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## Comments

Couple of more blog posts on nailing down the mechanism for ENSO:

(1)This one contextualizes the ENSO behavior in terms of a common forcing governing ENSO, QBO, and the Chandler wobblehttp://contextearth.com/2017/05/21/the-lunar-geophysical-connection/

Here are a few charts from the post:

What are the odds that the fundamental frequencies of all these behaviors are the same to that precision?

As it turns out NASA JPL were on this lunar-forcing path several years ago, but elected not to fund the proposed research and so the progress stalled.

(2)Yesterday's post provides a historical context. Essentially all the geophysics applicable to the model was known by 1920.http://contextearth.com/2017/05/30/ocean-dynamics-history/

Could have done a decent job in predicting ENSO evolution with Pacific Ocean SST data up to 1920

(3)Posted on the Azimuth blog.https://johncarlosbaez.wordpress.com/2017/04/24/complexity-theory-and-evolution-in-economics/#comment-92343

ENSO means zero in Japanese and also has a Zen interpretation

`Couple of more blog posts on nailing down the mechanism for ENSO: --- --- **(1)** This one contextualizes the ENSO behavior in terms of a common forcing governing ENSO, QBO, and the Chandler wobble http://contextearth.com/2017/05/21/the-lunar-geophysical-connection/ Here are a few charts from the post: ![cw](http://imagizer.imageshack.us/a/img922/9128/U1BKZz.png) ![qbo](http://imageshack.com/a/img922/7145/G5zhhx.png) ![enso](http://imageshack.com/a/img924/7921/4KVw6j.png) What are the odds that the fundamental frequencies of all these behaviors are the same to that precision? ![ensoprec](http://imagizer.imageshack.us/a/img922/5296/vUfioS.png) As it turns out NASA JPL were on this lunar-forcing path several years ago, but elected not to fund the proposed research and so the progress stalled. --- --- **(2)** Yesterday's post provides a historical context. Essentially all the geophysics applicable to the model was known by 1920. http://contextearth.com/2017/05/30/ocean-dynamics-history/ Could have done a decent job in predicting ENSO evolution with Pacific Ocean SST data up to 1920 ![predict](https://imageshack.com/i/pnfzYLXqp) --- --- **(3)** Posted on the Azimuth blog. https://johncarlosbaez.wordpress.com/2017/04/24/complexity-theory-and-evolution-in-economics/#comment-92343 ENSO means zero in Japanese and also has a Zen interpretation > “The ensō symbolizes absolute enlightenment, strength, elegance, the universe, and mu (the void).”`

Trump pulled US out of the Paris Accord :(

If we actually had better knowledge of climate behavior and more emphasis on how AGW interacts with Peak Oil, we would likely have a different discourse. Just by having a real understanding of ENSO, we could compensate out the natural variability in the global temperature signal. The almost monotonic increase in temperature would be much more obvious and we wouldn't have to contend with the uncertainty players such as Curry controlling the political discussion.

This ENSO model is looking more solid. With virtually no free parameters, any automated fit to an ENSO interval does a good job of recreating the unfitted intervals. http://contextearth.com/2017/05/31/enso-model-fit-1880-1980/

The common criticism of these kinds of models is that they overfit and can not be tested with historical data, as any contamination of a model with available data will then taint the model and give a false impression that it actually works. This model is different and works with minimal set of parameters, just the 2 tidal cycles and a seasonal cycle.

`Trump pulled US out of the Paris Accord :( If we actually had better knowledge of climate behavior and more emphasis on how AGW interacts with Peak Oil, we would likely have a different discourse. Just by having a real understanding of ENSO, we could compensate out the natural variability in the global temperature signal. The almost monotonic increase in temperature would be much more obvious and we wouldn't have to contend with the uncertainty players such as Curry controlling the political discussion. This ENSO model is looking more solid. With virtually no free parameters, any automated fit to an ENSO interval does a good job of recreating the unfitted intervals. http://contextearth.com/2017/05/31/enso-model-fit-1880-1980/ ![](http://imageshack.com/a/img923/1430/gIh5df.png) The common criticism of these kinds of models is that they overfit and can not be tested with historical data, as any contamination of a model with available data will then taint the model and give a false impression that it actually works. This model is different and works with minimal set of parameters, just the 2 tidal cycles and a seasonal cycle.`

Couple more blog posts that demonstrate how well the ENSO model works in describing the observations and the geophysics ...

The first is evidence as to how such a simple model can produce such a rich Fourier spectra, contrary to people's preconceived notions: http://contextearth.com/2017/06/02/enso-and-fourier-analysis/

The second is an untainted match between the lunar forcing used to model ENSO and the lunar forcing obtained from the earth's Length-of-day (LOD) data http://contextearth.com/2017/06/03/enso-forcing-validation-via-lod-data/

Have to look at Fig.6 in the post to see the match closely.

This is untainted because the LOD is completely out-of-band with the fitting data and so becomes a strong validation test for the model.

`Couple more blog posts that demonstrate how well the ENSO model works in describing the observations and the geophysics ... The first is evidence as to how such a simple model can produce such a rich Fourier spectra, contrary to people's preconceived notions: http://contextearth.com/2017/06/02/enso-and-fourier-analysis/ The second is an untainted match between the lunar forcing used to model ENSO and the lunar forcing obtained from the earth's Length-of-day (LOD) data http://contextearth.com/2017/06/03/enso-forcing-validation-via-lod-data/ Have to look at Fig.6 in the post to see the match closely. ![fit](http://imageshack.com/a/img924/1548/KteCZ3.png) This is untainted because the LOD is completely [out-of-band](https://en.wikipedia.org/wiki/Out-of-band) with the fitting data and so becomes a strong validation test for the model.`

ENSO forcing match against digitized LOD variation

I took the correlation coefficient of this and its above 0.7. For the cycle factors applied, the fit doesn't get much better if the phases and amplitude are allowed to vary -- the correlation coefficient go up by a slight 0.02, and it reduces the ENSO fit only slightly.

`ENSO forcing match against digitized LOD variation ![](http://imageshack.com/a/img924/2977/ii7X3w.png) I took the correlation coefficient of this and its above 0.7. For the cycle factors applied, the fit doesn't get much better if the phases and amplitude are allowed to vary -- the correlation coefficient go up by a slight 0.02, and it reduces the ENSO fit only slightly.`

From the above agreement in forcing stimulii.

(A)The forcing for QBO is mainly Draconic

(B)(C) ENSO and LOD have the same tidal forcing

`From the above agreement in forcing stimulii. (A)The forcing for QBO is mainly Draconic (B)(C) ENSO and LOD have the same tidal forcing ![flow](http://imageshack.com/a/img924/1999/Y7pEf2.png)`

Difficult to believe that behaviors such as ENSO and QBO are not related to external forcing. I can't think of one large scale cyclic behavior that can't be pinned to some other regular cycle. Even the cycles of sunspots are known to be intimately tied to the sun's rotation. So even though they haven't quite nailed the predictability of sunspots yet, they know it isn't some spontaneous oscillation as the purveyors of the wind-only mechanism for ENSO seem to think.

Thus, much like sunspots, ENSO is likely sensitive to variations in the Earth's rotation speed. As the moon is known to cause cyclic variations in the speed, these same variations should be able to be picked up in an ENSO wave equation model. And what do we find but that the two most critical lunar periods, the Draconic 27.2122 days and Anomalistic 27.5545 days feed into a best-fit model to within 1 minute each.

http://contextearth.com/2017/06/08/scaling-el-nino/

Got a reply tweet from Andrew Dessler concerning this and he said

"Climate is a physics problem, not a statistics one. Looking at correlations is interesting, but not sufficient. Must have physical basis."Some of these guys do not realize that science deals with this situation automatically. They should be able to eventually reject the lunar forcing by coming up with evidence that rejects it. It shouldn't be hard, as all they have to do is show that the ENSO cycles are incommensurate with the lunar cycles. And show how there is not enough energy supplied by the lunisolar cycles to move volumes of water in a reduced effective gravity environment. If they can't, however, then the lunar model will remain as a potential ENSO driver.

`Difficult to believe that behaviors such as ENSO and QBO are not related to external forcing. I can't think of one large scale cyclic behavior that can't be pinned to some other regular cycle. Even the cycles of sunspots are known to be intimately tied to the sun's rotation. So even though they haven't quite nailed the predictability of sunspots yet, they know it isn't some spontaneous oscillation as the purveyors of the wind-only mechanism for ENSO seem to think. Thus, much like sunspots, ENSO is likely sensitive to variations in the Earth's rotation speed. As the moon is known to cause cyclic variations in the speed, these same variations should be able to be picked up in an ENSO wave equation model. And what do we find but that the two most critical lunar periods, the Draconic 27.2122 days and Anomalistic 27.5545 days feed into a best-fit model to within 1 minute each. http://contextearth.com/2017/06/08/scaling-el-nino/ Got a reply tweet from Andrew Dessler concerning this and he said *"Climate is a physics problem, not a statistics one. Looking at correlations is interesting, but not sufficient. Must have physical basis."* Some of these guys do not realize that science deals with this situation automatically. They should be able to eventually reject the lunar forcing by coming up with evidence that rejects it. It shouldn't be hard, as all they have to do is show that the ENSO cycles are incommensurate with the lunar cycles. And show how there is not enough energy supplied by the lunisolar cycles to move volumes of water in a reduced effective gravity environment. If they can't, however, then the lunar model will remain as a potential ENSO driver.`

This is a magnification of the fitting contour around the best forcing period values for ENSO. These pair of peak values are each found to be less than a minute apart from the known values of the Draconic cycle (27.2122 days) and Anomalistic cycle (27.5545 days).

The forcing comes directly from the angular momentum variations in the Earth's rotation. The comparison between what the ENSO model uses (from the Draconic and Anomalistic terms above) and what is measured via monitoring the length-of-day (LOD) is shown below

The lower LOD pane is a fit over 3 years, which is about 40 lunar months. These essentially get aliased in the upper ENSO pane, which only responds to the peak tidal forces at a specific time of the year -- around Nov/Dec.

So many numbers have to align perfectly for this model to work out, and it looks like it does.

`This is a magnification of the fitting contour around the best forcing period values for ENSO. These pair of peak values are each found to be less than a minute apart from the known values of the Draconic cycle (27.2122 days) and Anomalistic cycle (27.5545 days). ![tidal](http://imageshack.com/a/img922/3818/QuW4FT.png) The forcing comes directly from the angular momentum variations in the Earth's rotation. The comparison between what the ENSO model uses (from the Draconic and Anomalistic terms above) and what is measured via monitoring the length-of-day (LOD) is shown below ![lod](http://imageshack.com/a/img922/6999/yC88BH.png) The lower LOD pane is a fit over 3 years, which is about 40 lunar months. These essentially get aliased in the upper ENSO pane, which only responds to the peak tidal forces at a specific time of the year -- around Nov/Dec. So many numbers have to align perfectly for this model to work out, and it looks like it does.`

This is the physics of the tidal forcing -- imparting a 1 millisecond slowdown (or speedup) on the rotation of the earth with a surface velocity of almost 500 meters/second over the course of a couple of weeks (a fortnight) will result in an inertial lateral movement of ~ 1/2 a meter in the volume of the Pacific ocean due to Newton's first law.

This does not seem like a big deal until you realize that the thermocline can absorb this inertial impulse as a vertical sloshing, since the effective gravity is reduced by orders of magnitude due to the slight density differences above and below the thermocline. This is reflected as an Atwood number and shows up in Rayleigh-Taylor instability experiments, e.g. SEE THIS PAPER

With an Atwood number less than 0.001 which is ~0.1% density differences in a stratified fluid, the 0.5 meter displacement that occurs over two weeks now occurs effectively over half an hour. That's just an elementary scaling exercise.

So intuitively, one has to ask the question of what would happen if the ocean was translated laterally by 1/2 a meter over the course of a 1/2 an hour? We know what happens with earthquakes in something as simple as a swimming pool

or as threatening as a tsunami. But this is much more subtle because we can't obviously see it, and why it has likely been overlooked as a driver of ENSO.

All that math modeling of ENSO described here works backwards to this point. The

actual forcingworking on the earth's rotation can lead to the response shown here, both in the dynamic sense of tracing the measured path and now in terms of a physical order-of-magnitude justification.`This is the physics of the tidal forcing -- imparting a 1 millisecond slowdown (or speedup) on the rotation of the earth with a surface velocity of almost 500 meters/second over the course of a couple of weeks (a fortnight) will result in an inertial lateral movement of ~ 1/2 a meter in the volume of the Pacific ocean due to Newton's first law. This does not seem like a big deal until you realize that the thermocline can absorb this inertial impulse as a vertical sloshing, since the effective gravity is reduced by orders of magnitude due to the slight density differences above and below the thermocline. This is reflected as an Atwood number and shows up in Rayleigh-Taylor instability experiments, e.g. [SEE THIS PAPER](http://rsta.royalsocietypublishing.org/content/roypta/368/1916/1663.full.pdf) With an Atwood number less than 0.001 which is ~0.1% density differences in a stratified fluid, the 0.5 meter displacement that occurs over two weeks now occurs effectively over half an hour. That's just an elementary scaling exercise. So intuitively, one has to ask the question of what would happen if the ocean was translated laterally by 1/2 a meter over the course of a 1/2 an hour? We know what happens with earthquakes in something as simple as a swimming pool https://youtu.be/27GMnYEWL0M or as threatening as a tsunami. But this is much more subtle because we can't obviously see it, and why it has likely been overlooked as a driver of ENSO. All that math modeling of ENSO described here works backwards to this point. The *actual forcing* working on the earth's rotation can lead to the response shown here, both in the dynamic sense of tracing the measured path and now in terms of a physical order-of-magnitude justification.`

The supposedly simplest "toy" models of ENSO that we describe on the Azimuth Project wiki page here http://www.azimuthproject.org/azimuth/show/ENSO are the ones that remarkably work the best to describe the actual dynamics. If the delayed action oscillator (minus the cubic term) is combined with a seasonally-modulated lunar forcing that's essentially all that is needed to train the model.

`The supposedly simplest "toy" models of ENSO that we describe on the Azimuth Project wiki page here http://www.azimuthproject.org/azimuth/show/ENSO are the ones that remarkably work the best to describe the actual dynamics. If the delayed action oscillator (minus the cubic term) is combined with a seasonally-modulated lunar forcing that's essentially all that is needed to train the model.`

My last comment:

Elaborated further here: http://contextearth.com/2017/06/23/ensoqbo-elevator-pitch/

`My last comment: > "The supposedly simplest "toy" models of ENSO that we describe on the Azimuth Project wiki page here http://www.azimuthproject.org/azimuth/show/ENSO are the ones that remarkably work the best to describe the actual dynamics. If the delayed action oscillator (minus the cubic term) is combined with a seasonally-modulated lunar forcing that's essentially all that is needed to train the model." Elaborated further here: http://contextearth.com/2017/06/23/ensoqbo-elevator-pitch/`

tweet

http://contextearth.com/2017/08/14/solar-eclipse-2017-what-else/

`tweet <blockquote class="twitter-tweet" data-lang="en"><p lang="en" dir="ltr">Because lunar & solar cycles so accurately known, we can predict <a href="https://twitter.com/hashtag/SolarEclipse2017?src=hash">#SolarEclipse2017</a> precisely. Same for <a href="https://twitter.com/hashtag/ENSO?src=hash">#ENSO</a> <a href="https://twitter.com/hashtag/ElNino?src=hash">#ElNino</a> <br> <a href="https://t.co/M8xJ3DwOso">https://t.co/M8xJ3DwOso</a></p>— Paul Pukite (@WHUT) <a href="https://twitter.com/WHUT/status/896857059366494208">August 13, 2017</a></blockquote> <script async src="//platform.twitter.com/widgets.js" charset="utf-8"></script> http://contextearth.com/2017/08/14/solar-eclipse-2017-what-else/`

Thought to comment on what a long strange trip it's been. The journey to modeling ENSO and QBO has been circuitous and then essentially doubled back to the most basic kind of forcing and the simplest toy differential equations.

The ENSO behavior is modeled as 2 lunar tidal signals and an annual forcing impulse applied to a delay differential equation of 1 year delay. Could have started with this premise from day one, but nothing in the research literature indicated lunar forcing had any effect on ENSO.

Same goes for QBO except that it is essentially a single lunar tidal signal and a bi-annual seasonal forcing signal - one impulse per nodal crossing. Lindzen had considered lunar forcing early but apparently couldn't find any correlation and that's why no one followed up there.

Looking back my first blog post on this topic was early 2014, so it's been almost 4 years of spare-time effort. And even though this was anticipated to be a software coding project, the model is simple enough to express on a spreadsheet without the need for any macros or scripts except for a standard Solver plugin. It's essentially a little more complex than a basic tidal analysis program.

http://ContextEarth.com

`Thought to comment on what a long strange trip it's been. The journey to modeling ENSO and QBO has been circuitous and then essentially doubled back to the most basic kind of forcing and the simplest toy differential equations. The ENSO behavior is modeled as 2 lunar tidal signals and an annual forcing impulse applied to a delay differential equation of 1 year delay. Could have started with this premise from day one, but nothing in the research literature indicated lunar forcing had any effect on ENSO. Same goes for QBO except that it is essentially a single lunar tidal signal and a bi-annual seasonal forcing signal - one impulse per nodal crossing. Lindzen had considered lunar forcing early but apparently couldn't find any correlation and that's why no one followed up there. Looking back my first blog post on this topic was early 2014, so it's been almost 4 years of spare-time effort. And even though this was anticipated to be a software coding project, the model is simple enough to express on a spreadsheet without the need for any macros or scripts except for a standard Solver plugin. It's essentially a little more complex than a basic tidal analysis program. http://ContextEarth.com`

Can we look forward to a publication submission or arXiv addition, @WebHubTel?

`Can we look forward to a publication submission or arXiv addition, @WebHubTel?`

Have submitted a presentation to the AGU this December.

Already have an arXiv paper in place but need to update it.

`Have submitted a presentation to the AGU this December. Already have an arXiv paper in place but need to update it.`

+1. Congratulations. I've much enjoyed the epic and learned a lot. Btw what solver did you use?

`+1. Congratulations. I've much enjoyed the epic and learned a lot. Btw what solver did you use?`

Jim asked:

I used the builtin Excel Solver.

There is also an open Solver that I have yet to try: http://opensolver.org/

For awhile, I was using a Mathematica solver but it doesn't allow a correlation coefficient as a goal, only least squares. I think there is some property of using a correlation coefficient that allows a solver to avoid getting stuck in local minima. It may have to do with not having to worry about scaling at every step. The rescaling can always get done at the end.

There is also the Eureqa solver that I used early on, but that tool got bought out by another company and not sure of it's status. Eureqa allowed correlation coefficient and a whole range of optimization targets, including an interesting hybrid cc+leastSquares target. The architects of that tool understood the importance of providing different targets for optimization.

Probably should try using R and one of the solvers there, but I am in a mode of staying with what works for now

`Jim asked: > "Btw what solver did you use?" I used the builtin [Excel Solver](https://support.office.com/en-us/article/Define-and-solve-a-problem-by-using-Solver-9ed03c9f-7caf-4d99-bb6d-078f96d1652c). There is also an open Solver that I have yet to try: http://opensolver.org/ For awhile, I was using a Mathematica solver but it doesn't allow a correlation coefficient as a goal, only least squares. I think there is some property of using a correlation coefficient that allows a solver to avoid getting stuck in local minima. It may have to do with not having to worry about scaling at every step. The rescaling can always get done at the end. There is also the Eureqa solver that I used early on, but that tool got bought out by another company and not sure of it's status. Eureqa allowed correlation coefficient and a whole range of optimization targets, including an interesting hybrid cc+leastSquares target. The architects of that tool understood the importance of providing different targets for optimization. Probably should try using R and one of the solvers there, but I am in a mode of staying with what works for now`

Thanks for the opensolver.org link.

`Thanks for the opensolver.org link.`

Jim, If you use it, let me know how it works. I am also going for minimizing the barrier to usage.

Latest validation here http://contextearth.com/2017/09/27/enso-tidal-forcing-validated-by-lod-data/

`Jim, If you use it, let me know how it works. I am also going for minimizing the barrier to usage. Latest validation here http://contextearth.com/2017/09/27/enso-tidal-forcing-validated-by-lod-data/ ![chao](http://imageshack.com/a/img923/2029/tQgtea.png)`

So far I've failed to load an .xlsx file of NOAA sesimic data into the Google sheets opensolver plugin version. I'll have to try with whatever the openoffice spreadsheet is called. Opensolver needs chromium which won't run on my currently, semi-borked setup.

`So far I've failed to load an .xlsx file of NOAA sesimic data into the Google sheets opensolver plugin version. I'll have to try with whatever the openoffice spreadsheet is called. Opensolver needs chromium which won't run on my currently, semi-borked setup.`

After spending all that time with the ENSO model, this is the progress after a day of working the Atlantic version of ENSO, the AMO

http://contextearth.com/2017/10/03/amo/

`After spending all that time with the ENSO model, this is the progress after a day of working the Atlantic version of ENSO, the AMO http://contextearth.com/2017/10/03/amo/`

Quantum and ENSO connects!

Full paper:

TopologicaloriginsofequatorialwavesOh, JohnB's gonna be writing about this for

months!There's also a supplement and two movies, S1 and S2.

Naw, I don't understand with any depth whatsoever, but I think it's very cool.`[Quantum and ENSO connects](http://www.sciencemag.org/news/2017/10/waves-drive-global-weather-patterns-finally-explained-thanks-inspiration-bagel-shaped)! [Full paper: _Topological_ _origins_ _of_ _equatorial_ _waves_](http://science.sciencemag.org/content/early/2017/10/04/science.aan8819.full) Oh, JohnB's gonna be writing about this for _months_! There's also [a supplement](http://science.sciencemag.org/highwire/filestream/700023/field_highwire_adjunct_files/0/aan8819_Delplace_SM.pdf) and two movies, [S1](http://science.sciencemag.org/highwire/filestream/700023/field_highwire_adjunct_files/1/aan8819s1.mov) and [S2](http://science.sciencemag.org/highwire/filestream/700023/field_highwire_adjunct_files/2/aan8819s2.mov). _Naw_, I don't understand with any depth whatsoever, but I think it's very cool.`

Thanks Jan, that's looks a lot like what I solved for last year -- reducing the Coriolis forces at the equator.

http://contextEarth.com/2016/09/23/compact-qbo-derviation/

I simplified much more than what they did.

`Thanks Jan, that's looks a lot like what I solved for last year -- reducing the Coriolis forces at the equator. http://contextEarth.com/2016/09/23/compact-qbo-derviation/ I simplified much more than what they did. ![p](http://imageshack.com/a/img923/5449/Z7Da4w.png)`

@WebHubTel, a letter to

Scienceis in order!`@WebHubTel, a letter to _Science_ is in order!`

Jan, Perhaps I can try that.

Is the QBO a Berry monopole?

Or is it related to a Weyl point across a Lifshitz transitiion? https://inspirehep.net/record/1441222/plots

Yes, John will be interested in this stuff because it looks like the anti-vortex stuff that he had written about last year

https://johncarlosbaez.wordpress.com/2016/10/07/kosterlitz-thouless-transition/

I made a comment at the time:

A few years ago, I referenced this paper by Marston, which was a call-to-arms to solving climate problems:

https://physics.aps.org/articles/v4/20

And this paper by Vallis is a good inspiration to look at simplifying the physics before doing CFD

http://contextearth.com/2016/09/03/geophysical-fluid-dynamics-first-and-then-cfd/

`Jan, Perhaps I can try that. Is the QBO a Berry monopole? Or is it related to a Weyl point across a Lifshitz transitiion? https://inspirehep.net/record/1441222/plots ![eq](https://inspirehep.net/record/1441222/files/LifshitzTransitionChiralSF.png) Yes, John will be interested in this stuff because it looks like the anti-vortex stuff that he had written about last year https://johncarlosbaez.wordpress.com/2016/10/07/kosterlitz-thouless-transition/ I made a comment at the time: > "These curl equations are fascinating and are of course endemic in applications from electromagnetics to fluid dynamics. Perhaps there is some overlap with the model of the QBO equatorial winds that we are working on at John’s Azimuth Forum (see sidebar) and at my blog. Have some notes here: http://contextearth.com/2016/09/23/compact-qbo-derviation/#comment-199906 " --- A few years ago, I referenced this paper by Marston, which was a call-to-arms to solving climate problems: https://physics.aps.org/articles/v4/20 And this paper by Vallis is a good inspiration to look at simplifying the physics before doing CFD http://contextearth.com/2016/09/03/geophysical-fluid-dynamics-first-and-then-cfd/ --- https://youtu.be/JBY5iIfPgd0`

This is probably the best preliminary paper on the topic

ELASTIC WAVE EQUATION, Yves Colin de Verdière, Séminaire de théorie spectrale et géométrie, Grenoble Volume25 (2006-2007) 55-69

http://tsg.cedram.org/cedram-bin/article/TSG_2006-2007__25__55_0.pdf

`This is probably the best preliminary paper on the topic ELASTIC WAVE EQUATION, Yves Colin de Verdière, Séminaire de théorie spectrale et géométrie, Grenoble Volume25 (2006-2007) 55-69 http://tsg.cedram.org/cedram-bin/article/TSG_2006-2007__25__55_0.pdf`

Re: Topological origins of equatorial waves

Blog post here on this paper: http://contextEarth.com/2017/10/13/interface-inflection-geophysics/

`Re: Topological origins of equatorial waves Blog post here on this paper: http://contextEarth.com/2017/10/13/interface-inflection-geophysics/`

Couple of recent posts where the harmonic series approximation for the ENSO forcing is reduced to a closed-form expression:

http://contextearth.com/2017/10/27/reverse-engineering-the-moons-orbit-from-enso-behavior/

http://contextearth.com/2017/11/03/approximating-the-enso-forcing-potential/

This simplification was also applied to QBO

These are timely findings, as the presentation was accepted to the AGU next month

GC41B-1022: Biennial-Aligned Lunisolar-Forcing of ENSO: Implications for Simplified Climate Models

Check the sub-title :)

`Couple of recent posts where the harmonic series approximation for the ENSO forcing is reduced to a closed-form expression: http://contextearth.com/2017/10/27/reverse-engineering-the-moons-orbit-from-enso-behavior/ http://contextearth.com/2017/11/03/approximating-the-enso-forcing-potential/ This simplification was also applied to QBO ![QBO](https://i2.wp.com/imageshack.com/a/img922/3353/w1aSa0.png) These are timely findings, as the presentation was accepted to the AGU next month [GC41B-1022: Biennial-Aligned Lunisolar-Forcing of ENSO: Implications for Simplified Climate Models](https://agu.confex.com/agu/fm17/meetingapp.cgi/Paper/221914) ![mtg](https://i2.wp.com/fallmeeting.agu.org/2017/files/2017/08/FM_WebsiteLogo_483x177-1.jpg?w=300) Check the sub-title :)`