks for the explanation. 
• Options
78.
edited November 2016

BBC World Service, Science in Action has just carried an interview you might like. The British Antarctic Survey has now dated accelerated melting as starting with an El Nino in the 1940s via a teleconnection. I don't know if they've published a new time series for sub-icecap temperatures. If they have any correlations might be interesting. http://www.bbc.co.uk/programmes/p04gn2gd

Comment Source:BBC World Service, Science in Action has just carried an interview you might like. The British Antarctic Survey has now dated accelerated melting as starting with an El Nino in the 1940s via a teleconnection. I don't know if they've published a new time series for sub-icecap temperatures. If they have any correlations might be interesting. http://www.bbc.co.uk/programmes/p04gn2gd
• Options
79.

I don't trust teleconnections right now, apart from the teleconnection between the ocean and the lunisolar forcing. Think about it ... this is the teleconnection which has been the elephant in the room all this time.

All the other teleconnections I would think are essentially the dog wagging its tail. These work as the equivalent of weaker proxy measures for the impact of ENSO on other phenomena.

Above I also mentioned the MLR approaches used in tidal analysis and prediction. This is the tidal prediction bible from NOAA, which discusses the linear regression approaches. https://tidesandcurrents.noaa.gov/publications/Tidal_Analysis_and_Predictions.pdf

I would imagine that much of what I am doing is captured here, such as the biennial modulation leading to splitting of the spectral terms.

I haven't really taken advantage of the spectral frequency fits possible with the ENSO model so far, having preferred to stay in the time domain.

Comment Source:I don't trust teleconnections right now, apart from the teleconnection between the ocean and the lunisolar forcing. Think about it ... this is the teleconnection which has been the elephant in the room all this time. All the other teleconnections I would think are essentially the dog wagging its tail. These work as the equivalent of weaker proxy measures for the impact of ENSO on other phenomena. --- Above I also mentioned the MLR approaches used in tidal analysis and prediction. This is the tidal prediction bible from NOAA, which discusses the linear regression approaches. https://tidesandcurrents.noaa.gov/publications/Tidal_Analysis_and_Predictions.pdf I would imagine that much of what I am doing is captured here, such as the biennial modulation leading to splitting of the spectral terms. > ![](http://imageshack.com/a/img921/1899/KGSVnN.png) I haven't really taken advantage of the spectral frequency fits possible with the ENSO model so far, having preferred to stay in the time domain. 
• Options
80.

:)

Comment Source: :)
• Options
81.

This new blog article is the state of the art in blog science, FWIW

Spectral methods in GCMs - and some thoughts on CFD

I really get nothing out of this treatment because Nick Stokes is not addressing the fundamental physics at work. He's just getting sucker-punched by a climate troll from Boeing who knows something about aero modeling.

Comment Source:This new blog article is the state of the art in blog science, FWIW [Spectral methods in GCMs - and some thoughts on CFD](https://moyhu.blogspot.com/2016/11/spectral-methods-in-gcms-and-some.html) I really get nothing out of this treatment because Nick Stokes is not addressing the fundamental physics at work. He's just getting sucker-punched by a climate troll from Boeing who knows something about aero modeling. 
• Options
82.
edited November 2016

This is a table comparing the ENSO and QBO models. It's interesting that the QBO model is a good training for the ENSO model even though the basic formulation differs between the two.

The most significant similarity between the two is their reliance on lunisolar forcing, and in particular, a lunar forcing that is accentuated with a seasonal cycle such that the effective forcing periods are aliased versions of the known lunar monthly periods.

One of the lessons learned from this exercise is to avoid any kind of filtering on the data during the fitting process since a priori one can't distinguish noise from signal. The only filtering was done prior to accessing the data from the repository. This is unnerving because for the SOI data, the anti-correlation between the Darwin and Tahiti data appears poor (see the figure below), yet when the dipole is extracted, it must effectively remove the noise. The great unknown is how much noise remains after the model is fitted to the data. I

The ultimate reason for the relative simplicity of the QBO model was that it only has one significant lunar driver in terms of the nodal or draconic tide. In contrast, the ENSO model requires all three of the lunar terms -- nodal, synodic, and anomalistic, in addition to two Earth wobble terms.

Comment Source:This is a table comparing the ENSO and QBO models. It's interesting that the QBO model is a good training for the ENSO model even though the basic formulation differs between the two. ![](http://imageshack.com/a/img923/6785/IBz3R6.jpg) The most significant similarity between the two is their reliance on lunisolar forcing, and in particular, a lunar forcing that is accentuated with a seasonal cycle such that the effective forcing periods are aliased versions of the known lunar monthly periods. One of the lessons learned from this exercise is to avoid any kind of filtering on the data during the fitting process since a priori one can't distinguish noise from signal. The only filtering was done prior to accessing the data from the repository. This is unnerving because for the SOI data, the anti-correlation between the Darwin and Tahiti data appears poor (see the figure below), yet when the dipole is extracted, it must effectively remove the noise. The great unknown is how much noise remains after the model is fitted to the data. I ![](http://imageshack.com/a/img924/2168/xrmGNv.png) The ultimate reason for the relative simplicity of the QBO model was that it only has one significant lunar driver in terms of the nodal or draconic tide. In contrast, the ENSO model requires all three of the lunar terms -- nodal, synodic, and anomalistic, in addition to two Earth wobble terms. 
• Options
83.

The figures aren't showing up in my browser (chromium). hth.

Comment Source:The figures aren't showing up in my browser (chromium). hth. 
• Options
84.

Thanks, Jim. Think its fixed.

Comment Source:Thanks, Jim. Think its fixed.
• Options
85.
edited November 2016

Yep. +1

Comment Source:Yep. +1 
• Options
86.

This new blog post is the state-of-the-art in blog climate science, FWIW

Spectral methods in GCMs - and some thoughts on CFD

It's all so very pointless rope-a-dope of technical jargon, IMO. And worse are the comments by an aero guy at Boeing named Young. They can't seem to see the forest for the trees.

Comment Source:This new blog post is the state-of-the-art in blog climate science, FWIW [Spectral methods in GCMs - and some thoughts on CFD](https://moyhu.blogspot.com/2016/11/spectral-methods-in-gcms-and-some.html) It's all so very pointless rope-a-dope of technical jargon, IMO. And worse are the comments by an aero guy at Boeing named Young. They can't seem to see the forest for the trees. 
• Options
87.

The model I have for ENSO is a Mathieu equation for sloshing, which is analogous to the inverted pendulum on a moving cart experiment. As long as the cart shows a specific back-and-forth oscillation (or an up-and-down -- see below), the pendulum can stably invert.

So check out this cool video related to the inverted pendulum on a cart -- the "rolling" inverted pendulum. Last year Brian Josephson made a YouTube video where he demonstrates how the inverted pendulum property manifested itself in his kitchen !

No wonder he won a Nobel prize -- he is just a plainly intellectually curious fellow.

This is the up-and-down cyclically forced inverted pendulum

Again, it uses a Mathieu formulation as described in the video summary:

"This inverted pendulum was realized by only oscillating its pivot in 58Hz. It can be done without feedback control. You can find the principle explained in some textbooks about non-linear dynamics. This equation of motion is classfied as Mathieu equation. By solving this equation, you can find the stable condition to keep an inverted pendulum upright. See also WikiPedia, which has short description about how it works; http://en.wikipedia.org/wiki/Inverted_pendulum"

To many people, this looks counter-intuitive because they can't imagine what keeps the rod upright.

Comment Source:The model I have for ENSO is a Mathieu equation for sloshing, which is analogous to the inverted pendulum on a moving cart experiment. As long as the cart shows a specific back-and-forth oscillation (or an up-and-down -- see below), the pendulum can stably invert. So check out this cool video related to the inverted pendulum on a cart -- the "rolling" inverted pendulum. Last year Brian Josephson made a YouTube video where he demonstrates how the inverted pendulum property manifested itself in his kitchen ! <iframe width="854" height="480" src="https://www.youtube.com/embed/0tvpxxz8Sk8" frameborder="0" allowfullscreen></iframe> No wonder he won a Nobel prize -- he is just a plainly intellectually curious fellow. --- This is the up-and-down cyclically forced inverted pendulum <iframe width="854" height="480" src="https://www.youtube.com/embed/rwGAzy0noU0" frameborder="0" allowfullscreen></iframe> Again, it uses a Mathieu formulation as described in the video summary: >"This inverted pendulum was realized by only oscillating its pivot in 58Hz. It can be done without feedback control. You can find the principle explained in some textbooks about non-linear dynamics. This equation of motion is classfied as Mathieu equation. By solving this equation, you can find the stable condition to keep an inverted pendulum upright. See also WikiPedia, which has short description about how it works; http://en.wikipedia.org/wiki/Inverted_pendulum" To many people, this looks counter-intuitive because they can't imagine what keeps the rod upright. 
• Options
88.

This is the most impressive set of graphs that I have ever created based on the result of an automated solver:

The significance is that the red and blue time-series profiles individually decompose from non-overlapping intervals of the ENSO record from 1880 to 2013. Each set of curves took over an hour to calculate by running the Excel solver. This is an agonizingly slow process as it iterates over likely billions of calculations, yet, in the end the sets of curves asymptotically converge to essentially the same result, apart from a ~1.5 month shift between the red and the blue sets.

The coloration pattern is an artifact of the blue lines being drawn in the foreground and the red lines in the background. As the fit is highly coincident, the red lines hide behind the blue lines and only pop out where there is a discrepancy between the values. So the more blue you see, the higher the correlation coefficient. And based on my experiences, any coefficient above 0.8 is excellent agreement for highly oscillatory waveforms such as these.

This is one of those results that is well beyond requiring a statistical significance check since a coincidentally random convergence would obviously be extremely remote.

At the very least, this result should put to rest that ENSO is either a (1) chaotic or (2) random behavior. I am convinced the reason that the deterministic behavior has not been revealed in previous research is because applying a Mathieu modulation of a two-year period was never considered. This never occurred to anyone despite the fact that every text on the hydrodynamics of liquid sloshing asserts that a Mathieu-based wave equation represents the physics of the process, although not considered on an oceanic scale. Each of the following books and papers touches on that aspect:

• Benjamin, T. Brooke, and F. Ursell. "The stability of the plane free surface of a liquid in vertical periodic motion." Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. Vol. 225. No. 1163. The Royal Society, 1954.
• O. M. Faltinsen and A. N. Timokha, Sloshing. Cambridge University Press, 2009.
• O. M. Faltinsen, “A numerical nonlinear method of sloshing in tanks with two-dimensional flow,” Journal of Ship Research, vol. 22, no. 3, 1978.
• G. Wu, Q. Ma, and R. Eatock Taylor, “Numerical simulation of sloshing waves in a 3D tank based on a finite element method,” Applied Ocean Research, vol. 20, no. 6, pp. 337–355, 1998.
• R. A. Ibrahim, Liquid sloshing dynamics: theory and applications. Cambridge University Press, 2005.
• J. B. Frandsen, “Sloshing motions in excited tanks,” Journal of Computational Physics, vol. 196, no. 1, pp. 53–87, 2004.
• F. Dubois and D. Stoliaroff, “Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics,” arXiv preprint arXiv:1407.1829, 2014.

The one recent paper that may be the supporting groundbreaker, along with a cited paper, is

• J. Rajchenbach and D. Clamond, “Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited,” Journal of Fluid Mechanics, vol. 777, p. R2, 2015. PDF here
• Skeldon, A. C., and A. M. Rucklidge. "Can weakly nonlinear theory explain Faraday wave patterns near onset?." Journal of Fluid Mechanics 777 (2015): 604-632.

They make some strong claims concerning our understanding of the dispersion of waves, in particular when placed in the context of a Mathieu modulation.

Comment Source:This is the most impressive set of graphs that I have ever created based on the result of an automated solver: ![](http://imageshack.com/a/img922/1315/DAs5W0.png) The significance is that the red and blue time-series profiles individually decompose from **non-overlapping intervals** of the ENSO record from 1880 to 2013. Each set of curves took over an hour to calculate by running the Excel solver. This is an agonizingly slow process as it iterates over likely billions of calculations, yet, in the end the sets of curves asymptotically converge to essentially the same result, apart from a ~1.5 month shift between the red and the blue sets. The coloration pattern is an artifact of the blue lines being drawn in the foreground and the red lines in the background. As the fit is highly coincident, the red lines hide behind the blue lines and only pop out where there is a discrepancy between the values. So the more blue you see, the higher the correlation coefficient. And based on my experiences, any coefficient above 0.8 is excellent agreement for highly oscillatory waveforms such as these. This is one of those results that is well beyond requiring a statistical significance check since a coincidentally random convergence would obviously be extremely remote. At the very least, this result should put to rest that ENSO is either a (1) chaotic or (2) random behavior. I am convinced the reason that the deterministic behavior has not been revealed in previous research is because applying a Mathieu modulation of a two-year period was never considered. This never occurred to anyone despite the fact that every text on the hydrodynamics of liquid sloshing asserts that a Mathieu-based wave equation represents the physics of the process, although not considered on an oceanic scale. Each of the following books and papers touches on that aspect: * Benjamin, T. Brooke, and F. Ursell. "The stability of the plane free surface of a liquid in vertical periodic motion." Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. Vol. 225. No. 1163. The Royal Society, 1954. * O. M. Faltinsen and A. N. Timokha, *Sloshing*. Cambridge University Press, 2009. * O. M. Faltinsen, “A numerical nonlinear method of sloshing in tanks with two-dimensional flow,” Journal of Ship Research, vol. 22, no. 3, 1978. * G. Wu, Q. Ma, and R. Eatock Taylor, “Numerical simulation of sloshing waves in a 3D tank based on a finite element method,” Applied Ocean Research, vol. 20, no. 6, pp. 337–355, 1998. * R. A. Ibrahim, *Liquid sloshing dynamics: theory and applications*. Cambridge University Press, 2005. * J. B. Frandsen, “Sloshing motions in excited tanks,” Journal of Computational Physics, vol. 196, no. 1, pp. 53–87, 2004. * F. Dubois and D. Stoliaroff, “Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics,” arXiv preprint arXiv:1407.1829, 2014. The one recent paper that may be the supporting groundbreaker, along with a cited paper, is * J. Rajchenbach and D. Clamond, “Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited,” Journal of Fluid Mechanics, vol. 777, p. R2, 2015. [PDF here](http://contextearth.com/wp-content/uploads/2016/10/rajchenbach2015.pdf) * Skeldon, A. C., and A. M. Rucklidge. "Can weakly nonlinear theory explain Faraday wave patterns near onset?." Journal of Fluid Mechanics 777 (2015): 604-632. They make some strong claims concerning our understanding of the dispersion of waves, in particular when placed in the context of a Mathieu modulation. 
• Options
89.

This is another run with slightly different periods and with a mix of NINO34 and SOI data.

I continue to be amazed that the solver converges to the same set of forcing functions independent of the time interval chosen. Is this the sign of a stationary, deterministic, and ergodic time series? Stationary means that the properties of the time series don't change over time. Ergodic means that the properties of the time series can be deduced from a sufficiently long sample.

So this is all consistent with ENSO as the forced response to (1) the angular momentum wobble of the earth and (2) that of the gravitational pull of the moon and sun. Together, those are the only significant behaviors that also show the same long term stationary characteristics, and the fact that they share the same frequencies with the ENSO result makes the model difficult to invalidate.

I am posting these ideas because I am trying to articulate my arguments for the upcoming AGU presentation. Based on all the research I have searched on this topic, it's still the case that no one is close to settling on this same mechanism. The sense that I get is that most climate scientists believe that ENSO is driven primarily by the direction of the currently prevailing wind. But what causes the wind direction? One can look at QBO, which I will be presenting as well and see that has a primarily lunisolar forcing mechanism, so that's in my favor.

Comment Source:This is another run with slightly different periods and with a mix of NINO34 and SOI data. ![](http://imagizer.imageshack.us/a/img923/1209/sDD9C8.png) I continue to be amazed that the solver converges to the same set of forcing functions independent of the time interval chosen. Is this the sign of a stationary, deterministic, and ergodic time series? Stationary means that the properties of the time series don't change over time. Ergodic means that the properties of the time series can be deduced from a sufficiently long sample. So this is all consistent with ENSO as the forced response to (1) the angular momentum wobble of the earth and (2) that of the gravitational pull of the moon and sun. Together, those are the only significant behaviors that also show the same long term stationary characteristics, and the fact that they share the same frequencies with the ENSO result makes the model difficult to invalidate. I am posting these ideas because I am trying to articulate my arguments for the upcoming AGU presentation. Based on all the research I have searched on this topic, it's still the case that no one is close to settling on this same mechanism. The sense that I get is that most climate scientists believe that ENSO is driven primarily by the direction of the currently prevailing wind. But what causes the wind direction? One can look at QBO, which I will be presenting as well and see that has a primarily lunisolar forcing mechanism, so that's in my favor. 
• Options
90.

I hope you cause something of a kerfuffle, especially with the paper!

Comment Source:I hope you cause something of a kerfuffle, especially with the paper! 
• Options
91.
edited December 2016

Fascinating paper by Lord Rayleigh from 1883 whereby he clearly describes the phenomenon of period doubling. He also describes the Mathieu equation, which was not widely recognized at the time, having been described just a decade earlier.

Rayleigh, Lord. "XXXIII. on maintained vibrations." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 15, no. 94 (1883): 229-235.

Review by Dino Boccaletti, Giuseppe Pucacco in Theory of Orbits: Perturbative and Geometrical Methods (2013)

Rayleigh used the Mathieu equation in a much more general context than Mathieu, who only applied it to a specific elliptical-shaped membrane in 1868.

Comment Source:Fascinating paper by Lord Rayleigh from 1883 whereby he clearly describes the phenomenon of period doubling. He also describes the Mathieu equation, which was not widely recognized at the time, having been described just a decade earlier. > Rayleigh, Lord. "XXXIII. on maintained vibrations." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 15, no. 94 (1883): 229-235. ![](http://imageshack.com/a/img924/7483/g9Zavd.png) ![](http://imageshack.com/a/img923/3918/C9Da61.png) ![](http://imageshack.com/a/img924/1510/nA0LJu.png) Review by Dino Boccaletti, Giuseppe Pucacco in Theory of Orbits: Perturbative and Geometrical Methods (2013) ![](http://imageshack.com/a/img924/2370/4PgP14.png) Rayleigh used the Mathieu equation in a much more general context than Mathieu, who only applied it to a specific elliptical-shaped membrane in 1868. 
• Options
92.

Presented ENSO and QBO models at AGU 2016. Audience was stunned. No questions.

Here is a powerpoint of the slides https://1drv.ms/p/s!AgwV65SOPvtOm33jA5R_1fb6fcp1

Comment Source:Presented ENSO and QBO models at AGU 2016. Audience was stunned. No questions. Here is a powerpoint of the slides https://1drv.ms/p/s!AgwV65SOPvtOm33jA5R_1fb6fcp1 
• Options
93.

The slides link doesn't work for me on chromium: 'Server Error in '/p' Application' 'Runtime Error.'

Comment Source:The slides link doesn't work for me on chromium: 'Server Error in '/p' Application' 'Runtime Error.' 
• Options
94.

Thanks for checking. That link apparently runs a server-driven powerpoint viewer which is not platform independent

Comment Source:Thanks for checking. That link apparently runs a server-driven powerpoint viewer which is not platform independent Try this as the direct download link: https://vff5jg-bn1305.files.1drv.com/y3mYWy1zFfWGB9xB26pLc-J3u8NbJSb9t3e96A3K6l4M20MUiLj0K8V6f3Wn3jlo-Fwoa2_faOYBfWNquPPJU5kyX587rj3TknApwL5PNAMG-k/AnalyticalFormulationOfEquatorialStandingWavePhenomena2.ppsx?download&psid=1 
• Options
95.
edited December 2016

One of the reasons that the metastable biennial nature of ENSO is not particularly foreign to me is that I recall grappling with a similar issue during my grad school days.

We were trying to grow GaAs layers on Ge for the long-term objective of optoelectronic integration -- optical column III-V compounds on electronic column IV substrates. I have the paper here on my ResearchGate site:

Suppression of antiphase domains in the growth of GaAs on Ge(100) by molecular beam epitaxy, Journal of Crystal Growth 81(1):214-220 · February 1987

If you read this you will see how we were able to infer that the Ge surface flipped from a natural 2-atomic-step pattern to a double 4-atomic step pattern when elemental As was applied.

Now, consider that this situation is completely metastable because the initiation of period doubling could occur on even-numbered step sequences or on odd-numbered sequences (with the sequence starting from some arbitrary point on the surface). That's the spatial equivalent of ENSO potentially adopting a biennial period in the time domain.

Yet, because we had laboratory control over the environment, we were able to watch when this metastable transition became completely mixed with anti-phase domains, which are mixtures of even and odd-sequenced steps. (you can see from all the experimental data that I used to be quite the lab rat at one time)

The problem is that -- with climate data -- we have absolutely no experimental controls we can place on anything, so all we can do is infer. So this coherent metastable doubling phenomenon does occur in nature, but how to validate the effect is where it gets tricky.

Comment Source:One of the reasons that the metastable biennial nature of ENSO is not particularly foreign to me is that I recall grappling with a similar issue during my grad school days. We were trying to grow GaAs layers on Ge for the long-term objective of optoelectronic integration -- optical column III-V compounds on electronic column IV substrates. I have the paper here on my ResearchGate site: [Suppression of antiphase domains in the growth of GaAs on Ge(100) by molecular beam epitaxy](https://www.researchgate.net/publication/222779719_Suppression_of_antiphase_domains_in_the_growth_of_GaAs_on_Ge100_by_molecular_beam_epitaxy), Journal of Crystal Growth 81(1):214-220 · February 1987 If you read this you will see how we were able to infer that the Ge surface flipped from a natural 2-atomic-step pattern to a double 4-atomic step pattern when elemental As was applied. ![](http://imageshack.com/a/img923/6916/SkZUiq.png) Now, consider that this situation is completely metastable because the initiation of period doubling could occur on even-numbered step sequences or on odd-numbered sequences (with the sequence starting from some arbitrary point on the surface). That's the *spatial* equivalent of ENSO potentially adopting a biennial period in the time domain. Yet, because we had laboratory control over the environment, we were able to watch when this metastable transition became completely mixed with anti-phase domains, which are mixtures of even and odd-sequenced steps. (you can see from all the experimental data that I used to be quite the lab rat at one time) The problem is that -- with climate data -- we have absolutely no experimental controls we can place on anything, so all we can do is infer. So this coherent metastable doubling phenomenon does occur in nature, but how to validate the effect is where it gets tricky. ![](http://imageshack.com/a/img923/2548/oSnTsb.png) 
• Options
96.

Nope: blank screen. I suggest you just stick a pdf on github. 'Stunned' is a bit silent ;).
I'd like to know if you got any feedback at all?

Comment Source:Nope: blank screen. I suggest you just stick a pdf on github. 'Stunned' is a bit silent ;). I'd like to know if you got any feedback at all?
• Options
97.

Jim, I did talk to some of the poster presenters who are doing similar work. My opinion is that the only ones that will figure this out are the mathematicians that are doing non-linear geophysics, so I hit the posters doing that work. The ones I talked to are looking at higher-latitude turbulence models of the ocean and atmosphere. I tried to convince them that they have to look at the most fundamental models that have the fewest degrees of freedom and dimensionality and then work up from there. And that would include QBO and ENSO. I got the impression that they would rather work on the stochastic properties of the high vorticity turbulent behaviors rather than anything more deterministic and anti-vortex along the equator.

As for the others, I get the impression that they all think that ENSO is caused by the wind, and that QBO is caused by gravitational flows coming from the ocean. Yet they don't see the circular reasoning flaw in that argument. The analogy is that someone thinking that electrical current is due to the voltage, and that the voltage is due to IR drop, but not realizing that another potential drives the circuit externally.

Comment Source:Jim, I did talk to some of the poster presenters who are doing similar work. My opinion is that the only ones that will figure this out are the mathematicians that are doing non-linear geophysics, so I hit the posters doing that work. The ones I talked to are looking at higher-latitude turbulence models of the ocean and atmosphere. I tried to convince them that they have to look at the most fundamental models that have the fewest degrees of freedom and dimensionality and then work up from there. And that would include QBO and ENSO. I got the impression that they would rather work on the stochastic properties of the high vorticity turbulent behaviors rather than anything more deterministic and anti-vortex along the equator. As for the others, I get the impression that they all think that ENSO is caused by the wind, and that QBO is caused by gravitational flows coming from the ocean. Yet they don't see the circular reasoning flaw in that argument. The analogy is that someone thinking that electrical current is due to the voltage, and that the voltage is due to IR drop, but **not** realizing that another potential drives the circuit externally. 
• Options
98.

Having a few problems uploading the QBO+ENSO presentation because the PowerPoint wants to save it as a 60 MB file. It doesn't matter if I compress the pictures, it's still about 60 MB.

So in the meantime, I uploaded it as a YouTube video, which you can pause https://youtu.be/KlbfcZ-KF5I

Nothing new in the presentation which I haven't gone over before.

Comment Source:Having a few problems uploading the QBO+ENSO presentation because the PowerPoint wants to save it as a 60 MB file. It doesn't matter if I compress the pictures, it's still about 60 MB. So in the meantime, I uploaded it as a YouTube video, which you can pause https://youtu.be/KlbfcZ-KF5I Nothing new in the presentation which I haven't gone over before. 
• Options
99.

+1.

Comment Source:+1. 
• Options
100.

In retrospect, I think the reason I didn't get any questions was that there was no gap between the presentations and mine was done in 15 minutes and so they had to move to the next one.

BTW, the previous speaker was a Cornell U senior and she machine-gunned her talk on thermocline motion and so was done early. There were two questions from the audience and she had no clue on how to answer them. That's the state-of-the-art for you :)

Comment Source:In retrospect, I think the reason I didn't get any questions was that there was no gap between the presentations and mine was done in 15 minutes and so they had to move to the next one. BTW, the previous speaker was a Cornell U senior and she machine-gunned her [talk on thermocline motion](https://agu.confex.com/agu/fm16/meetingapp.cgi/Paper/142091) and so was done early. There were two questions from the audience and she had no clue on how to answer them. That's the state-of-the-art for you :) 
This discussion has been closed.