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## Comments

On the QBO and ENSO, I was musing about the question of what, in terms of tectonic movements, it would take to distrupt the oscillation and, shy of that, to perturb it detectably? Is there some aspect of the signal that could, say, detect the formation of a new volcanic island in the equatorial band?

`On the QBO and ENSO, I was musing about the question of what, in terms of tectonic movements, it would take to distrupt the oscillation and, shy of that, to perturb it detectably? Is there some aspect of the signal that could, say, detect the formation of a new volcanic island in the equatorial band?`

Jan, If it adds a significant delta angular momentum push, it would probably be detectable. Krakatoa was the formation of an island in a narrow strait which would impact the oceanic flow east-to-west and west-to-east

Discussed this aspect here: http://contextearth.com/2018/05/08/unified-enso-proxy/#comment-208496

Added info: On 12/22/2018, a killer tsunami related to Krakatoa hit this same Sunda Strait region depicted above. It may again transiently impact the flow hydrodynamics as the original Krakatoa eruption did.

`Jan, If it adds a significant delta angular momentum push, it would probably be detectable. Krakatoa was the formation of an island in a narrow strait which would impact the oceanic flow east-to-west and west-to-east Discussed this aspect here: http://contextearth.com/2018/05/08/unified-enso-proxy/#comment-208496 ![](http://imagizer.imageshack.us/a/img924/6189/v9hxUR.png) Added info: On 12/22/2018, a killer tsunami related to Krakatoa hit this same Sunda Strait region depicted above. It may again transiently impact the flow hydrodynamics as the original Krakatoa eruption did.`

The book Mathematical Geoenergy is out and available on the Wiley/AGU site and other online sellers such as Amazon.

Everything you wanted to know about fossil fuels, renewable energy, and climate science treated in simple mathematical terms is presented in around 400 pages with several hundred figures.

This forum was of some help in ironing out some of the models, so thanks to Jim S, Jan, and a few other diehards who have commented in these threads over the past several years.

`The book Mathematical Geoenergy is out and available on the Wiley/AGU site and other online sellers such as Amazon. Everything you wanted to know about fossil fuels, renewable energy, and climate science treated in simple mathematical terms is presented in around 400 pages with several hundred figures. This forum was of some help in ironing out some of the models, so thanks to Jim S, Jan, and a few other diehards who have commented in these threads over the past several years.`

Congrats Paul! https://smile.amazon.com/Mathematical-Geoenergy-Discovery-Depletion-Geophysical-ebook/dp/B07L8CZSP9/

`Congrats Paul! https://smile.amazon.com/Mathematical-Geoenergy-Discovery-Depletion-Geophysical-ebook/dp/B07L8CZSP9/`

Thanks Jan.

Also wanted to mention that the final peer-reviewed models for QBO and ENSO are included in the text

`Thanks Jan. Also wanted to mention that the final peer-reviewed models for QBO and ENSO are included in the text`

The experts might know about this already, but the following paper just "hit the streets":

T. Bolton, L. Zanna, "Applications of deep learning to ocean data inference and sub‐grid parameterisation",

JournalofAdvancesinModelingEarthSystems, 4 January 2019.This is (at least) the second paper which applies machine learning to the deep technical problem of convective fluids. The first I know of is

P. A. O'Gorman, J. G. Dwyer, "Using machine learning to parameterize moist convection: potential for modeling of climate, climate change and extreme events",

JournalofAdvancesinModelingEarthSystems, 3 October 2018.To me this is very heartening, and I'm having a solid chuckle because a well-known expert in such systems announced at the Lorenz-Cheney symposium last year in response to a question that such techniques offered no promised at all for work in this area.

`The experts might know about this already, but the following paper just "hit the streets": T. Bolton, L. Zanna, "[Applications of deep learning to ocean data inference and sub‐grid parameterisation](https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2018MS001472)", _Journal_ _of_ _Advances_ _in_ _Modeling_ _Earth_ _Systems_, 4 January 2019. This is (at least) the second paper which applies machine learning to the deep technical problem of convective fluids. The first I know of is P. A. O'Gorman, J. G. Dwyer, "[Using machine learning to parameterize moist convection: potential for modeling of climate, climate change and extreme events](https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2018MS001351)", _Journal_ _of_ _Advances_ _in_ _Modeling_ _Earth_ _Systems_, 3 October 2018. To me this is very heartening, and I'm having a solid chuckle because a well-known expert in such systems announced at the Lorenz-Cheney symposium last year in response to a question that such techniques offered no promised at all for work in this area.`

I think machine learning will help explain some surprising features in the future.

Also some of the condensed matter ideas will filter in:

Mathematicians Tame Turbulence in Flattened Fluids | Quanta Magazine https://t.co/ZyyQRqCiI1

`I think machine learning will help explain some surprising features in the future. Also some of the condensed matter ideas will filter in: Mathematicians Tame Turbulence in Flattened Fluids | Quanta Magazine https://t.co/ZyyQRqCiI1 ![](https://pbs.twimg.com/media/DweI4r3X4AEBg4y.jpg)`

Speaking of all this neat stuff, have you seen the Vertex Ring Collision videos out of Harvard? I first saw them at said Lorenz-Charney Symposium:

`Speaking of all this neat stuff, have you seen the Vertex Ring Collision videos out of Harvard? I first saw them at said Lorenz-Charney Symposium: [![Vortex ring collisions](http://img.youtube.com/vi/_z099yZzQik/0.jpg)](http://www.youtube.com/watch?v=_z099yZzQik)`

Thanks Jan,

As Brad Marston says, the key to modeling toroids (vortex rings), such as QBO, is to assume two equators. The one equator is the physical equator of the earth, while the other equator is the characteristic internal twisting of the vortex ring.

The lower dimensionality must lead to the inverse cascade and the preservation of order at the expense of further finer-levels of turbulence. That must be why QBO and ENSO are not at all turbulent flows. Gravity and the Coriolis forces around the physical equator are constraining the dimensionality of the system, while the gravitational influence of the moon and sun are providing additonal perturbing driving tidal forces.

On the other hand, the Vortex Ring Collision videos show what happens when the flow is unconstrained in the outward direction.

There's this guy on the blogs, David P. Young, who is apparently an aero engineer at Boeing, who claims that further modeling of fluid dynamics is hopeless for climate science, because the turbulence cascade always goes toward finer levels of details. And because the finer levels take more and more computational power, then it becomes impossible to make any progress and any further research, such as with climate GCMs, is doomed to failure.

But as we see with an inverse cascade, quite the opposite occurs -- the behaviors may become more ordered and occur at larger spatial scales. This is exactly what Brad Marston is pointing out and counters what naysayers like Young are claiming.

`Thanks Jan, As Brad Marston says, the key to modeling toroids (vortex rings), such as QBO, is to assume two equators. The one equator is the physical equator of the earth, while the other equator is the characteristic internal twisting of the vortex ring. ![](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Vortex_ring.gif/250px-Vortex_ring.gif) The lower dimensionality must lead to the inverse cascade and the preservation of order at the expense of further finer-levels of turbulence. That must be why QBO and ENSO are not at all turbulent flows. Gravity and the Coriolis forces around the physical equator are constraining the dimensionality of the system, while the gravitational influence of the moon and sun are providing additonal perturbing driving tidal forces. On the other hand, the Vortex Ring Collision videos show what happens when the flow is unconstrained in the outward direction. --- There's this guy on the blogs, David P. Young, who is apparently an aero engineer at Boeing, who claims that further modeling of fluid dynamics is hopeless for climate science, because the turbulence cascade always goes toward finer levels of details. And because the finer levels take more and more computational power, then it becomes impossible to make any progress and any further research, such as with climate GCMs, is doomed to failure. But as we see with an inverse cascade, quite the opposite occurs -- the behaviors may become more ordered and occur at larger spatial scales. This is exactly what Brad Marston is pointing out and counters what naysayers like Young are claiming.`

Hi..following and thinking about it..

`Hi..following and thinking about it..`

Paul, something I've been interested in for a time -- but I don't have the physics or math skills to seriously pursue it -- is the idea of oceanic eddies as dissipators of energy, ultimately as heat. As you probably know, but for the benefit of readership, eddies in oceans break off of major currents and they can persist a long time, on the order of months, They also spin off subsidiary eddies. Such turbulent rings, on a much smaller scale, are created and exploited by many swimming creatures, such as fish and whales. Because they persist, they offer a volume of water such swimmers can push against. Indeed, the common observation of dolphins swimming along ships near their bows is their stealing energy from the ship to help propel them.

In the case of oceanic currents, given that the driving engine for transport equator-to-pole is heat transport, resulting in moving, I wonder how much of the big eddies to small eddies to smaller eddies to smallest eddies results in transfer of that same energy to locales, quite apart from conduction.

`Paul, something I've been interested in for a time -- but I don't have the physics or math skills to seriously pursue it -- is the idea of oceanic eddies as dissipators of energy, ultimately as heat. As you probably know, but for the benefit of readership, eddies in oceans break off of major currents and they can persist a long time, on the order of months, They also spin off subsidiary eddies. Such turbulent rings, on a much smaller scale, are created and exploited by many swimming creatures, such as fish and whales. Because they persist, they offer a volume of water such swimmers can push against. Indeed, the common observation of dolphins swimming along ships near their bows is their stealing energy from the ship to help propel them. In the case of oceanic currents, given that the driving engine for transport equator-to-pole is heat transport, resulting in moving, I wonder how much of the big eddies to small eddies to smaller eddies to smallest eddies results in transfer of that same energy to locales, quite apart from conduction.`

I think the behavior known as Tropical Instability Waves is the hot topic right now. Massive vortices with sharply delineated edges created along the equator and aligned with phases of ENSO.

`I think the behavior known as Tropical Instability Waves is the hot topic right now. Massive vortices with sharply delineated edges created along the equator and aligned with phases of ENSO.`

Paul,"aligned with phases of ENSO", could you please detail it a bit?As I understood it you spotted lunar and solar influence that indeed allow to predict ENSO but the underlayer mechanism remains to be totally described ( possible related to topology), isnt it?

`Paul,"aligned with phases of ENSO", could you please detail it a bit?As I understood it you spotted lunar and solar influence that indeed allow to predict ENSO but the underlayer mechanism remains to be totally described ( possible related to topology), isnt it?`

The general connection is that the Tropical Instability Waves appear stronger during La Nina phases.

Precisely along the equator there is no preferred vortex behavior as Coriolis cancels, which forms the premise for an ENSO standing wave solution. Jan was asking about vortices, so that's why I mentioned the TIW.

My ENSO+QBO equatorial solution is here which is a Google books excerpt. Start with equations 11.1, then apply the 2-equator topological ansatz of 11.2 when simplifying, and the analytical standing wave solution of 11.3 results.

I propose that high wave-number solutions when perturbed off-equator will lead to the TIW.

`The general connection is that the Tropical Instability Waves appear stronger during La Nina phases. Precisely along the equator there is no preferred vortex behavior as Coriolis cancels, which forms the premise for an ENSO standing wave solution. Jan was asking about vortices, so that's why I mentioned the TIW. My ENSO+QBO equatorial solution is [here](https://books.google.ch/books?id=xb17DwAAQBAJ&printsec=frontcover&dq=mathematical+geoenergy&hl=en&sa=X&ved=0ahUKEwjYqpO8iPXfAhUNYVAKHZSZCGcQ6AEIKDAA#v=onepage&q=Yanai&f=false) which is a Google books excerpt. Start with equations 11.1, then apply the 2-equator topological ansatz of 11.2 when simplifying, and the analytical standing wave solution of 11.3 results. I propose that high wave-number solutions when perturbed off-equator will lead to the TIW.`

Recent strong Tropical Instability Waves apparent along the eastern equatorial Pacific https://youtu.be/rS95rPXZMUg

`Recent strong Tropical Instability Waves apparent along the eastern equatorial Pacific https://youtu.be/rS95rPXZMUg`

Pacific seems to be an isolated system..?TIW doenst relate to El nino, only la nina?

`Pacific seems to be an isolated system..?TIW doenst relate to El nino, only la nina?`

@Pierre Prado ... The Pacific is an

unusualsystem: It's the only place (presently) on the planet where you have a big stretch of ocean where the Coriolis force is essentially zero, along the Equator.`@Pierre Prado ... The Pacific is an _unusual_ system: It's the only place (presently) on the planet where you have a big stretch of ocean where the Coriolis force is essentially zero, along the Equator.`

The TIW may be there for El Nino, it is just more obvious with La Nina.

I talked to one presenter at AGU about the sharpness of the front

Over a short lateral distance the vortex front shows a sharp change in temperature. The vortices are large in diameter and regularly spaced like traveling waves

Warner, S. J., Holmes, R. M., M. Hawkins, E. H., S. Hoecker-Martínez, M., Savage, A. C., & Moum, J. N. (2018). Buoyant Gravity Currents Released from Tropical Instability Waves. Journal of Physical Oceanography, 48(2), 361–382. doi:10.1175/jpo-d-17-0144.1

`The TIW may be there for El Nino, it is just more obvious with La Nina. I talked to one presenter at AGU about the sharpness of the front ![](https://imageshack.com/a/img923/889/vPMRNO.gif) Over a short lateral distance the vortex front shows a sharp change in temperature. The vortices are large in diameter and regularly spaced like traveling waves ![](https://imageshack.com/a/img921/7118/yi86cs.gif) Warner, S. J., Holmes, R. M., M. Hawkins, E. H., S. Hoecker-Martínez, M., Savage, A. C., & Moum, J. N. (2018). Buoyant Gravity Currents Released from Tropical Instability Waves. Journal of Physical Oceanography, 48(2), 361–382. doi:10.1175/jpo-d-17-0144.1`

Thanks Jan...Interesting Paul..

`Thanks Jan...Interesting Paul..`

Jan said:

Agree. What I want to do next is perturb the equatorial-only analytic solution with the off-latitude Coriolis factors and see if there is an elegant way to capture the TIW vortices. It may not remain analytic but there is still the travelling wave part of the solution I haven't explored.

As with electromagnetic modelling, looking at quasi-static behavior - standing wave vs travelling wave, etc - is always enlightening.

`Jan said: > The Pacific is an unusual system: It's the only place (presently) on the planet where you have a big stretch of ocean where the Coriolis force is essentially zero, along the Equator. Agree. What I want to do next is perturb the equatorial-only analytic solution with the off-latitude Coriolis factors and see if there is an elegant way to capture the TIW vortices. It may not remain analytic but there is still the travelling wave part of the solution I haven't explored. As with electromagnetic modelling, looking at quasi-static behavior - standing wave vs travelling wave, etc - is always enlightening.`

looking at quasi-static behavior - standing wave vs travelling wave, etc - is always enlightening...do you have a ref to recommend (fluid context)?

`looking at quasi-static behavior - standing wave vs travelling wave, etc - is always enlightening...do you have a ref to recommend (fluid context)?`

Pierre, It looks as if the quasi-static approximation has a different meaning in fluid dynamics vs electromagnetism. What I want to explore is how a standing wave such as ENSO can be related to a travelling wave such as TIW. The analytic solution is a standing wave, as the temporal and spatial dimensions are seperable. I am thinking about the math involved in casting it back into a travelling wave. There is likely a slow or quasi-static approximation that blends the temporal and spatial and thus creating a travelling wave.

`Pierre, It looks as if the quasi-static approximation has a different meaning in fluid dynamics vs electromagnetism. What I want to explore is how a standing wave such as ENSO can be related to a travelling wave such as TIW. The analytic solution is a standing wave, as the temporal and spatial dimensions are seperable. I am thinking about the math involved in casting it back into a travelling wave. There is likely a slow or quasi-static approximation that blends the temporal and spatial and thus creating a travelling wave.`

What's intriguing about the equatorial behaviors of ENSO and QBO is that they should be driven by congruent forcing factors. ENSO is much more complex so it has a few factors to choose from, but the QBO is driven primarily by the Draconic cycle (with a small seasonal factor).

After modeling ENSO to a rather precise degree, the Draconic forcing was used as a starting point to model QBO. It was then allowed to vary slightly to enable a gradient-search fit.

This is how the two forcing factors align:

This is how well the QBO model fits the 30 hPa QBO data using essentially the same Draconic profile as was used for ENSO:

The odds of that happening by chance is exceedingly low. When fitting the ENSO data, there was no training against QBO, yet very close to the same factor worked for QBO. That's a type of cross-validation that's vitally important for climate modeling.

`What's intriguing about the equatorial behaviors of ENSO and QBO is that they should be driven by congruent forcing factors. ENSO is much more complex so it has a few factors to choose from, but the QBO is driven primarily by the Draconic cycle (with a small seasonal factor). After modeling ENSO to a rather precise degree, the Draconic forcing was used as a starting point to model QBO. It was then allowed to vary slightly to enable a gradient-search fit. This is how the two forcing factors align: ![cc](https://imageshack.com/a/img922/8009/7EyJNs.gif) This is how well the QBO model fits the 30 hPa QBO data using essentially the same Draconic profile as was used for ENSO: ![qbo](https://imageshack.com/a/img922/6417/tDYJxZ.gif) The odds of that happening by chance is exceedingly low. When fitting the ENSO data, there was no training against QBO, yet very close to the same factor worked for QBO. That's a type of cross-validation that's vitally important for climate modeling.`

Of possible interest ...

Muis, S., Haigh, I. D., Guimarães Nobre, G., Aerts, J. C. J. H., & Ward, P. J. (2018). Influence of El Niño‐Southern Oscillation on global coastal flooding. Earth's Future, 6, 1311–1322. https://doi.org/10.1029/2018EF000909

I've corresponded with Professor Haigh when I misunderstood and was wrong about something ...

`Of possible interest ... Muis, S., Haigh, I. D., Guimarães Nobre, G., Aerts, J. C. J. H., & Ward, P. J. (2018). Influence of El Niño‐Southern Oscillation on global coastal flooding. Earth's Future, 6, 1311–1322. https://doi.org/10.1029/2018EF000909 I've corresponded with Professor Haigh [when I misunderstood and was wrong about something](https://667-per-cm.net/2014/04/21/comment-on-timescales-for-detecting-a-significant-acceleration-in-sea-level-rise-by-haigh-et-al/) ...`

We presented models for ENSO (i.e. the El Nino behavior), QBO, and a few other climate indices at the American Geophysical Union meeting last month in DC and the presentation PDF is now available from the ESSOAR archive here:

https://www.essoar.org/doi/10.1002/essoar.10500568.1

`We presented models for ENSO (i.e. the El Nino behavior), QBO, and a few other climate indices at the American Geophysical Union meeting last month in DC and the presentation PDF is now available from the ESSOAR archive here: https://www.essoar.org/doi/10.1002/essoar.10500568.1`

Great..

`Great..`

The presentation included an analysis of the North Atlantic Oscillation (NAO) which is a more rapid dipole.

What I am working on right now appears more promising, as can now sustain a correlation over a wider range

`The presentation included an analysis of the North Atlantic Oscillation (NAO) which is a more rapid dipole. ![](http://imagizer.imageshack.us/a/img923/2047/9qN1Ui.png) What I am working on right now appears more promising, as can now sustain a correlation over a wider range`

The differences between ENSO and NAO are stark. ENSO has stronger cycles from 2 to 7 years, while NAO can show fluctuations at 2 or more per year. Yet, essentially the same forcing (as shown in the cross-correlation plots along the lower pane above) can model each time-series effectively.

Why this happens is that the higher-K standing waves are producing the NAO behavior, while the lower-K produce the ENSO standing waves. Because of dispersion, high-K correspond to a higher temporal frequency and thus NAO are higher frequency.

Qualitatively, the fit to NAO doesn't look as good as to ENSO, but this is at least partly due to the large number of cycles we are trying to fit to and the fact that we are at a monthly sampling which adds temporal uncertainty.

`The differences between ENSO and NAO are stark. ENSO has stronger cycles from 2 to 7 years, while NAO can show fluctuations at 2 or more per year. Yet, essentially the same forcing (as shown in the cross-correlation plots along the lower pane above) can model each time-series effectively. Why this happens is that the higher-K standing waves are producing the NAO behavior, while the lower-K produce the ENSO standing waves. Because of dispersion, high-K correspond to a higher temporal frequency and thus NAO are higher frequency. ![](https://imageshack.com/a/img923/5189/EYB0Uy.png) Qualitatively, the fit to NAO doesn't look as good as to ENSO, but this is at least partly due to the large number of cycles we are trying to fit to and the fact that we are at a monthly sampling which adds temporal uncertainty.`

I wonder, Paul (@WebHubTel), whether this kind of analysis could be adapted to studies of the North Wall of the Gulf Stream, per https://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-18-0212.1 or Wolfe, Hameed, Chi, "On the drivers of decadal variability of the Gulf Stream North Wall",

JournalofClimate, February 2019. They are looking at frequency domain connections between North Wall variability and NAO as well as the less well known Atlantic meridional mode (AMM) which is an equatorial process in the Atlantic. The Stream and its North Wall are important for Northeastern U.S. weather as well as primary production in the region.`I wonder, Paul (@WebHubTel), whether this kind of analysis could be adapted to studies of the North Wall of the Gulf Stream, per https://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-18-0212.1 or Wolfe, Hameed, Chi, "On the drivers of decadal variability of the Gulf Stream North Wall", *Journal* *of* *Climate*, February 2019. They are looking at frequency domain connections between North Wall variability and NAO as well as the less well known Atlantic meridional mode (AMM) which is an equatorial process in the Atlantic. The Stream and its North Wall are important for Northeastern U.S. weather as well as primary production in the region.`

Jan, There may be something to that. I split up the NAO time series into 3 separate 23 year intervals and found that the same lunar forcing works for each, but that the standing wave indices differ for each interval. This may be the variability in the North Wall over time. I am playing around with this while writing this response. thanks

`Jan, There may be something to that. I split up the NAO time series into 3 separate 23 year intervals and found that the same lunar forcing works for each, but that the standing wave indices differ for each interval. This may be the variability in the North Wall over time. I am playing around with this while writing this response. thanks`

Welcome, Paul. Glad. Hope there is a demonstrable connection. That would be cool. To me, it also means it seems there's something more universal in the mechanism you've calibrated and discovered for ENSO.

`Welcome, Paul. Glad. Hope there is a demonstrable connection. That would be cool. To me, it also means it seems there's something more universal in the mechanism you've calibrated and discovered for ENSO.`

this link seems to connect gulf stream/enso/nao https://en.m.wikipedia.org/wiki/Latitude_of_the_Gulf_Stream_and_the_Gulf_Stream_north_wall_index

`this link seems to connect gulf stream/enso/nao https://en.m.wikipedia.org/wiki/Latitude_of_the_Gulf_Stream_and_the_Gulf_Stream_north_wall_index`

From Pierre's link:

The distinction between a teleconnection and a common-mode forcing is important. Why would a behavior in one ocean basin remotely impact a behavior in a geographically separated one? I know it's possible to create a mathematical network connection between the two, but is that physically plausible? (note: this was in fact the motivation for the Azimuth Project ENSO study if you look back to much earlier discussion threads)

Much more plausible that a common-mode forcing mechanism links the two. The forcing is much the same, but the difference in the basin structure creates a different standing wave pattern. Now, and imho, it is possible that 2nd order factors are influencing the North Wall, which would presumably then impact the standing wave boundary conditions.

`From Pierre's link: ![](https://pbs.twimg.com/media/DyUlPNBUwAET2oV.jpg) The distinction between a teleconnection and a common-mode forcing is important. Why would a behavior in one ocean basin remotely impact a behavior in a geographically separated one? I know it's possible to create a mathematical network connection between the two, but is that physically plausible? (note: this was in fact the motivation for the Azimuth Project ENSO study if you look back to much earlier discussion threads) Much more plausible that a common-mode forcing mechanism links the two. The forcing is much the same, but the difference in the basin structure creates a different standing wave pattern. Now, and imho, it is possible that 2nd order factors are influencing the North Wall, which would presumably then impact the standing wave boundary conditions.`

This is matching the NAO model on 3 different training intervals. The lunisolar tidal forcing is essentially the same across each. What does change slightly is the high-K wavenumber coefficient (and its amplitude scaling) that is used to sharpen the peaks. The lower K coefficients vary as well but do not contribute as much to the amplitude.

1950-1973CC in training 0.761973-1996CC in training 0.721996-2019CC in training 0.78This is the value of the K wavenumber which is in arbitrary units. The NAO amplitude does seem to grow over time.

It's impossible to tell at the moment whether this variation is overfitting to noise or whether it is actually capturing the variation. Yet the transitions slightly outside the training intervals does suggest that the fit for the training does extend a few years outside each interval.

`This is matching the NAO model on 3 different training intervals. The lunisolar tidal forcing is essentially the same across each. What does change slightly is the high-K wavenumber coefficient (and its amplitude scaling) that is used to sharpen the peaks. The lower K coefficients vary as well but do not contribute as much to the amplitude. **1950-1973** CC in training 0.76 ![lo](https://imageshack.com/a/img922/1694/vqJytE.png) **1973-1996** CC in training 0.72 ![mid](https://imageshack.com/a/img924/7513/VKJs7U.png) **1996-2019** CC in training 0.78 ![hi](https://imageshack.com/a/img924/3373/TB8Fmm.png) This is the value of the K wavenumber which is in arbitrary units. The NAO amplitude does seem to grow over time. <pre> Low Mid High k(wn) 61429 61614 62660 Amp 1.370 1.697 2.161 </pre> It's impossible to tell at the moment whether this variation is overfitting to noise or whether it is actually capturing the variation. Yet the transitions slightly outside the training intervals does suggest that the fit for the training does extend a few years outside each interval.`

Interesting...

`Interesting...`

Outside of this forum, there seems to be a rote allegiance to the belief that these standing-wave climate behaviors are due to chaos. Apparently no one wants to get involved in the analysis because they have been told that chaos leads to the butterfly effect and the butterfly effect makes long-range predictions impossible.

As an example, this post on the "AND THEN THERE'S PHYSICS" blog considers a concept called the Hawkmoth Effect, which is complementary to the Butterfly Effect, and supposedly just as difficult to deal with.

I've found it very tricky to engage in any kind of sustained discussion with this group. There's very little give and take.

`Outside of this forum, there seems to be a rote allegiance to the belief that these standing-wave climate behaviors are due to chaos. Apparently no one wants to get involved in the analysis because they have been told that chaos leads to the butterfly effect and the butterfly effect makes long-range predictions impossible. As an example, this [post](https://andthentheresphysics.wordpress.com/2019/01/31/the-hawkmoth-effect/) on the "AND THEN THERE'S PHYSICS" blog considers a concept called the Hawkmoth Effect, which is complementary to the Butterfly Effect, and supposedly just as difficult to deal with. I've found it very tricky to engage in any kind of sustained discussion with this group. There's very little give and take.`

Paul, the people which you cite apparently don't understand chaos at all, or its implications, or what it does not imply. If they are going to invoke it, they should understand it. An approachable text is Prof Lenny Smith's [

Chaos: A very short introduction|https://global.oup.com/academic/product/chaos-a-very-short-introduction-9780192853783] is one source, and he's [written a more technical discussion atPNAS|https://www.pnas.org/content/99/suppl_1/2487]. Interestingly he mentions [Prof Myles Allen's|https://www.eci.ox.ac.uk/people/mallen.html]climateprediction.comwhich became the highly successful [climateprediction.net|https://www.climateprediction.net/] and offers the cheapest means of modeling long range climate available, based as it is on the [BOINC platform|https://boinc.berkeley.edu/]. (Disclosure: I am an enthusiast and run this in my spare cycles at home.) I have [wrestled with some member of this group|https://667-per-cm.net/2014/12/01/tsonis-swanson-chaos-and-s__t-happens/] myself.`Paul, the people which you cite apparently don't understand chaos at all, or its implications, or what it does not imply. If they are going to invoke it, they should understand it. An approachable text is Prof Lenny Smith's [*Chaos: A very short introduction*|https://global.oup.com/academic/product/chaos-a-very-short-introduction-9780192853783] is one source, and he's [written a more technical discussion at *PNAS*|https://www.pnas.org/content/99/suppl_1/2487]. Interestingly he mentions [Prof Myles Allen's|https://www.eci.ox.ac.uk/people/mallen.html] *climateprediction.com* which became the highly successful [*climateprediction.net*|https://www.climateprediction.net/] and offers the cheapest means of modeling long range climate available, based as it is on the [BOINC platform|https://boinc.berkeley.edu/]. (Disclosure: I am an enthusiast and run this in my spare cycles at home.) I have [wrestled with some member of this group|https://667-per-cm.net/2014/12/01/tsonis-swanson-chaos-and-s__t-happens/] myself.`

Jan, Thanks, but I steer clear of any chaotic formulations. I think it's a no-win situation -- once a behavior is deemed chaotic (or turbulent), there's not much you can do about it (especially in terms of predictability)

That's why I am concentrating on the equatorial behaviors. For example, it’s known from condensed matter physics that vortices cannot form directly along an equator. The vortex-free zone is known as a Meissner belt in superconducting spherical shells. No vortices means there is no turbulence, so there is hope in analyzing the behavior.

`Jan, Thanks, but I steer clear of any chaotic formulations. I think it's a no-win situation -- once a behavior is deemed chaotic (or turbulent), there's not much you can do about it (especially in terms of predictability) That's why I am concentrating on the equatorial behaviors. For example, it’s known from condensed matter physics that vortices cannot form directly along an equator. The vortex-free zone is known as a Meissner belt in superconducting spherical shells. No vortices means there is no turbulence, so there is hope in analyzing the behavior. > Coexistence of the Meissner and vortex states on a nanoscale superconducting spherical shell, April 2009, Physical review. B, Condensed matter 79(13) > ![](https://imageshack.com/a/img922/7214/NUmjF4.png)`

Even the originator of chaotic contributions to climate and the Butterfly Effect, Edward Lorenz, had concerns about claiming all aspects of climate were chaotic. This included of course tides :

but also QBO & ENSO (El Nino) :

A recent article even read more into the Lorenz quote statement:

I always had an optimistic goal of making climate variability as easy to predict as the tides, to where we could train the past time-series and use that as a predictor similarly to tidal analysis. This seems like an obvious approach to try, but I can see how researchers would miss the necessary ansatz required to make it work.

`Even the originator of chaotic contributions to climate and the Butterfly Effect, Edward Lorenz, had concerns about claiming all aspects of climate were chaotic. This included of course tides : > ![](https://pbs.twimg.com/media/DzJDNZ-WsAAWINX.jpg) but also QBO & ENSO (El Nino) : > ![](https://pbs.twimg.com/media/DzJDNqTWwAAh0Su.jpg) A recent article even read more into the Lorenz quote statement: > "After seminal researches of E.N. Lorenz, almost all meteorologists agree that weather variations are chaotic, i.e., they are unstable to small disturbances, and so unpredictable for more or less distant future. Moreover, the same opinion is widely accepted among climatologists concerning climatic variations (Ghil 1985 ); however, Lorenz himself said (Lorenz 2006 ): *"I do not know: whether are climatic variations chaotic or nonchaotic."* The main aim in this paper is to demonstrate that short-term climatic variations (the periods from 2 years to about one decade), even if they look to be very complex(strange in terms of mathematics), are nonchaotic. Instead of chaos, a mutual order exists in these variations. The existence of such an order admits to predict the variations with no predictability limit in principle." --I. V. Serykh and D. M. Sonechkin, “Nonchaotic and globally synchronized short-term climatic variations and their origin,” Theor Appl Climatol, pp. 1–18, Jan. 2019. I always had an optimistic goal of making climate variability as easy to predict as the tides, to where we could train the past time-series and use that as a predictor similarly to tidal analysis. This seems like an obvious approach to try, but I can see how researchers would miss the necessary ansatz required to make it work.`

Here is an animated view of how the ENSO model is trained.

The general idea is that Laplace's tidal equation solution evolves with wavenumber applied. The initial forcing is from the lunar tidal pattern modulated by an annual impulse. The wavenumber is then increased from a low-value to higher value to better match the standing wave pattern of ENSO, spanning the years 1880 to present.

https://youtu.be/Y466JuI2MPQ

The ENSO model requires three crucial ingredients: (1) An annual impulse (2) A monthly/fortnightly tidal forcing (3) Analytical solution to Laplace's tidal equations. The last one is the non-intuitive master puzzle piece, worked in via a nonlinear fitting procedure which is essentially animated in the video.

This all seems complicated largely because there are these 3 pieces that have to be simultaneously applied. That may be a fact of solving challenging problems:

`Here is an animated view of how the ENSO model is trained. The general idea is that Laplace's tidal equation solution evolves with wavenumber applied. The initial forcing is from the lunar tidal pattern modulated by an annual impulse. The wavenumber is then increased from a low-value to higher value to better match the standing wave pattern of ENSO, spanning the years 1880 to present. https://youtu.be/Y466JuI2MPQ The ENSO model requires three crucial ingredients: (1) An annual impulse (2) A monthly/fortnightly tidal forcing (3) Analytical solution to Laplace's tidal equations. The last one is the non-intuitive master puzzle piece, worked in via a nonlinear fitting procedure which is essentially animated in the video. This all seems complicated largely because there are these 3 pieces that have to be simultaneously applied. That may be a fact of solving challenging problems: ![](https://imagizer.imageshack.com/img924/3435/NlwY71.png)`

I just learned that Prof Wally Broecker, great scientist and oceanographer, died Monday at 87. Among his many, many other accomplishments, he wrote about and spoke about clear air capture of CO

_{2}, and emphasized the need to pursue something like that.`I just learned that Prof Wally Broecker, <a href="https://www.ldeo.columbia.edu/~broecker/CO2%20Earths%20Climate%20Driver.pdf">great scientist</a> and oceanographer, <a href="https://www.nytimes.com/2019/02/19/obituaries/wallace-broecker-dead.html">died Monday at 87</a>. Among his many, many other accomplishments, he wrote about and spoke about <a href="https://www.researchgate.net/publication/270622925_Does_air_capture_constitute_a_viable_backstop_against_a_bad_CO2_trip/download">clear air capture of CO<sub>2</sub></a>, and emphasized the need to pursue something like that.`

From the NY Times

From what I have read, Broecker modeled the climate system as an "angry beast", with the meaning that it was highly unpredictable. I wonder if this was the scientific interpretation that he was trying to convince people of.

`From the NY Times > "Dr. Broecker could be combative and even curmudgeonly, recalled Michael E. Mann, a climate scientist at Pennsylvania State University. “He was quite opinionated and often fought hard to make his scientific interpretations the prevailing doctrine,” he said. From what I have read, Broecker modeled the climate system as an "angry beast", with the meaning that it was highly unpredictable. I wonder if this was the scientific interpretation that he was trying to convince people of.`

Not exactly sure what you meant there, Paul.

But, I think Prof Broecker was suggesting we don't really know what the climate system and weather are capable of doing. There are paleoclimate records, but it's not like they have good enough temporal resolution to see how fast they play out. There is some evidence these could be abrupt.I suspect, although do not have a quote from him, that because the support for Equilibrium Climate Sensitivity is constrained to be positive, with a potentially long tail, and because I think Prof Broecker's standards of evidence were high (just

readthis, for example, particularly the last section), he was honestly insisting we don't know what will happen in the sequel of what he called "Revelle's experiment". The closest he came to a warning was in his "Chaotic climate" article forScientificAmerican. Surely, the energy balance tallies suggest there's alotof energy being stored, and there is not, at present, an equilibrium between ocean reservoir and atmosphere for it. How will it arrive at equilibrium? Will it be fast or slow? I think Dr Broecker was suggesting we probably don't want to find out.`> I wonder if this was the scientific interpretation that he was trying to convince people of. Not exactly sure what you meant there, Paul. _But_, I think [Prof Broecker](https://www.nytimes.com/1998/03/17/science/scientist-at-work-wallace-s-broecker-iconoclastic-guru-of-the-climate-debate.html) was suggesting we don't really know what the climate system and weather are capable of doing. There are paleoclimate records, but it's not like they have good enough temporal resolution to see how fast they play out. There is [some evidence](http://faculty.sites.uci.edu/erignot/files/2017/06/Ice-melt-sea-level-rise-and-superstorms-evidence-from-paleoclimate-data-climate-modeling-and-modern-observations-that-2C-global-warming-is-highly-dangerous.pdf) these could be abrupt. I suspect, although do not have a quote from him, that because the support for Equilibrium Climate Sensitivity is constrained to be positive, with a potentially long tail, and because I think Prof Broecker's standards of evidence were high ([just _read_ this, for example](https://www.ldeo.columbia.edu/~broecker/CO2%20Earths%20Climate%20Driver.pdf), particularly the last section), he was honestly insisting we don't know what will happen in the sequel of what he called "Revelle's experiment". The closest he came to a warning was in his ["Chaotic climate" article](http://lustiag.pp.fi/__CC-Art/__USRes/1995_broecker_chaotic_%20climate.pdf) for _Scientific_ _American_. Surely, the energy balance tallies suggest there's a _lot_ of energy being stored, and there is not, at present, an equilibrium between ocean reservoir and atmosphere for it. How will it arrive at equilibrium? [Will it be fast or slow?](https://www.mit.edu/~pog/src/gertler_changing_energy_summer_2019.pdf) I think Dr Broecker was suggesting we probably don't want to find out.`

By the way, Prof Paul O'Gorman, co-author of that

PNASpaper I just cited above ("Changing available energy for extratropical cyclones and associated convection in Northern Hemisphere summer"), has an extended interview about his earlier work on machine learning and climate modeling.`By the way, Prof Paul O'Gorman, co-author of that _PNAS_ paper I just cited above ("[Changing available energy for extratropical cyclones and associated convection in Northern Hemisphere summer](https://www.mit.edu/~pog/src/gertler_changing_energy_summer_2019.pdf)"), has [an extended interview](http://news.mit.edu/2019/mit-3q-paul-o-gorman-machine-learning-for-climate-modeling-0213) about his earlier work on machine learning and climate modeling.`

Also from NY Times

I would say maybe since the 1960's, and since that breakthrough was on chaos and the butterfly effect, that was more of a dead-end.

`Also from NY Times > "Ken Caldeira, a climate scientist at the Carnegie Institution for Science in Stanford, Calif., has a habit of asking new graduate students to name the largest fundamental breakthrough in climate physics since 1979. It’s a trick question. There has been no breakthrough." I would say maybe since the 1960's, and since that breakthrough was on chaos and the butterfly effect, that was more of a dead-end.`

@WebHubTel, I attended the Lorenz-Charney symposium at MIT, in their memory and honor, and there was a historical talk about how Lorenz realized the influence of chaos and the Butterfly Effect. To the degree the talk was accurate, I was underwhelmed. The phenomenon he "discovered" has been well known to numerical analysts for substantially longer than a century. And Mandelbrot had been working on chaos for a while, as well as a plethora of Russians we tend to forget. Now, sure, I don't want to take anything away from Lorenz: There is a definite role for a scientific leader to take an idea that's already out there and emphasize its importance and show how it is very relevant. Bradley Efron did that with the Bootstrap in Statistics, although he'll be the first to tell you -- and has noted it in his books -- that he did not "invent" the idea at all. So, to some extent, the Lorenz thing is a favorite son being cheered. And you're right, the typical things which are concluded from Butterfly, various impossibility results and No Free Lunch theorems, are largely incorrect.

`@WebHubTel, I attended the Lorenz-Charney symposium at MIT, in their memory and honor, and there was a historical talk about how Lorenz realized the influence of chaos and the Butterfly Effect. To the degree the talk was accurate, I was underwhelmed. The phenomenon he "discovered" has been well known to numerical analysts for substantially longer than a century. And Mandelbrot had been working on chaos for a while, as well as a plethora of Russians we tend to forget. Now, sure, I don't want to take anything away from Lorenz: There is a definite role for a scientific leader to take an idea that's already out there and emphasize its importance and show how it is very relevant. Bradley Efron did that with the Bootstrap in Statistics, although he'll be the first to tell you -- and has noted it in his books -- that he did not "invent" the idea at all. So, to some extent, the Lorenz thing is a favorite son being cheered. And you're right, the typical things which are concluded from Butterfly, various impossibility results and No Free Lunch theorems, are largely incorrect.`

Cool. Besides Efron's Bootstrapping, I am reminded of Simulated Annealing which was popularized by Kirkpatrick but was known from physics.

Interesting that the solution to Navier-Stokes that I apply to these models have a non-linear and nearly chaotic look to them. They have solutions of the form

sin( A sin(Bt)), whereAandBare unknown and constrained, respectively. There are few good ways to deduce whatAis, but if you can do it by iterative brute force, the results obtained fit the data very well. (see the video in #240 above)The point is that someone else looking at the output would immediately dismiss it by saying it was chaotic, which is essentially the legacy of Lorenz. Perhaps a similar response that many had with Broecker's claim of climate being an "angry beast", in that it was impossible to predict what climate would do after being forced by an arbitrary input.

`Cool. Besides Efron's Bootstrapping, I am reminded of Simulated Annealing which was popularized by Kirkpatrick but was known from physics. Interesting that the solution to Navier-Stokes that I apply to these models have a non-linear and nearly chaotic look to them. They have solutions of the form *sin( A sin(Bt))*, where *A* and *B* are unknown and constrained, respectively. There are few good ways to deduce what *A* is, but if you can do it by iterative brute force, the results obtained fit the data very well. (see the video in #240 above) The point is that someone else looking at the output would immediately dismiss it by saying it was chaotic, which is essentially the legacy of Lorenz. Perhaps a similar response that many had with Broecker's claim of climate being an "angry beast", in that it was impossible to predict what climate would do after being forced by an arbitrary input.`

@WebHubTel, Broecker's "angry beast" came up in another context in which I was involved. I think that was Prof Broecker's way of urging humility and caution, given the paleoclimate record, on what exactly climate might do being exposed to essentially a Green's function test. He did write a thing titled "Chaotic Climate" for

Scientific Americanin 1995, but my copy doesn't suggest much concern with Chaos. A similar sentiment is expressed in a much more mathematical context, one which I like very much: The 1974 Hirsch and Smale,Differential Equations, Dynamical Systems, and Linear Algebra. In Chapter 12, devoted to applications in Ecology, featuring Predator-Prey, they end withI love that statement. It has always guided me ever since I learned of the promise and eventual success of mathematical methods in Biology which I first discovered in the 1978 monograph by Peter Yodzis,

Competition for Space and the Structure of Ecological Communities, inLecture Notes in Biomathematics. That was a true eye-opener. I hope to return to that some day, when I retire.That approach to Biological and especially Ecology is now definitive. See, for example, L. Pásztor,

et al,Theory-Based Ecology: A Darwinian approach, Oxford University Press, 2016.`@WebHubTel, Broecker's "angry beast" came up in another context in which I was involved. I think that was Prof Broecker's way of urging humility and caution, given the paleoclimate record, on what exactly climate might do being exposed to essentially a Green's function test. He did write a thing titled ["Chaotic Climate" for *Scientific American* in 1995](http://earth.columbia.edu/ac/bios/broecker.html), but [my copy](https://user.fm/files/v2-2eead6387387ba22b4da8b56819b0004/ChaoticClimate--Broecker1995.pdf) doesn't suggest much concern with Chaos. A similar sentiment is expressed in a much more mathematical context, one which I like very much: The 1974 Hirsch and Smale, *Differential Equations, Dynamical Systems, and Linear Algebra*. In Chapter 12, devoted to applications in Ecology, featuring Predator-Prey, they end with >The moral is clear: in the absence of comprehensive knowledge, a deliberate change in the ecology, even an apparently minor one, is a very risky proposition. I love that statement. It has always guided me ever since I learned of the promise and eventual success of mathematical methods in Biology which I first discovered in the 1978 monograph by Peter Yodzis, [*Competition for Space and the Structure of Ecological Communities*](https://books.google.com/books?hl=en&lr=&id=zoPqCAAAQBAJ&oi=fnd&pg=PA1&dq=yodzis+competition+for+space+and+the+structure+of+ecological+communities&ots=i2DW4MVcVi&sig=DbdFUfnlx-6_StPtNSu1jj2FORU#v=onepage&q=yodzis%20competition%20for%20space%20and%20the%20structure%20of%20ecological%20communities&f=false), in *Lecture Notes in Biomathematics*. That was a true eye-opener. I hope to return to that some day, when I retire. That approach to Biological and especially Ecology is now definitive. See, for example, L. Pásztor, *et al*, *Theory-Based Ecology: A Darwinian approach*, Oxford University Press, 2016.`

Here's a very convincing data analysis that shows that ENSO (and El Nino) is far from being chaotic. I ran an auto-correlation of the the Fourier series of an ENSO time-series and found a strongly correlated one-year shift from all spectral components

This can only happen if the Green's function (impulse) response derives from a frequency modulation or mixing of an annual impulse with another external forcing.

It's crucial to understand that this is not an auto-correlation in the time-domain but an auto-correlation in the frequency-domain, which is rarely used under most circumstances.

The analysis is related to deciding whether a detected radio signal is noisy/ chaotic or AM/FM -- it's really a matter of demodulating the signal with the carrier (mixing) signal to find out what the information content is. What the auto-correlation in the frequency-domain does is to demodulate the signal; i.e. convolution in the frequency domain leads to multiplication in the time domain. In other words, this analysis is directly showing what the time-domain mixing or multiplication function, which is an annual impulse train.

I also haven't found anything close to this kind of analysis in the research literature.

`Here's a very convincing data analysis that shows that ENSO (and El Nino) is far from being chaotic. I ran an auto-correlation of the the Fourier series of an ENSO time-series and found a strongly correlated one-year shift from all spectral components ![](https://geoenergymath.files.wordpress.com/2019/02/3rffsl.png) This can only happen if the Green's function (impulse) response derives from a frequency modulation or mixing of an annual impulse with another external forcing. It's crucial to understand that this is not an auto-correlation in the time-domain but an auto-correlation in the frequency-domain, which is rarely used under most circumstances. The analysis is related to deciding whether a detected radio signal is noisy/ chaotic or AM/FM -- it's really a matter of demodulating the signal with the carrier (mixing) signal to find out what the information content is. What the auto-correlation in the frequency-domain does is to demodulate the signal; i.e. convolution in the frequency domain leads to multiplication in the time domain. In other words, this analysis is directly showing what the time-domain mixing or multiplication function, which is an annual impulse train. I also haven't found anything close to this kind of analysis in the research literature.`

@WebHubTel, Publish it? If not in a geophysics journal, why not try IOP or a statistics journal like JTSA or even Computational and Graphical Statistics? Or at least put it out there in arXiv?

`@WebHubTel, Publish it? If not in a geophysics journal, why not try IOP or a statistics journal like JTSA or even Computational and Graphical Statistics? Or at least put it out there in arXiv?`