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QBO and ENSO

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251.

"@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?"

Jan, All this is published in the Mathematical Geoenergy textbook, which goes into quite a bit of detail describing the annual impulse modulation model.

IMO, that worked out better than a research paper in that it was able to present a comprehensive view of all these related phenomena -- ENSO, QBO, Chandler Wobble, LOD, tidal forcing, and the topological math along the equator.

Comment Source:> "@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?" Jan, All this is published in the Mathematical Geoenergy textbook, which goes into quite a bit of detail describing the annual impulse modulation model. IMO, that worked out better than a research paper in that it was able to present a comprehensive view of all these related phenomena -- ENSO, QBO, Chandler Wobble, LOD, tidal forcing, and the topological math along the equator. 
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252.

10 Things: What We Learn About Earth by Studying the Moon

By Isabelle Yan, NASA’s Goddard Space Flight Center

"Earth’s magnetic field is our shield, constantly protecting us from harmful solar wind or cosmic ray particles. This important buffer is generated by the fast-flowing movement of liquid iron and nickel in Earth’s outer core.

One thing that makes this molten ocean of metal move is the Moon’s gravity. Recent research suggests that the Moon’s gravity tugs on Earth’s mantle layer (which sits on top of the outer core). This causes the liquid, outer core to slosh around, helping to generate the energy needed to maintain our magnetic field."

The moon's gravity also tugs on the non-spherical Earth's axis as well, leading to a perfectly synchronized Chandler Wobble.

"Earth’s signature 23.5-degree tilt on its axis is due to the Moon keeping it in check. The 23.5-degree angle ensures our planet is safe to live on, as a more exaggerated tilt would cause more extreme seasons.

Without the Moon’s gravity, Earth would wobble more violently on its axis, drastically altering the climate. Besides maintaining climate stability, the Moon also sets the rhythm of Earth — the highs and lows of our tides — which affects the variety of ways we use the ocean for food, travel and recreation. Precisely measuring the mass, size and orbital properties of the Moon is essential for predicting these rhythms of tides and seasons. "

The Moon's gravity likely has an equally significant effect on both tides and on the dynamic rhythm of ENSO. This equivalence is too similar to ignore.

That NASA post on the Moon's influence on the Earth's behavior is timely as it intersects nicely with three blog posts I have written in the past month.

Comment Source:> [10 Things: What We Learn About Earth by Studying the Moon](https://solarsystem.nasa.gov/news/812/10-things-what-we-learn-about-earth-by-studying-the-moon/?utm_source=TWITTER&utm_medium=NASA&utm_campaign=NASASocial&linkId=64050866) > By Isabelle Yan, NASA’s Goddard Space Flight Center > *"Earth’s magnetic field is our shield, constantly protecting us from harmful solar wind or cosmic ray particles. This important buffer is generated by the fast-flowing movement of liquid iron and nickel in Earth’s outer core.* > *One thing that makes this molten ocean of metal move is the Moon’s gravity. Recent research suggests that the Moon’s gravity tugs on Earth’s mantle layer (which sits on top of the outer core). This causes the liquid, outer core to slosh around, helping to generate the energy needed to maintain our magnetic field."* The moon's gravity also tugs on the non-spherical Earth's axis as well, leading to a [perfectly synchronized Chandler Wobble](https://geoenergymath.com/2019/02/09/71/). ![](https://geoenergymath.files.wordpress.com/2019/02/cw_time_series_1950_1962.png) and of course the [lunisolar tug largely controls the modulation in the Earth's length-of-day (LOD)](https://geoenergymath.com/2019/02/13/length-of-day/) ![](https://geoenergymath.files.wordpress.com/2019/02/baa0cq.png) > *"Earth’s signature 23.5-degree tilt on its axis is due to the Moon keeping it in check. The 23.5-degree angle ensures our planet is safe to live on, as a more exaggerated tilt would cause more extreme seasons.* > *Without the Moon’s gravity, Earth would wobble more violently on its axis, drastically altering the climate. Besides maintaining climate stability, the Moon also sets the rhythm of Earth — the highs and lows of our tides — which affects the variety of ways we use the ocean for food, travel and recreation. Precisely measuring the mass, size and orbital properties of the Moon is essential for predicting these rhythms of tides and seasons. "* The Moon's gravity likely has an equally significant effect on both tides and on the dynamic rhythm of ENSO. [This equivalence is too similar to ignore](https://geoenergymath.com/2019/02/25/long-period-tides/). ![](https://geoenergymath.files.wordpress.com/2019/02/rkzhjm.png) That NASA post on the Moon's influence on the Earth's behavior is timely as it intersects nicely with three blog posts I have written in the past month. 
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253.

Came across a self-published book "The Deep Pull: A Major Advance in the Science of Tides" by Walter Hayduk, a retired chemical engineer. His claim is that the moon and the sun have major influences on ocean dynamics, and in a way that is outside of the consensus. His premise is that that the tractive lunisolar force on the ocean is horizontally along the surface and not the vertical bulge picture that is typically intuited. So when the moon is setting or rising over the horizon, that is when the strongest forces occur, primarily because it is not competing against the much stronger gravitational force of the earth itself. This also occurs as an impulse over a specific band:

"We can surmise that the force on the water is very brief but intense, sometimes termed as an impulse force"

I am not sure that this is such an original idea as I have read elsewhere in the literature on tides that this tangential force is a way to rationalize a cumulative effect in tidal displacement.

He also asserts that the wind is largely driven by lunisolar gravitational forces with the same horizontal influence, which is not really the consensus understanding.

Overall it's very qualitative with simple order-of-magnitude calculations thrown in and some references at the end.

"then the moon and sun have much, much more influence on our world than we have heretofore ever believed.
Clearly, there is a lot of innovation to mull over for well-motivated, knowledgeable, scientifically inclined researchers."

Comment Source:Came across a self-published book ["The Deep Pull: A Major Advance in the Science of Tides"](https://books.google.com/books?id=RN1iDwAAQBAJ) by Walter Hayduk, a retired chemical engineer. His claim is that the moon and the sun have major influences on ocean dynamics, and in a way that is outside of the consensus. His premise is that that the tractive lunisolar force on the ocean is horizontally along the surface and not the vertical bulge picture that is typically intuited. So when the moon is setting or rising over the horizon, that is when the strongest forces occur, primarily because it is not competing against the much stronger gravitational force of the earth itself. This also occurs as an impulse over a specific band: > "We can surmise that the force on the water is very brief but intense, sometimes termed as an impulse force" ![](https://imagizer.imageshack.com/img924/3947/ymWcWA.gif) I am not sure that this is such an original idea as I have read elsewhere in the literature on tides that this tangential force is a way to rationalize a cumulative effect in tidal displacement. He also asserts that the wind is largely driven by lunisolar gravitational forces with the same horizontal influence, which is not really the consensus understanding. Overall it's very qualitative with simple order-of-magnitude calculations thrown in and some references at the end. > "then the moon and sun have much, much more influence on our world than we have heretofore ever believed. >Clearly, there is a lot of innovation to mull over for well-motivated, knowledgeable, scientifically inclined researchers." 
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254.
edited March 2019

In red is the input forcing $$F(t)$$ which is an annual impulse-driven tidal signal. After each impulse, the signal is integrated into the previous value, thus creating an up-and-down stair-step appearing time series. This approach works well to fit to the QBO time-series, but obviously doesn't match to the ENSO time-series shown in blue.

This is the response $$G(t)$$ after applying the LTE solution $$G(t) = sin(A F(t) + B)$$. Now the model aligns to the data (whereas QBO works with a wavenumber=0 LTE solution). The art is in how to select the values of $$A$$ and $$B$$.

How to do that straightforwardly is the challenge. Can't simply apply an $$\arcsin$$ to $$G(T)$$ because that function is multi-valued and a variation of amplitude folding occurs when $$A$$ is large enough.

May need some advice from the Category Theorists on how to develop a decent adjoint functor approach to seamlessly transition between $$F(t)$$ and $$G(t)$$ without resorting to a brute force iteration scheme.

Comment Source:In red is the input forcing \$$F(t) \$$ which is an annual impulse-driven tidal signal. After each impulse, the signal is integrated into the previous value, thus creating an up-and-down stair-step appearing time series. This approach works well to fit to the [QBO time-series](https://contextearth.com/2018/10/16/asymptotic-qbo-period/), but obviously doesn't match to the ENSO time-series shown in blue. ![impulse](https://imagizer.imageshack.com/img922/4549/s82nWL.png) <br> <br> <br> <br> This is the response \$$G(t) \$$ after applying the LTE solution \$$G(t) = sin(A F(t) + B) \$$. Now the model aligns to the data (whereas QBO works with a wavenumber=0 LTE solution). The art is in how to select the values of \$$A \$$ and \$$B \$$. ![response](https://imagizer.imageshack.com/img924/4781/x088fz.png) How to do that straightforwardly is the challenge. Can't simply apply an \$$\arcsin \$$ to \$$G(T) \$$ because that function is multi-valued and a variation of [amplitude folding](https://hal.archives-ouvertes.fr/hal-01007007/document) occurs when \$$A \$$ is large enough. May need some advice from the Category Theorists on how to develop a decent adjoint functor approach to seamlessly transition between \$$F(t) \$$ and \$$G(t) \$$ without resorting to a brute force iteration scheme. 
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255.
edited March 2019

Since ENSO is the result of a standing wave dipole and that the SOI measure takes the difference between two SW extremes, then we should see something like $$G(t) = C_1 sin(AF(t)+B_1) + C_2 sin(-AF(t)+B_2)$$ as the best fit. And that's exactly what happens. The peak locations are not perfectly aligned to extrema so the fit guesses on how it is partitioned. This can be observed as the iteration process converges.

Comment Source:<br> Since ENSO is the result of a standing wave dipole and that the SOI measure takes the difference between two SW extremes, then we should see something like \$$G(t) = C_1 sin(AF(t)+B_1) + C_2 sin(-AF(t)+B_2) \$$ as the best fit. And that's exactly what happens. The peak locations are not perfectly aligned to extrema so the fit guesses on how it is partitioned. This can be observed as the iteration process converges.
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256.

Have a several year head-start on modeling the geophysics of ENSO, from the low wavenumber dipole to the high wavenumber harmonics.

Comment Source:Have a several year head-start on modeling the geophysics of ENSO, from the low wavenumber dipole to the high wavenumber harmonics. ![](https://imagizer.imageshack.com/img921/2203/kzM347.png)
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257.

The last comment here is elaborated more completely on the blog https://geoenergymath.com/2019/03/23/high-resolution-analysis-of-soi/

Comment Source:The last comment here is elaborated more completely on the blog https://geoenergymath.com/2019/03/23/high-resolution-analysis-of-soi/
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258.

From a physics blog thread on public involvement in science: https://andthentheresphysics.wordpress.com/2019/03/20/public-involvement-in-science/

this lecture from a geologist :

"Paul, as we’ve discussed before, there are real-world situations which can’t be reduced to a closed-form solution of an accepted physics model. In most of what we’re talking about, the problem is not the use or non-use of accepted physics, but that properly representing the real world using accepted physics results in situations where there is no closed-form solution. Sometimes Nature just doesn’t cooperate. I’ve seen such situations in my professional life, and others have been discussed on this blog, where strict adherence to models with closed-form solutions restricts you to a world that is so different from the real world it might as well be an undiscovered exoplanet. For example, economic models that don’t properly represent the carbon cycle and reach optimistic conclusions about mitigation requirements because their model draws down atmospheric CO2 an order of magnitude faster than the real world can.

In those situations you have three choices:

1) Model the fantasy world using closed-form solutions and congratulate yourself you’ve avoided any numerical pitfalls. But not be surprised when people say “meh, it’s just a fantasy world, look how drastically it diverges from reality if we omit the most recent ten years of data and hindcast”.

2) Model the real world numerically, while being cognisant of the potential pitfalls and taking steps to mitigate them. Independent reproduction is better for that than Auditing, for the pitfalls as well as the science. Auditing will just find the pitfalls that reside in library solvers or hardware differences. Different code, different solver, same or similar data may identify structural problems.

3) Give up."

(sigh)

What's interesting about this argument is how backward the logic is. A closed-form analytical solution can significantly help in checking the validity of a numerical solution to a given set of equations. One of the current issues of Navier-Stokes numerical solutions is in how sensitive they are to grid and time-stepping resolution. The closed-form solution can provide a guide to how close the numerical solution is converging to the set of asymptotic values.

I can see how this would evolve given a solution such as cos(A sin(w t)). If A gets large enough, the numerical solution requires the equivalent of a long Taylor's series of coefficients to capture the full resolution in time that the folding dynamics will reveal. One will know what time-stepping delta is required quite quickly.

Comment Source:From a physics blog thread on public involvement in science: https://andthentheresphysics.wordpress.com/2019/03/20/public-involvement-in-science/ this lecture from a geologist : > "Paul, as we’ve discussed before, there are real-world situations which can’t be reduced to a closed-form solution of an accepted physics model. In most of what we’re talking about, the problem is not the use or non-use of accepted physics, but that properly representing the real world using accepted physics results in situations where there is no closed-form solution. Sometimes Nature just doesn’t cooperate. I’ve seen such situations in my professional life, and others have been discussed on this blog, where strict adherence to models with closed-form solutions restricts you to a world that is so different from the real world it might as well be an undiscovered exoplanet. For example, economic models that don’t properly represent the carbon cycle and reach optimistic conclusions about mitigation requirements because their model draws down atmospheric CO2 an order of magnitude faster than the real world can. > In those situations you have three choices: > 1) Model the fantasy world using closed-form solutions and congratulate yourself you’ve avoided any numerical pitfalls. But not be surprised when people say “meh, it’s just a fantasy world, look how drastically it diverges from reality if we omit the most recent ten years of data and hindcast”. > 2) Model the real world numerically, while being cognisant of the potential pitfalls and taking steps to mitigate them. Independent reproduction is better for that than Auditing, for the pitfalls as well as the science. Auditing will just find the pitfalls that reside in library solvers or hardware differences. Different code, different solver, same or similar data may identify structural problems. > 3) Give up." (sigh) What's interesting about this argument is how backward the logic is. A closed-form analytical solution can significantly help in checking the validity of a numerical solution to a given set of equations. One of the current issues of Navier-Stokes numerical solutions is in how sensitive they are to grid and time-stepping resolution. The closed-form solution can provide a guide to how close the numerical solution is converging to the set of asymptotic values. I can see how this would evolve given a solution such as cos(*A* sin(*w t*)). If *A* gets large enough, the numerical solution requires the equivalent of a long Taylor's series of coefficients to capture the full resolution in time that the folding dynamics will reveal. One will know what time-stepping delta is required quite quickly. 
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259.

@WebHubTel, I think there is a fourth possibility to those the geologist suggested:

4) Model the real world using one or more closed-form solutions which have no (known) physical connections to the real world yet predicts observations well.

5) Model the real world as a tree (perhaps but not necessarily a random tree or random forest) with closed-form solutions applicable to tiny bits of physics at each of its leaves. In practice, there would be a large library of tiny-bit-of-physics models, and the trees would be grown based upon data using boosting, and possibly model averaging.

Comment Source:@WebHubTel, I think there is a fourth possibility to those the geologist suggested: 4) Model the real world using one or more closed-form solutions which have no (known) physical connections to the real world yet predicts observations well. 5) Model the real world as a tree (perhaps but not necessarily a random tree or random forest) with closed-form solutions applicable to tiny bits of physics at each of its leaves. In practice, there would be a large library of tiny-bit-of-physics models, and the trees would be grown based upon data using boosting, and possibly model averaging. 
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260.

Jan,

4) i.e. a heuristic

5) a category theory connection ? with the bits of physics connected by a network diagram.

Comment Source:Jan, 4) i.e. a heuristic 5) a category theory connection ? with the bits of physics connected by a network diagram. 
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261.
edited March 2019

Thinking about Jan's comment some more, emphasizing the trees and leaves analogy.

"Model the real world as a tree (perhaps but not necessarily a random tree or random forest) with closed-form solutions applicable to tiny bits of physics at each of its leaves"

ENSO is likely a forest of trees with leaves. If the ENSO model is fit to the coarser time resolution of the NINO34 time-series, that represents the trees in the forest. Yet the leaves can be observed if we take the finer resolution of the SOI time-series into consideration. The physics is the same but the SOI reveals the high-wave-number components of the standing wave modes.

The upper panel with the brown background is the NINO34 complete time-series from 1880 to the present time. This represents the forest of trees

The middle panel with the green background is the SOI time-series representing a high-resolution view since 1992. This represents the leaves on each tree. The inset shows how it is represented on the lower-resolution scale. Some low-pass filtering was applied to reveal how the model matches to the data, which was fit only to the upper graph. In other words, the leaves are emergent based on needing some high-K components to fit the sharpness of the low-res time-series.

The lower panel is the frequency spectrum of SOI in the high-resolution mode.

The question is how much information we can glean from continuing to characterize the high-res leaves of ENSO. These high-K standing wave nodes may not be particularly stationary in time, as they are not as bound to the container as the primary low-K standing wave dipole of ENSO is. The diagram below indicates how the characterization works along the step edge of the 1998 El Nino transition.

Comment Source:Thinking about Jan's comment some more, emphasizing the trees and leaves analogy. > "Model the real world as a *tree* (perhaps but not necessarily a random tree or random forest) with closed-form solutions applicable to tiny bits of physics at each of its *leaves*" ENSO is likely a forest of trees with leaves. If the ENSO model is fit to the coarser time resolution of the NINO34 time-series, that represents the trees in the forest. Yet the leaves can be observed if we take the finer resolution of the SOI time-series into consideration. The physics is the same but the SOI reveals the high-wave-number components of the standing wave modes. ![enso](https://imagizer.imageshack.com/img921/1737/OG2RdX.png) The upper panel with the brown background is the NINO34 complete time-series from 1880 to the present time. This represents the forest of trees The middle panel with the green background is the SOI time-series representing a high-resolution view since 1992. This represents the leaves on each tree. The inset shows how it is represented on the lower-resolution scale. Some low-pass filtering was applied to reveal how the model matches to the data, which was fit only to the upper graph. In other words, the leaves are emergent based on needing some high-K components to fit the sharpness of the low-res time-series. The lower panel is the frequency spectrum of SOI in the high-resolution mode. The question is how much information we can glean from continuing to characterize the high-res leaves of ENSO. These high-K standing wave nodes may not be particularly stationary in time, as they are not as bound to the container as the primary low-K standing wave dipole of ENSO is. The diagram below indicates how the characterization works along the step edge of the 1998 El Nino transition. ![diagram](https://imagizer.imageshack.com/img921/9246/Dl9HE4.png) 
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262.

Need to invest some time to understand your comment. It has also prompted me to look and understand this: https://sci-hub.tw/10.1016/j.oceaneng.2015.05.038

Comment Source:Need to invest some time to understand your comment. It has also prompted me to look and understand this: https://sci-hub.tw/10.1016/j.oceaneng.2015.05.038
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263.

This is the network model evolution of wave dynamics in the ENSO model. The first response is a damping from the impulse-driven tidal forcing. This response is then constrained according to the solution of Laplace's Tidal Equations as a set of standing wave modes.

In terms of category theory, there is a possibility of developing adjoint methods to simultaneously solve the composition.

Comment Source:This is the network model evolution of wave dynamics in the ENSO model. The first response is a damping from the impulse-driven tidal forcing. This response is then constrained according to the solution of Laplace's Tidal Equations as a set of standing wave modes. ![](https://imagizer.imageshack.com/img924/8215/ycXSL8.png) In terms of category theory, there is a possibility of developing adjoint methods to simultaneously solve the composition. 
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264.
Comment Source:Tamino does ENSO: https://tamino.wordpress.com/2019/04/06/new-kid-in-town/
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265.

From comment #254 above, I was asking about coming up with an adjoint approach to inverting the sin(f(t)) model for ENSO. It's obvious that we can try arcsin to do the inversion, but this generates a solution only on a limited domain. I tried guessing what an additional phase shift can be, but this induces a correlation that is highly artificial (e.g. it's very easy to get a CC over 0.9 without trying hard). After normalizing the input ENSO to peaks of ±1 across the time-series range, this is what the fit looks like:

The level shifts derive from the harmonic folding frequency that was inferred from the forward sin(f(t)) fit. From that fit, there is a strong fundamental frequency shown below that reproduces several phase shifts in the fit above.

So I think it is interesting that an inversion might be possible, although it might not help in making the fit process more efficient as its parametric selectivity is low in comparison to the high sensitivity of the forward fit.

Comment Source:From comment #254 above, I was asking about coming up with an adjoint approach to inverting the sin(f(t)) model for ENSO. It's obvious that we can try arcsin to do the inversion, but this generates a solution only on a limited domain. I tried guessing what an additional phase shift can be, but this induces a correlation that is highly artificial (e.g. it's very easy to get a CC over 0.9 without trying hard). After normalizing the input ENSO to peaks of ±1 across the time-series range, this is what the fit looks like: ![](https://imagizer.imageshack.com/img923/7788/RHvlfO.png) The level shifts derive from the harmonic folding frequency that was inferred from the forward sin(f(t)) fit. From that fit, there is a strong fundamental frequency shown below that reproduces several phase shifts in the fit above. ![](https://imagizer.imageshack.com/img923/2287/FR05an.png) So I think it is interesting that an inversion might be possible, although it might not help in making the fit process more efficient as its parametric selectivity is low in comparison to the high sensitivity of the forward fit. 
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266.

from Arbic, Brian K, Matthew H Alford, Joseph K Ansong, Maarten C Buijsman, Robert B Ciotti, J Thomas Farrar, Robert W Hallberg, Christopher E Henze, Christopher N Hill, and Conrad A Luecke. “Primer on Global Internal Tide and Internal Gravity Wave Continuum Modeling in HYCOM and MITgcm.” New Frontiers in Operational Oceanography, 2018, 307–92.

Look at the standing wave along the equatorial Pacific. The nodes generally align at different times corresponding to the start of a Tropical Instability Wave train. They do eventually start travelling but the caption indicates that they emerge from what looks like a relatively fixed location.

"The six TIWs had periods ranging from 15 to 24 days, with the sixth TIW being the longest and the second and fourthTIWs being the shortest."

Having a spatially fixed standing wave pattern says nothing about what the temporal dynamics are, same as with the spatially fixed but erratic ENSO time-series.

Comment Source: ![](https://imagizer.imageshack.com/img921/2429/4649pZ.png) > from Arbic, Brian K, Matthew H Alford, Joseph K Ansong, Maarten C Buijsman, Robert B Ciotti, J Thomas Farrar, Robert W Hallberg, Christopher E Henze, Christopher N Hill, and Conrad A Luecke. [“Primer on Global Internal Tide and Internal Gravity Wave Continuum Modeling in HYCOM and MITgcm.”](https://www.aoml.noaa.gov/phod/docs/Inoue_et_al-2019-Journal_of_Geophysical_Research__Oceans.pdf) New Frontiers in Operational Oceanography, 2018, 307–92. Look at the standing wave along the equatorial Pacific. The nodes generally align at different times corresponding to the start of a Tropical Instability Wave train. They do eventually start travelling but the caption indicates that they emerge from what looks like a relatively fixed location. > "The six TIWs had periods ranging from 15 to 24 days, with the sixth TIW being the longest and the second and fourthTIWs being the shortest." Having a spatially fixed standing wave pattern says nothing about what the temporal dynamics are, same as with the spatially fixed but erratic ENSO time-series. 
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267.

Continuing with the forest & trees analysis of ENSO, there's just an incredible amount of fluctuation detail in the time-series.

If a fit is concentrated on the daily SOI data from 2010 to 2013 (high-resolution training range shown in yellow):

Next, if that fit is relaxed and the lower-resolution monthly data from 1880-present is simultaneously fit with the 2010-2013 daily data fit:

The possibilities for cross-validation of models are endless with such a rich data set. I think I am just scratching the surface with the possibilities.

Comment Source:Continuing with the forest & trees analysis of ENSO, there's just an incredible amount of fluctuation detail in the time-series. If a fit is concentrated on the daily SOI data from 2010 to 2013 (high-resolution training range shown in yellow): ![focussed](https://imagizer.imageshack.com/img924/9486/jEkbt1.png) Next, if that fit is relaxed and the lower-resolution monthly data from 1880-present is simultaneously fit with the 2010-2013 daily data fit: ![balanced](https://imagizer.imageshack.com/img923/4691/sqGsyS.png) The possibilities for cross-validation of models are endless with such a rich data set. I think I am just scratching the surface with the possibilities. 
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268.
edited April 2019

As an addendum to comment #249, note that being able to extract the 1-year delay is a consequence of Floquet (math) or Bloch theory (condensed matter physics), concisely expressed as F(t) = exp(-iωt)P(t), whereby a clear periodic function can be extracted from a signal. The other underlying signal, P(t), can be complex in nature, for example, a Mathieu function due to fluid sloshing in a container or it can be a set of multi-factored sinusoidal signals further modified by the solution to Laplace’s tidal equations as described in the book chapter.

Comment Source:As an addendum to comment #249, note that being able to extract the 1-year delay is a consequence of Floquet (math) or Bloch theory (condensed matter physics), concisely expressed as F(t) = exp(-iωt)P(t), whereby a clear periodic function can be extracted from a signal. The other underlying signal, P(t), can be complex in nature, for example, a Mathieu function due to fluid sloshing in a container or it can be a set of multi-factored sinusoidal signals further modified by the solution to Laplace’s tidal equations as described in the book chapter. 
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269.

If anyone is interested in this paper:

"Manifestation of the topological index formula in quantum waves and geophysical waves" https://arxiv.org/pdf/1901.10592.pdf

"Abstract: Using semi-classical analysis in R^n we present a quite general model for which the topological index formula of Atiyah-Singer predicts a spectral flow with a transition of a finite number of eigenvalues transitions between clusters (energy bands). This model corresponds to physical phenomena that are well observed for quantum energy levels of small molecules [9, 10] but also in geophysics for the oceanic or atmospheric equatorial waves [20, 4]."

Comment Source:If anyone is interested in this paper: "Manifestation of the topological index formula in quantum waves and geophysical waves" https://arxiv.org/pdf/1901.10592.pdf > "Abstract: Using semi-classical analysis in R^n we present a quite general model for which the topological index formula of Atiyah-Singer predicts a spectral flow with a transition of a finite number of eigenvalues transitions between clusters (energy bands). This model corresponds to physical phenomena that are well observed for quantum energy levels of small molecules [9, 10] but also in geophysics for the oceanic or atmospheric equatorial waves [20, 4]." 
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270.

thanks

Comment Source:thanks
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271.

For #265, I reviewed an adjoint approach to unwinding a model from the data. In a related context, here is a recent paper based on the methods of Errico(1997).

Wang, Qiang, Mu Mu, and Guodong Sun. “A Useful Approach to Sensitivity and Predictability Studies in Geophysical Fluid Dynamics: Conditional Nonlinear Optimal Perturbation.” National Science Review, 2019.

The one part I described in comments #254 & #265 is the arcsin() inversion, but another aspect is the forcing time integration which creates the erratic square wave basis. Since that is close to a strongly lagged time integration, the straight-forward adjoint of that is the time derivative!

After applying these two adjoint functions, it may be possible to linearize the problem and invert for the set of tidal factors. I think the only iteration term left is the arcsin transfer factor period.

This would provide the quickest path to an initial guess to the actual forcing, which can then be refined through the non-adjoint forward iteration optimization process. That is essentially the rationale for the work of Wang and before that Errico. The problem there (i.e. with GCMs) is to come up with a plausible set of initial conditions among the infinite number of possibilities. In contrast, for the ENSO model here, it is to efficiently estimate the tidal forcing factors from the much more limited set of possible factors -- and many of these are further constrained by known ephemeris.

Comment Source:For #265, I reviewed an adjoint approach to unwinding a model from the data. In a related context, here is a recent paper based on the methods of Errico(1997). > Wang, Qiang, Mu Mu, and Guodong Sun. “A Useful Approach to Sensitivity and Predictability Studies in Geophysical Fluid Dynamics: Conditional Nonlinear Optimal Perturbation.” National Science Review, 2019. > ![text](https://imagizer.imageshack.com/img923/701/2cpgAD.png) > ![chart](https://imagizer.imageshack.com/img924/9655/hE3GbH.png) The one part I described in comments #254 & #265 is the arcsin() inversion, but another aspect is the forcing time integration which creates the erratic square wave basis. Since that is close to a strongly lagged time integration, the straight-forward adjoint of that is the time derivative! After applying these two adjoint functions, it may be possible to linearize the problem and invert for the set of tidal factors. I think the only iteration term left is the arcsin transfer factor period. This would provide the quickest path to an initial guess to the actual forcing, which can then be refined through the non-adjoint forward iteration optimization process. That is essentially the rationale for the work of Wang and before that Errico. The problem there (i.e. with GCMs) is to come up with a plausible set of initial conditions among the infinite number of possibilities. In contrast, for the ENSO model here, it is to efficiently estimate the tidal forcing factors from the much more limited set of possible factors -- and many of these are further constrained by known ephemeris. 
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272.

What's unnerving about the model is that although the results are excellent, there may be a few minor factors that are missing. I kind of miss having an automated symbolic reasoning tool like Eureqa available. It would definitely help to find any second-order modifications.

(from what I can tell, Eureqa went from being a proprietary but free tool, to being a paid subscription-only service, and then sold off to an AI business forecasting company. The author of the software probably made a mint)

Comment Source:New blog post here: https://geoenergymath.com/2019/05/02/implicit-interpolating-cross-validation-of-enso/ What's unnerving about the model is that although the results are excellent, there may be a few minor factors that are missing. I kind of miss having an automated symbolic reasoning tool like Eureqa available. It would definitely help to find any second-order modifications. (from what I can tell, Eureqa went from being a proprietary but free tool, to being a paid subscription-only service, and then sold off to an AI business forecasting company. The author of the software probably made a mint) 
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273.

@WebHubTel: Paul, have you looked at Mathematica instead of Eureqa perchance?

Comment Source:@WebHubTel: Paul, have you looked at _Mathematica_ instead of _Eureqa_ perchance?
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274.

Jan, Yes, I experimented with the symbolic reasoning of Mathematica a bit a few years ago. I could not get it to converge to a solution as easily as it seemed to get hung up on local minima. There also weren't a lot of options for objective functions, IIRC. It really is much like the solver in Excel, which has user-defined objective functions and is less prone to local minima (Excel also doesn't crash or lock up your computer as often as Mathematica, but that may just be me). Neither is as free-form as Eureqa as both Mathematica and Excel Solver require a structured symbolic template to run.

I really should look at this again as I have a Alpha Pro subscription for a little while yet. There is a also a pricey subscription to a Pro version of Excel Solver, which is also kind of tempting -> https://www.solver.com/pricing-excel-product-software-and-support. Curious as to what this will do beyond the free version.

Do you have any strategies for success based on your experience? Thanks!

Comment Source:Jan, Yes, I experimented with the symbolic reasoning of Mathematica a bit a few years ago. I could not get it to converge to a solution as easily as it seemed to get hung up on local minima. There also weren't a lot of options for objective functions, IIRC. It really is much like the solver in Excel, which has user-defined objective functions and is less prone to local minima (Excel also doesn't crash or lock up your computer as often as Mathematica, but that may just be me). Neither is as free-form as Eureqa as both Mathematica and Excel Solver require a structured symbolic template to run. I really should look at this again as I have a Alpha Pro subscription for a little while yet. There is a also a pricey subscription to a Pro version of Excel Solver, which is also kind of tempting -> https://www.solver.com/pricing-excel-product-software-and-support. Curious as to what this will do beyond the free version. Do you have any strategies for success based on your experience? Thanks! 
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275.

Actually, yes, I subscribed to the Mathematica Home Edition version last month, using the Wolfram Cloud installation. It's $20 per month. I've been doing a bunch of work with cumulants and Edgeworth Expansions of late, and wanted to check my algebra and such. I'm also getting into tensors for multivariate statistics. I found the biggest challenge was the notion, which I'm not used to. The last symbolic maths system I put any time on was the old Macsyma. I did try the newer Maxima, but wasn't happy with it. So there's a learning curve but I'm glad I subscribe. It's like the central Wordpress: When they update, you update. I've also found the Overleaf ShareLaTeX to be pretty nice, having exclusively used MikTeX before. I'm also trying Wolfram Alpha Pro for a year. I'm less impressed with that. Comment Source:Actually, yes, I subscribed to the _Mathematica_ Home Edition version last month, using the Wolfram Cloud installation. It's$20 per month. I've been doing a bunch of work with cumulants and Edgeworth Expansions of late, and wanted to check my algebra and such. I'm also getting into tensors for multivariate statistics. I found the biggest challenge was the notion, which I'm not used to. The last symbolic maths system I put any time on was the old Macsyma. I did try the newer Maxima, but wasn't happy with it. So there's a learning curve but I'm glad I subscribe. It's like the central Wordpress: When they update, you update. I've also found the _Overleaf_ ShareLaTeX to be pretty nice, having exclusively used MikTeX before. I'm also trying Wolfram Alpha Pro for a year. I'm less impressed with that.
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276.

Yes, the Mathematica notation seems a little write-only.

Comment Source:Yes, the Mathematica notation seems a little write-only.
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277.

I looked at the recent online documentation for the Mathematica symbolic regression function called FindFormula. It seems a bit more free-form now than I recall. I should be able to do FindFormula with a set of paired data points and it now appears to return a formula.

So the test would be to create a set of points from Table[{x, N[Sin[10 * Sin[x]]]}, {x, 0, 20, .1}] and see if it returns Sin(10 * Sin(x)). Eureqa could do this, which I always found amazing. It would try any combination no matter how weird.

But FindFormula doesn't seem to work with Alpha Pro like it does with Mathematica, so I can't test it.

Comment Source:I looked at the recent online documentation for the Mathematica symbolic regression function called FindFormula. It seems a bit more free-form now than I recall. I should be able to do FindFormula with a set of paired data points and it now appears to return a formula. So the test would be to create a set of points from Table[{x, N[Sin[10 * Sin[x]]]}, {x, 0, 20, .1}] and see if it returns Sin(10 * Sin(x)). Eureqa could do this, which I always found amazing. It would try any combination no matter how weird. But FindFormula doesn't seem to work with Alpha Pro like it does with Mathematica, so I can't test it.
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278.

Yes, Alpha Pro is too oriented towards casual inquiries about general knowledge. I've put in well formed Mathematica-like expressions, and found that Alpha looked up some obscure fact about something random. That's why I switched to the Mathematica edition.

Comment Source:Yes, Alpha Pro is too oriented towards casual inquiries about general knowledge. I've put in well formed Mathematica-like expressions, and found that Alpha looked up some obscure fact about something random. That's why I switched to the Mathematica edition.
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279.
edited May 2019

I downloaded an evaluation version of Mathematica, and FindFormula was able to find an exact solution to Sin(3 * Sin(x)) after around 30 seconds of computation. That probably explains why it's not available on Alpha -- way too computation intensive.

My concern now is that with anything more complicated, you can't tell if the calculation is making progress to a solution. There is also the function NonlinearModelFit which requires a structured parametric model and also works with the same limitations.

So the benefit of Eureqa is that it will provide a continuous updating of the solution, allowing one to evaluate progress. And the Excel Solver also has a continual updating of the objective function value, at any which point in time the computation can be terminated.

Comment Source:I downloaded an evaluation version of Mathematica, and FindFormula was able to find an exact solution to Sin(3 * Sin(x)) after around 30 seconds of computation. That probably explains why it's not available on Alpha -- way too computation intensive. My concern now is that with anything more complicated, you can't tell if the calculation is making progress to a solution. There is also the function NonlinearModelFit which requires a structured parametric model and also works with the same limitations. So the benefit of Eureqa is that it will provide a continuous updating of the solution, allowing one to evaluate progress. And the Excel Solver also has a continual updating of the objective function value, at any which point in time the computation can be terminated.
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280.

I think there are options and hooks, some global, which control the verbosity of progress reports. But I have no experience with FindFormula in particular.

Comment Source:I think there are options and hooks, some global, which control the verbosity of progress reports. But I have no experience with FindFormula in particular.
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281.

I didn't see any options for progress, just for time out and max iterations.

Example of an evaluation

The NonlinearModelFit works well but FindFormula looks hopeless for this case. As I recall, Eureqa would try much harder than this.

Comment Source:I didn't see any options for progress, just for time out and max iterations. Example of an evaluation ![](https://imagizer.imageshack.com/img921/3982/hJIsLH.png) The NonlinearModelFit works well but FindFormula looks hopeless for this case. As I recall, Eureqa would try much harder than this. 
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282.

Some of those options, as I said, are global, not parameters. (I dislike that kind of computational language design.) There are skillions of 'em.

Comment Source:Some of those options, as I said, are global, not parameters. (I dislike that kind of computational language design.) There are skillions of 'em.
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283.

Thanks, I see that there are wrapper functions called Monitor and EvaluationMonitor, which may be useful.

As for a gold standard, this is the kind of dynamic monitoring that Eureqa provided before it disappeared. One could move the cursor to any entry to evaluate as the computation was running.

This is modeling a reaction network.

Comment Source:Thanks, I see that there are wrapper functions called Monitor and EvaluationMonitor, which may be useful. As for a gold standard, this is the kind of dynamic monitoring that Eureqa provided before it disappeared. One could move the cursor to any entry to evaluate as the computation was running. https://www.youtube.com/watch?v=HpuaY1huYlw This is modeling a reaction network. 
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284.

Since I had a few days left on my evaluation copy of Mathematica, here are a few wavelet transform comparisons of ENSO model and data.

Result :

The wavelet scalogram doesn't show much order yet the model fits well.

The model forcing for the daily time-series fit shows more of a periodic order:

Arcsin of data (see comment #265 above) compared to model forcing on the 130 year monthly time-series is shown below:

That is a beautiful scalogram showing a biennial period in the center and more clearly stratified harmonics. This again demonstrates how order emerges from chaos with the arcsin adjoint applied

And this is what sin(k sin(wt)) looks like on a scalogram for increasing values of k

Applying an arcsin on any one of these will result in a clearly delineated horizontal line. But in general, with multiple sinusoidal factors involved, the the transform will reduce the amount of harmonic structure.

Comment Source:Since I had a few days left on my evaluation copy of Mathematica, here are a few wavelet transform comparisons of ENSO model and data. Result : ![result](https://imagizer.imageshack.com/img923/2553/Ixx65T.png) The wavelet scalogram doesn't show much order yet the model fits well. The model forcing for the daily time-series fit shows more of a periodic order: ![dailyforce](https://imagizer.imageshack.com/img921/3852/E1bL7Y.png) Arcsin of data (see [comment #265](#Comment_21110) above) compared to model forcing on the 130 year monthly time-series is shown below: ![arcsin](https://imagizer.imageshack.com/img923/778/4XgJ5d.png) That is a beautiful scalogram showing a biennial period in the center and more clearly stratified harmonics. This again demonstrates how order emerges from chaos with the arcsin adjoint applied --- And this is what *sin(k sin(wt))* looks like on a scalogram for increasing values of k ![sinsin](https://imagizer.imageshack.com/img923/6501/2ZF6ap.png) Applying an arcsin on any one of these will result in a clearly delineated horizontal line. But in general, with multiple sinusoidal factors involved, the the transform will reduce the amount of harmonic structure. 
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285.
edited May 2019

great..perhaps study coherence data x model also pays off... https://rpubs.com/ibn_abdullah/rwcoher

Comment Source:great..perhaps study coherence data x model also pays off... https://rpubs.com/ibn_abdullah/rwcoher
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286.

Thanks Pierre, I should have known the R library has some good wavelet algorithms. I was looking for a good wavelet correlation for awhile so this biwavelet approach should be useful

Comment Source:Thanks Pierre, I should have known the R library has some good wavelet algorithms. I was looking for a [good wavelet correlation for awhile](https://forum.azimuthproject.org/discussion/comment/14549/#Comment_14549) so this biwavelet approach should be useful
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287.
edited May 2019

The last model forcing to arcsin comparison had low resolution. This is with a shorter Gabor window in the wavelet. The y-axis scaling is in octaves (each unit is a period doubling based on the fundamental unit, in months or days)

Here you can see the stationary periodic components more clearly and how the detail matches. The third chart changes the resolution from monthly to daily. The fourth removes the annual impulse multiplier which essentially unaliases the signal, showing the underlying monthly and fortnightly tidal signals (the top 2 lines). Those can be compared to the tidal signals in the differential LOD signal in the earth's rotation, which is the chart to the far right taken from the following paper:

[1]B. F. Chao, W. Chung, Z. Shih, and Y. Hsieh, “Earth’s rotation variations: a wavelet analysis,” Terra Nova, vol. 26, no. 4, pp. 260–264, 2014.

Comment Source:The last model forcing to arcsin comparison had low resolution. This is with a shorter Gabor window in the wavelet. The y-axis scaling is in octaves (each unit is a period doubling based on the fundamental unit, in months or days) ![scalogram](https://imagizer.imageshack.com/img924/7994/w1fumU.png) Here you can see the stationary periodic components more clearly and how the detail matches. The third chart changes the resolution from monthly to daily. The fourth removes the annual impulse multiplier which essentially unaliases the signal, showing the underlying monthly and fortnightly tidal signals (the top 2 lines). Those can be compared to the tidal signals in the differential LOD signal in the earth's rotation, which is the chart to the far right taken from the following paper: [1]B. F. Chao, W. Chung, Z. Shih, and Y. Hsieh, “Earth’s rotation variations: a wavelet analysis,” Terra Nova, vol. 26, no. 4, pp. 260–264, 2014. 
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288.

This is my last use of Mathematica before my license expires

In addition to applying the modeling steps to ENSO, we can apply the same procedure to QBO. This is more straightforward because the arcsin transformation for QBO is very mild in comparison to ENSO,

The general wavelet scalogram patterns are very similar to ENSO. The distinction is that the QBO model uses a Draconic factor as the fundamental forcing (since the standing wave is longitudinally symmetric) while ENSO uses Synodic (as the forcing is strongest in the Pacific). Also QBO is forced semiannually, as can be guessed from the Semi-annual Oscillation that exists at higher altitudes than QBO, while ENSO is forced annually.

Comment Source:This is my last use of Mathematica before my license expires In addition to applying the modeling steps to ENSO, we can apply the same procedure to QBO. This is more straightforward because the arcsin transformation for QBO is very mild in comparison to ENSO, ![qbo](https://imagizer.imageshack.com/img923/2821/DSbiWw.png) The general wavelet scalogram patterns are very similar to ENSO. The distinction is that the QBO model uses a Draconic factor as the fundamental forcing (since the standing wave is longitudinally symmetric) while ENSO uses Synodic (as the forcing is strongest in the Pacific). Also QBO is forced semiannually, as can be guessed from the Semi-annual Oscillation that exists at higher altitudes than QBO, while ENSO is forced annually. 
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289.

(a) Consider using SiZer, which was recently highlighted in a recent article about Atlantic planktonic decline. (b) There's another worthwhile area of application for your techniques: recurrent and blocking Rossby wave patterns which lock long duration cold and hot spells in the Northern Temperate.

Comment Source:(a) Consider using [SiZer](https://www.tandfonline.com/doi/abs/10.1080/0266476042000270554), which was recently highlighted in [a recent article about Atlantic planktonic decline](https://www.nature.com/articles/s41586-019-1181-8). (b) There's [another worthwhile area of application](https://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-18-0664.1) for your techniques: recurrent and blocking Rossby wave patterns which lock long duration cold and hot spells in the Northern Temperate.
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290.

Thanks

(a) Looks good, capturing zero crossings is something I've been thinking about, having used a simple metric that compares sign changes between two time series. SiZer should be similarly useful as well.

(b) That looks interesting in terms of certain wavenumbers being involved in the standing wave patterns -- come to think of it, that's why the standing waves stand out. Perhaps similar in origin to the Tropical Instability Waves along the equator.

The frustrating aspect of spatiotemporal patterns is that while the spatial aspect is stable (due to the fixed boundary conditions), the temporal aspect can be highly erratic. So finding a stable standing wave pattern is just the start.

Here's another view of the wavelet scalograms that I've created from the last few days. I didn't get a chance to experiment with all the different wavelet windowing functions. The Gabor appears best for isolating periodicities but the MexicanHat or DGaussian may be better for pattern matching.

Comment Source:Thanks (a) Looks good, capturing zero crossings is something I've been thinking about, having used [a simple metric that compares sign changes between two time series](http://contextearth.com/2017/10/25/improved-solver-target-error-metric/). SiZer should be similarly useful as well. (b) That looks interesting in terms of certain wavenumbers being involved in the standing wave patterns -- come to think of it, that's why the standing waves stand out. Perhaps similar in origin to the Tropical Instability Waves along the equator. The frustrating aspect of spatiotemporal patterns is that while the spatial aspect is stable (due to the fixed boundary conditions), the temporal aspect can be highly erratic. So finding a stable standing wave pattern is just the start. Here's another view of the wavelet scalograms that I've created from the last few days. I didn't get a chance to experiment with all the different wavelet windowing functions. The Gabor appears best for isolating periodicities but the MexicanHat or DGaussian may be better for pattern matching. ![sin](https://imagizer.imageshack.com/img923/7122/QECvns.png) 
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291.
edited May 2019

One aspect of ENSO that I haven't analyzed too deeply is the spatial part of the standing wave dipole. In solving Laplace's Tidal Equations, the separation of the partial diffEq leads to a family of terms

 sin(a f(t)) sin(k a x) + sin(b f(t)) sin(k b x)  + ...

where f(t) is the tidal forcing and a & b are the two strongest terms in any model fit, corresponding to the main dipole and a higher wavenumber solution.

Each spatial term is simply a sine wave with a wavenumber proportional to the a,b,.. value, with a phase aligned to the oceanic boundary conditions.

So the a term is locked to a dipole that spans the equatorial Pacific, with the node crossing at a location in between Darwin and Tahiti denoted by the arrow. All the other factors scale from the temporal to spatial domain accordingly (via the fixed k term).

Thus, when I do the time-domain fit of ENSO while keeping track of all terms a,b,... , the spatial result is automatically set in place, apart from aligning the phase and selecting the spatial scaling factor k.

This is an initial stab at a spatially aligned fit of the generated standing wave as a Hovmoller diagram, with the left side measurements from [1] and the right side the model:

The node is right around 160E. The faster cycles in the model are what I think are related to the formation of Tropical Instability Waves, which has some character of a traveling wave. Will see if any of that fine structure is revealed in the other sources of data.

[1] Pinker, R. T., S. A. Grodsky, B. Zhang, A. Busalacchi, and W. Chen. “ENSO Impact on Surface Radiative Fluxes as Observed from Space: ENSO IMPACT ON SURFACE RADIATIVE FLUXES.” Journal of Geophysical Research: Oceans 122, no. 10 (October 2017): 7880–96. https://doi.org/10.1002/2017JC012900.

EDITED 5/22: A possible source of the fine structure via ocean currents, from https://earth.nullschool.net/#current/ocean/surface/currents/orthographic=-158.02,2.78,819

If the data is available, a Hovmoller diagram could be created from a time-series sequence

Comment Source:One aspect of ENSO that I haven't analyzed too deeply is the spatial part of the standing wave dipole. In solving Laplace's Tidal Equations, the separation of the partial diffEq leads to a family of terms <pre> sin(a f(t)) sin(k a x) + sin(b f(t)) sin(k b x) + ... </pre> where f(t) is the tidal forcing and a & b are the two strongest terms in any model fit, corresponding to the main dipole and a higher wavenumber solution. Each spatial term is simply a sine wave with a wavenumber proportional to the a,b,.. value, with a phase aligned to the oceanic boundary conditions. ![soi](https://imagizer.imageshack.com/img924/4284/T6Vj8Y.png) So the a term is locked to a dipole that spans the equatorial Pacific, with the node crossing at a location in between Darwin and Tahiti denoted by the arrow. All the other factors scale from the temporal to spatial domain accordingly (via the fixed k term). Thus, when I do the time-domain fit of ENSO while keeping track of all terms a,b,... , the spatial result is automatically set in place, apart from aligning the phase and selecting the spatial scaling factor k. This is an initial stab at a spatially aligned fit of the generated standing wave as a [Hovmoller diagram](https://en.wikipedia.org/wiki/Hovm%C3%B6ller_diagram), with the left side measurements from [1] and the right side the model: ![sw](https://imagizer.imageshack.com/img924/2853/qRlhIH.png) The node is right around 160E. The faster cycles in the model are what I think are related to the formation of Tropical Instability Waves, which has some character of a traveling wave. Will see if any of that fine structure is revealed in the other sources of data. [1] Pinker, R. T., S. A. Grodsky, B. Zhang, A. Busalacchi, and W. Chen. “ENSO Impact on Surface Radiative Fluxes as Observed from Space: ENSO IMPACT ON SURFACE RADIATIVE FLUXES.” Journal of Geophysical Research: Oceans 122, no. 10 (October 2017): 7880–96. https://doi.org/10.1002/2017JC012900. EDITED 5/22: A possible source of the fine structure via ocean currents, from https://earth.nullschool.net/#current/ocean/surface/currents/orthographic=-158.02,2.78,819 ![nullschool](https://imagizer.imageshack.com/img924/8424/TjDW0v.png) If the data is available, a Hovmoller diagram could be created from a time-series sequence 
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292.

This is also a good ref on Hovmoller diagrams

Fedorov, Alexey V, and Jaclyn N Brown. “Equatorial Waves.” In Ocean Currents, Ed. J. H. Steele, S. A. Thorpe, and K. K. Turekian. Elsevier Science, 2010. http://people.earth.yale.edu/sites/default/files/files/Fedorov/50_Fedorov_EqWaves_Encyclopedia_2009.pdf.

This diagram is an idealized version of what would happen if the standing wave was closer to a pure sine wave without harmonics. From Federov, on the left is their ocean response to oscillatory winds in a shallow-water model, shown as a Hovmoller diagram of the thermocline depth anomalies on the equator. On the right is my simple LTE model with a single forcing cycle.

So Hovmoller is essentially plotting the spatiotemporal standing wave as a surface countour plot

Comment Source:This is also a good ref on Hovmoller diagrams Fedorov, Alexey V, and Jaclyn N Brown. “Equatorial Waves.” In Ocean Currents, Ed. J. H. Steele, S. A. Thorpe, and K. K. Turekian. Elsevier Science, 2010. http://people.earth.yale.edu/sites/default/files/files/Fedorov/50_Fedorov_EqWaves_Encyclopedia_2009.pdf. This diagram is an idealized version of what would happen if the standing wave was closer to a pure sine wave without harmonics. From Federov, on the left is their ocean response to oscillatory winds in a shallow-water model, shown as a Hovmoller diagram of the thermocline depth anomalies on the equator. On the right is my simple LTE model with a single forcing cycle. ![hov](https://imagizer.imageshack.com/img924/8536/XLW5qV.png) So Hovmoller is essentially plotting the spatiotemporal standing wave as a surface countour plot <center> ![sw](https://imagizer.imageshack.com/img921/3500/V9p6Vg.png) </center> 
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293.
edited May 2019

the last several comments captured in 2 blog posts:

https://geoenergymath.com/2019/05/20/applying-wavelet-scalograms/

https://geoenergymath.com/2019/05/22/characterizing-spatial-standing-waves-of-enso/

Getting to the point that I may be able to perturb the solution of LTE off the equator with a latitudinal delta and capture the dynamics of tropical instability waves

Added: Here is an animation of a surface contour representation of the standing wave ENSO dipole over the 1880-2013 period

Comment Source:the last several comments captured in 2 blog posts: https://geoenergymath.com/2019/05/20/applying-wavelet-scalograms/ https://geoenergymath.com/2019/05/22/characterizing-spatial-standing-waves-of-enso/ Getting to the point that I may be able to perturb the solution of LTE off the equator with a latitudinal delta and capture the dynamics of tropical instability waves ![](https://ars.els-cdn.com/content/image/1-s2.0-S1463500311000849-gr6.jpg) Added: Here is an animation of a surface contour representation of the standing wave ENSO dipole over the 1880-2013 period https://youtu.be/3NR7dV422OY 
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294.

Here is a recent update to the quantitative QBO model. The telling point about these kinds of alignments based on known fixed inputs is that they either work or they don't -- there is no middle ground. For example, two sine waves that have slightly different frequencies will have zero correlation over the long term.

If they do match with a high correlation, it's a matter of either treating it as a plausible model OR rejecting it as a coincidence given that there is something fundamentally wrong in the physics. In every atmospheric or oceanic model, the null hypothesis should always be either diurnal, annual, semi-annual, or tidal forcing. To reject the null hypothesis should take an equally convincing argument.

https://geoenergymath.com/2019/05/27/detailed-forcing-of-qbo/

Comment Source:Here is a recent update to the quantitative QBO model. The telling point about these kinds of alignments based on known fixed inputs is that they either work or they don't -- there is no middle ground. For example, two sine waves that have slightly different frequencies will have zero correlation over the long term. If they do match with a high correlation, it's a matter of either treating it as a plausible model *OR* rejecting it as a coincidence given that there is something fundamentally wrong in the physics. In every atmospheric or oceanic model, the null hypothesis should always be either diurnal, annual, semi-annual, or tidal forcing. To reject the null hypothesis should take an equally convincing argument. https://geoenergymath.com/2019/05/27/detailed-forcing-of-qbo/ 
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295.

From the outset, the forcing to the modeled behaviors of ENSO and QBO were split into a few categories. There is the declination of the lunar cycles, the proximity of the lunar cycles, and the sun's orbit. Each of these categories has similar complexity in terms of characterizing the orbital cycle to enough detail.

As it turns out, the declination forcing for ENSO, when isolated on it's own and retaining the 2d-order detail, provides an almost exact match as a QBO forcing.

The anomalistic cycle due to the ellipticity of the lunar orbit does not impact QBO like it does ENSO. That extra factor leads to much of the complexity of ENSO, while QBO is deceptively simple. It's deceptive because the 2nd-order details need to be right to capture the variation of the QBO's rather periodic pattern, and the ENSO model fit appears to parameterize this variation quite accurately.

Comment Source:From the outset, the forcing to the modeled behaviors of ENSO and QBO were split into a few categories. There is the declination of the lunar cycles, the proximity of the lunar cycles, and the sun's orbit. Each of these categories has similar complexity in terms of characterizing the orbital cycle to enough detail. As it turns out, the declination forcing for ENSO, when isolated on it's own and retaining the 2d-order detail, provides an almost exact match as a QBO forcing. ![](https://imagizer.imageshack.com/img921/7735/Y3B3y9.png) The anomalistic cycle due to the ellipticity of the lunar orbit does not impact QBO like it does ENSO. That extra factor leads to much of the complexity of ENSO, while QBO is deceptively simple. It's deceptive because the 2nd-order details need to be right to capture the variation of the QBO's rather periodic pattern, and the ENSO model fit appears to parameterize this variation quite accurately.
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296.

The last message was a teaser to this longer description:

https://geoenergymath.com/2019/05/31/teleconnections-vs-common-mode-mechanisms/

Comment Source:The last message was a teaser to this longer description: https://geoenergymath.com/2019/05/31/teleconnections-vs-common-mode-mechanisms/ 
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297.

It's newsworthy when a climate science paper appears in the prestigious Physical Review Letters. This paper concerns the anomaly observed in the QBO in 2016

Synopsis:A Missing Beat in Earth’s Oscillating Wind Patterns

https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.122.214504

I had been mostly analyzing the QBO data for altitudes corresponding to 30hPa, but the data at a lower altitude of 70hPa shows much more fine structure -- which the tidal model also shows. The positive peaks (westerlies) show an interesting pattern in how they alternate with even and odd subpeaks, which is what the model does -- i.e. the annual impulses create that same pattern when multiplied by the monthly draconic cycles.

(The negative peaks (easterlies) appear not quite as regular, so those excursions are suppressed here)

The other aspect concerns the anomaly of 2016. Based on fitting the tidal model, I think the anomaly instead occurred in 2005 as a slight shift in the annual impulse, and that this shift reverted back in 2016. This is supported by the upper and lower chart. The upper chart shows excellent agreement up to 2005, whereby it needs a slight shift (see lower panel) in the annual impulse to get back in phase.

But this shift disappeared in 2016 (i.e. actually an anomaly reversal). The impulse trains are shown as insets, indicating how subtle this shift needs to be to influence the phase of the model.

The PRL paper recognizes only the anomaly of 2016, which is obvious to the eye (see below the indicated "disruption"), whereas the anomaly of 2005 is only obvious to the model

What I find disappointing about the full PRL paper is that they can explain an anomaly without truly understanding what causes the QBO. One of the PRL authors is of the same team that published in Science the Topological Origin of Equatorial Waves paper, so they should have done more than just a qualitative study.

Not only this paper but there was another PRL paper on QBO that I missed from last year:

"Nonlinear saturation of the large scale flow in a laboratory model of the quasibiennial oscillation"

https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.121.134502

This one is problematic because they create a laboratory experiment that has nothing close to the topological environment of the atmospheric QBO. A rotating upright cylinder with downward gravity is not even close to a spinning sphere with radial gravitational forces and a Coriolis effect

Comment Source:It's newsworthy when a climate science paper appears in the prestigious Physical Review Letters. This paper concerns the anomaly observed in the QBO in 2016 >Synopsis:A Missing Beat in Earth’s Oscillating Wind Patterns >https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.122.214504 I had been mostly analyzing the QBO data for altitudes corresponding to 30hPa, but the data at a lower altitude of 70hPa shows much more fine structure -- which the tidal model also shows. The positive peaks (westerlies) show an interesting pattern in how they alternate with even and odd subpeaks, which is what the model does -- i.e. the annual impulses create that same pattern when multiplied by the monthly draconic cycles. ![](https://imagizer.imageshack.com/img922/115/U7ZmzE.png) (The negative peaks (easterlies) appear not quite as regular, so those excursions are suppressed here) The other aspect concerns the anomaly of 2016. Based on fitting the tidal model, I think the anomaly instead occurred in 2005 as a slight shift in the annual impulse, and that this shift reverted back in 2016. This is supported by the upper and lower chart. The upper chart shows excellent agreement up to 2005, whereby it needs a slight shift (see lower panel) in the annual impulse to get back in phase. But this shift disappeared in 2016 (i.e. actually an anomaly reversal). The impulse trains are shown as insets, indicating how subtle this shift needs to be to influence the phase of the model. The PRL paper recognizes only the anomaly of 2016, which is obvious to the eye (see below the indicated "disruption"), whereas the anomaly of 2005 is only obvious to the model ![prl](https://imagizer.imageshack.com/img923/6269/GDhlia.png) What I find disappointing about the [full PRL paper](https://doi.org/10.1103/PhysRevLett.122.214504) is that they can explain an anomaly without truly understanding what causes the QBO. One of the PRL authors is of the same team that published in Science the [Topological Origin of Equatorial Waves](https://science.sciencemag.org/content/358/6366/1075) paper, so they should have done more than just a qualitative study. Not only this paper but there was another PRL paper on QBO that I missed from last year: > "Nonlinear saturation of the large scale flow in a laboratory model of the quasibiennial oscillation" > https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.121.134502 This one is problematic because they create a laboratory experiment that has nothing close to the topological environment of the atmospheric QBO. A rotating upright cylinder with downward gravity is not even close to a spinning sphere with radial gravitational forces and a Coriolis effect ![prl2](https://imagizer.imageshack.com/img924/8301/Y15hSh.png) 
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298.
edited June 2019

This is a great, never-ending detective story where there are always new suspects for you to nail :) Perhaps a letter(s) to PRL might help raise the profile of your model? I'm suffering from the paradox of choice as to what's the best open source tool to code a solid geometry earth model simulation? I've not been able to find out what nullschool's programmed in. Fenics and Blender are 2 other possibles.

Comment Source:This is a great, never-ending detective story where there are always new suspects for you to nail :) Perhaps a letter(s) to PRL might help raise the profile of your model? I'm suffering from the paradox of choice as to what's the best open source tool to code a solid geometry earth model simulation? I've not been able to find out what nullschool's programmed in. Fenics and Blender are 2 other possibles. 
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299.

Jim, thanks. I burned my bridges at PRL when I tried submitting something earlier. Wish I could find the response, but it was something to the effect that PRL is mainly looking for models that improve the fundamental understanding of physics and that climate science does not fit into that category. That's why I was surprised that they published these last two papers on QBO.

"I'm suffering from the paradox of choice as to what's the best open source tool to code a solid geometry earth model simulation? I've not been able to find out what nullschool's programmed in. Fenics and Blender are 2 other possibles."

Is this solid geometry or more concerning fluid geometry?

Comment Source:Jim, thanks. I burned my bridges at PRL when I tried submitting something earlier. Wish I could find the response, but it was something to the effect that PRL is mainly looking for models that improve the fundamental understanding of physics and that climate science does not fit into that category. That's why I was surprised that they published these last two papers on QBO. > "I'm suffering from the paradox of choice as to what's the best open source tool to code a solid geometry earth model simulation? I've not been able to find out what nullschool's programmed in. Fenics and Blender are 2 other possibles." Is this solid geometry or more concerning fluid geometry? 
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300.

I naively jumped to the idea of overlaying simulation of flows using some CSG model as constraints. I had to look up what fluid geometry is. I've heard of Kahler manifolds but not Monge-Ampere equations so I think I should learn something about them. I was thinking FEM might be what I should try and use for a fruit fly ocean model.

Comment Source:I naively jumped to the idea of overlaying simulation of flows using some CSG model as constraints. I had to look up what fluid geometry is. I've heard of Kahler manifolds but not Monge-Ampere equations so I think I should learn something about them. I was thinking FEM might be what I should try and use for a fruit fly ocean model.