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# QBO and ENSO

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301.
edited June 2019

Lots of the fluid dynamics is done through OpenFOAM and VisualCFD https://www.openfoam.com/products/visualcfd.php

As for solid geometry simulations, I would also consider something like PTC Creo. https://en.wikipedia.org/wiki/PTC_Creo_Elements/Pro

(I worked on these kinds of solid models for over 10 years and learned how to animate them in real-time through external software. )

So there are two aspects to this: (1) doing the morphing, articulations, and rotations of solid body features, which is what PTC CREO is good at, and (2) doing the visualization of nullschool-like features of fluid turbulence, etc, which I don't have as much experience with. Of course, leveraging all the gaming software essentially combines the two because they need all the interesting articulation and morphing motions and fluid scenarios.

Comment Source:Lots of the fluid dynamics is done through OpenFOAM and VisualCFD https://www.openfoam.com/products/visualcfd.php As for solid geometry simulations, I would also consider something like PTC Creo. https://en.wikipedia.org/wiki/PTC_Creo_Elements/Pro (I worked on these kinds of solid models for over 10 years and learned how to animate them in real-time through external software. ) So there are two aspects to this: (1) doing the morphing, articulations, and rotations of solid body features, which is what PTC CREO is good at, and (2) doing the visualization of nullschool-like features of fluid turbulence, etc, which I don't have as much experience with. Of course, leveraging all the gaming software essentially combines the two because they need all the interesting articulation and morphing motions and fluid scenarios.
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302.

Thanks for the detailed answers. This will keep me busy! I've not kept up with CAD since 1991 when I worked in an AEC language called Eagle.

Re your 2005 observation: NPL accepted the original paper so an update to their model must fit their T&C scope. If not, I bet the authors would want to know.

Comment Source:Thanks for the detailed answers. This will keep me busy! I've not kept up with CAD since 1991 when I worked in an AEC language called Eagle. Re your 2005 observation: NPL accepted the original paper so an update to their model must fit their T&C scope. If not, I bet the authors would want to know.
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303.

Jim said:

"Re your 2005 observation: NPL accepted the original paper so an update to their model must fit their T&C scope. If not, I bet the authors would want to know."

The scientific discussion is fascinating as is -- I have my model published so can wait it out. What mystifies me is how they can make assertions on what appears to be a 2nd-order effect (the 2016 QBO anomaly) when they clearly don't have a good handle on what's causing the primary oscillation in the first place. In other words, how can they be certain it is an anomaly unless they are certain that the underlying phenomena is deterministic? The best explanation that they are able to come up with is that the anomaly is an additional transient bifurcation of the original bifurcation

Comment Source:Jim said: > "Re your 2005 observation: NPL accepted the original paper so an update to their model must fit their T&C scope. If not, I bet the authors would want to know." The scientific discussion is fascinating as is -- I have my model published so can wait it out. What mystifies me is how they can make assertions on what appears to be a 2nd-order effect (the 2016 QBO anomaly) when they clearly don't have a good handle on what's causing the primary oscillation in the first place. In other words, how can they be certain it is an anomaly unless they are certain that the underlying phenomena is deterministic? The best explanation that they are able to come up with is that the anomaly is an additional transient bifurcation of the original bifurcation 
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304.

+1

Comment Source:+1 
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305.
edited June 2019

Interesting data from this paper on the equatorial-only Semi-Annual Oscillation (SAO) of the upper stratosphere and lower mesosphere wind pattern.

[1] T. Hirooka, T. Ohata, and N. Eguchi, “Modulation of the Semiannual Oscillation Induced by Sudden Stratospheric Warming Events,” in ISWA2016, Tokyo, Japan, 2016, p. 16.

The SAO flips by 180 degrees between the stratosphere (the SSAO) and the mesosphere (the MSAO). You can see this in the upper panel below where the intense westerlies (in RED) occur during the beginning and middle of each year for the MSAO, and they occur between these times (Spring and Fall) for the SSAO. The direction times are complementary for the easterlies in BLUE. At altitudes between the MSAO and SSSAO, the strength of the SAO is significantly reduced as you can see in the lower panel showing the spectral lines. The QBO starts at altitudes below the SSAO.

This may be explained by the Laplace's Tidal Equation analytic solution that I have been applying to the ENSO and QBO models.

The equation applied is $$\sin( A \sin(4 \pi t + \phi) + \theta(z)) )$$

If the LTE phase varies in altitude (z) due to differing characteristics of the atmospheric density, the sense of the sinusoidal modulation will flip. This is for a value of A that is large enough to cause a strong modulation. For phases halfway between where the sign flips, the modulation bifurcates the semi-annual oscillation such that the 1/2-year period disappears and is replaced by (in-theory) a 1/4-year or 90-day cycle. Can kind of see that in the power-spectra above.

So below is the theoretical LTE plot alongside the paper's plot. The contour colors don't quite match up, and I don't have Mathematica any longer to get a matching color density plot

Comment Source:Interesting data from this paper on the equatorial-only Semi-Annual Oscillation (SAO) of the upper stratosphere and lower mesosphere wind pattern. [1] T. Hirooka, T. Ohata, and N. Eguchi, “Modulation of the Semiannual Oscillation Induced by Sudden Stratospheric Warming Events,” in ISWA2016, Tokyo, Japan, 2016, p. 16. The SAO flips by 180 degrees between the stratosphere (the SSAO) and the mesosphere (the MSAO). You can see this in the upper panel below where the intense westerlies (in RED) occur during the beginning and middle of each year for the MSAO, and they occur between these times (Spring and Fall) for the SSAO. The direction times are complementary for the easterlies in BLUE. At altitudes between the MSAO and SSSAO, the strength of the SAO is significantly reduced as you can see in the lower panel showing the spectral lines. The QBO starts at altitudes below the SSAO. ![sao1](https://imagizer.imageshack.com/img923/1894/WJ8gOf.png) This may be explained by the Laplace's Tidal Equation analytic solution that I have been applying to the ENSO and QBO models. The equation applied is \$$\sin( A \sin(4 \pi t + \phi) + \theta(z)) ) \$$ If the LTE phase varies in altitude (z) due to differing characteristics of the atmospheric density, the sense of the sinusoidal modulation will flip. This is for a value of *A* that is large enough to cause a strong modulation. For phases halfway between where the sign flips, the modulation bifurcates the semi-annual oscillation such that the 1/2-year period disappears and is replaced by (in-theory) a 1/4-year or 90-day cycle. Can kind of see that in the power-spectra above. So below is the theoretical LTE plot alongside the paper's plot. The contour colors don't quite match up, and I don't have Mathematica any longer to get a matching color density plot ![sao2](https://imagizer.imageshack.com/img921/1654/dLwB8A.png) 
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306.

This opinion piece by Hossenfelder in the NYT makes the claim that "Only Supercomputers Can Do the Math" of modeling the global climate:

https://www.nytimes.com/2019/06/12/opinion/climate-change-supercomputers.html

I recall that Hossenfelder wrote a book called "Lost in Math: How Beauty Leads Physics Astray"

"Whether pondering black holes or predicting discoveries at CERN, physicists believe the best theories are beautiful, natural, and elegant, and this standard separates popular theories from disposable ones. This is why, Sabine Hossenfelder argues, we have not seen a major breakthrough in the foundations of physics for more than four decades. The belief in beauty has become so dogmatic that it now conflicts with scientific objectivity: observation has been unable to confirm mindboggling theories, like supersymmetry or grand unification, invented by physicists based on aesthetic criteria. Worse, these "too good to not be true" theories are actually untestable and they have left the field in a cul-de-sac. "

I don't know if the latter helps explain the former view.

Next week's geophysics fluid dynamics conference presentations illustrates the potential simplicity

Infinite U(1) Symmetry of the Quasi-Linear Approximation : "Particle-relabeling symmetry of inviscid fluid equations, equivalent in the case of incompressible fluids to the infinite dimensional group of volume-preserving diffeomorphisms, is broken by the quasi-linear approximation. Instead the equations of motion are invariant under an infinite U(1) symmetry as the phase of each wave may be independently varied, reflecting the absence of wave + wave —> wave interactions. " "The infinite U(1) symmetry of linear waves manifests, by Noether’s theorem, as separate conservation of the pseudomomenta for each zonal wavenumber. The pseudomomenta are approximately conserved for quasilinear dynamics due to the separation in time scales between the evolution of the zonal mean and the waves. Whether or not an action principle or a Hamiltonian can be found that generates the quasilinear dynamics remains an open question; if one can be found then it should be possible to find exactly conserved pseudomomenta as the quasilinear system retains the infinite U(1) symmetry. Pseudomomenta are not conserved by the fully nonlinear dynamics"

U(1) corresponds to the unitary group of dimension 1, i.e. complex numbers of norm 1 residing on the unit circle. This is a global symmetry and corresponds to conserved quantities via Noether's theorem.

Not surprising that the solution used in ENSO and QBO is appropriately sin(a f(t) + θ). This appears fairly elegant to me, resulting from the real (observable) part of the complex number.

And also this presentation from the conference:

The Kelvin and Mixed Rossby Gravity Waves on the Spherical Earth : "While the theory developed by Matsuno for the equatorial \beta-plane allows for exact analytic solutions, the corresponding theory developed by Longuet-Higgins on the sphere can only be solved analytically at some asymptotic limits. In the present work we revisit the Kelvin and MRG waves on the sphere using two complimentary forms of analysis: (i) Special ad hoc analytic solutions that yield accurate approximations for the latitude-dependent amplitudes and dispersion relations of the Kelvin and MRG waves over a wide range of the parameters space. (ii) A Schrodinger formulation that provides a classification for the waves in terms of the mode numbers of the associated Sturm-Liouville problem. "

The ENSO and QBO solution above comes directly from transforming the complete Navier-Stokes through the Laplace's Tidal Equation linearizing simplification into a Sturm-Liouville formulation that could be analytically solved.

They might be getting close, perhaps a year or two they will catch on.

Comment Source:This opinion piece by Hossenfelder in the NYT makes the claim that "Only Supercomputers Can Do the Math" of modeling the global climate: https://www.nytimes.com/2019/06/12/opinion/climate-change-supercomputers.html I recall that Hossenfelder wrote a book called "Lost in Math: How Beauty Leads Physics Astray" > "Whether pondering black holes or predicting discoveries at CERN, physicists believe the best theories are beautiful, natural, and elegant, and this standard separates popular theories from disposable ones. This is why, Sabine Hossenfelder argues, we have not seen a major breakthrough in the foundations of physics for more than four decades. The belief in beauty has become so dogmatic that it now conflicts with scientific objectivity: observation has been unable to confirm mindboggling theories, like supersymmetry or grand unification, invented by physicists based on aesthetic criteria. Worse, these "too good to not be true" theories are actually untestable and they have left the field in a cul-de-sac. " I don't know if the latter helps explain the former view. Next week's geophysics fluid dynamics conference presentations illustrates the potential simplicity > [*Infinite U(1) Symmetry of the Quasi-Linear Approximation*](https://ams.confex.com/ams/22FLUID/meetingapp.cgi/Paper/360248 ) : > "Particle-relabeling symmetry of inviscid fluid equations, equivalent in the case of incompressible fluids to the infinite dimensional group of volume-preserving diffeomorphisms, is broken by the quasi-linear approximation. Instead the equations of motion are invariant under an infinite U(1) symmetry as the phase of each wave may be independently varied, reflecting the absence of wave + wave —> wave interactions. " > "The infinite U(1) symmetry of linear waves manifests, by Noether’s theorem, as separate conservation of the pseudomomenta for each zonal wavenumber. The pseudomomenta are approximately conserved for quasilinear dynamics due to the separation in time scales between the evolution of the zonal mean and the waves. Whether or not an action principle or a Hamiltonian can be found that generates the quasilinear dynamics remains an open question; if one can be found then it should be possible to find exactly conserved pseudomomenta as the quasilinear system retains the infinite U(1) symmetry. Pseudomomenta are not conserved by the fully nonlinear dynamics" > ![](https://imagizer.imageshack.com/img924/5267/NLgOou.png) U(1) corresponds to the unitary group of dimension 1, i.e. complex numbers of norm 1 residing on the unit circle. This is a global symmetry and corresponds to conserved quantities via Noether's theorem. Not surprising that the solution used in ENSO and QBO is appropriately sin(a f(t) + θ). This appears fairly elegant to me, resulting from the real (observable) part of the complex number. And also this presentation from the conference: > [*The Kelvin and Mixed Rossby Gravity Waves on the Spherical Earth*](https://ams.confex.com/ams/22FLUID/meetingapp.cgi/Paper/359154) : >"While the theory developed by Matsuno for the equatorial \beta-plane allows for exact analytic solutions, the corresponding theory developed by Longuet-Higgins on the sphere can only be solved analytically at some asymptotic limits. In the present work we revisit the Kelvin and MRG waves on the sphere using two complimentary forms of analysis: (i) Special ad hoc analytic solutions that yield accurate approximations for the latitude-dependent amplitudes and dispersion relations of the Kelvin and MRG waves over a wide range of the parameters space. (ii) A Schrodinger formulation that provides a classification for the waves in terms of the mode numbers of the associated Sturm-Liouville problem. " The ENSO and QBO solution above comes directly from transforming the complete Navier-Stokes through the Laplace's Tidal Equation linearizing simplification into a Sturm-Liouville formulation that could be analytically solved. They might be getting close, perhaps a year or two they will catch on. 
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307.
edited June 2019

A recent climate science review paper covers "Inferring causation from time series in Earth system sciences", Runge et al 2019, Nature Communications

Unfortunately they do not describe common-mode factor causation. I took one of their figures and added that class of mechanism in yellow

Common-mode factors are very common and well-known to experimentalists and trouble-shooters. In the figure, a mechanism linking two regions, which is often classified as a teleconnection may actually be common-mode -- in this case a correlation due to a shared lunar forcing that may tie the regions together.

So let's try a particular geospatial correlation. If we take the left region and shift it to the ENSO region in the Pacific and the right region to AMO in the Atlantic, we can evaluate the common mode between these two oceanic indices.

These two indices on first glance show no time correlation (even with lag shifts applied) -- the scatter plot correlation looks like a blob.

But what happens when we apply the same forcing used in modeling ENSO and then apply that to an AMO model? The Laplace's Tidal Equation (LTE) formulation is essentially normalized to sin (E f(t) + θ) for ENSO and sin (A f(t) + φ) for AMO, where the E and A factors calibrate the distinct standing wave number boundary conditions for the two regions, and f(t) is the common forcing.

Because of the way that the iteration works in fitting the model to the data (need to fiddle the parameters to avoid getting stuck at local minima) the two forcings aren't precisely the same, R=0.96, but can be considered to be virtually aligned.

The fitted models to the ENSO and AMO with common forcing are below

So the distinction between the two oscillating dipoles resides completely in the LTE sinusoidal modulation applied (the E and A factors), which differs for the two by a non-trivial amount. In other words, this is a single degree of freedom adjustment corresponding to a global wavenumber of 5 for ENSO and 4 for AMO.

The causative mechanism is thus simply a scaling in LTE space accounting for the difference in the geometry of the Atlantic and Pacific basins for a common-mode lunisolar forcing. As quoted in the previous post

"physicists believe the best theories are beautiful, natural, and elegant, and this standard separates popular theories from disposable ones."

This is as natural and elegant as it can get -- just a twist added to conventional tidal analysis

N.B. If you think it odd that a single parameter modification can change completely the character of a solution, consider the case of how band structures in materials can change completely with slight lattice transformations. This has a related explanation in terms of Brillouin zone folding. Solid state physicists treat this complication as a cost of doing business

Comment Source:A recent climate science review paper covers "Inferring causation from time series in Earth system sciences", Runge et al 2019, Nature Communications Unfortunately they do not describe common-mode factor causation. I took one of their figures and added that class of mechanism in yellow ![twitter](https://pbs.twimg.com/media/D9SG2G2WwAUzfvW.png) Common-mode factors are very common and well-known to experimentalists and trouble-shooters. In the figure, a mechanism linking two regions, which is often classified as a teleconnection may actually be common-mode -- in this case a correlation due to a shared lunar forcing that may tie the regions together. So let's try a particular geospatial correlation. If we take the left region and shift it to the ENSO region in the Pacific and the right region to AMO in the Atlantic, we can evaluate the common mode between these two oceanic indices. These two indices on first glance show no time correlation (even with lag shifts applied) -- the scatter plot correlation looks like a blob. ![](https://imagizer.imageshack.com/img921/4896/VDd7c8.png) But what happens when we apply the same forcing used in modeling ENSO and then apply that to an AMO model? The Laplace's Tidal Equation (LTE) formulation is essentially normalized to *sin (E f(t) + θ)* for ENSO and *sin (A f(t) + φ)* for AMO, where the *E* and *A* factors calibrate the distinct standing wave number boundary conditions for the two regions, and *f(t)* is the common forcing. Because of the way that the iteration works in fitting the model to the data (need to fiddle the parameters to avoid getting stuck at local minima) the two forcings aren't precisely the same, R=0.96, but can be considered to be virtually aligned. ![](https://imagizer.imageshack.com/img921/3833/jL1bUW.png) The fitted models to the ENSO and AMO with common forcing are below ![](https://imagizer.imageshack.com/img922/36/ArYWl7.png) So the distinction between the two oscillating dipoles resides completely in the LTE sinusoidal modulation applied (the *E* and *A* factors), which differs for the two by a non-trivial amount. In other words, this is a single degree of freedom adjustment corresponding to a global wavenumber of 5 for ENSO and 4 for AMO. ![](https://imagizer.imageshack.com/img924/6623/hTOiPU.png) The causative mechanism is thus simply a scaling in LTE space accounting for the difference in the geometry of the Atlantic and Pacific basins for a common-mode lunisolar forcing. As quoted in the previous post > "physicists believe the best theories are beautiful, natural, and elegant, and this standard separates popular theories from disposable ones." This is as natural and elegant as it can get -- just a twist added to conventional tidal analysis --- N.B. If you think it odd that a single parameter modification can change completely the character of a solution, consider the case of how band structures in materials can change completely with slight lattice transformations. This has a related explanation in terms of Brillouin zone folding. Solid state physicists treat this complication as a cost of doing business ![](https://i.stack.imgur.com/dS2lz.png) 
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308.

Finding this common-mode forcing between ENSO and AMO is either the correct scientific interpretation or the most unlikely rabbit hole to fall into. There is always a concern with regards to over-fitting. The ENSO fit could be fortuitous, based on adjusting the parameters and tweaking for the best alignment between data and model. Yet using the same set of parameters as a starting point is squared as unlikely if each process was based on random chance.

Comment Source:Finding this common-mode forcing between ENSO and AMO is either the correct scientific interpretation or the most unlikely rabbit hole to fall into. There is always a concern with regards to over-fitting. The ENSO fit could be fortuitous, based on adjusting the parameters and tweaking for the best alignment between data and model. Yet using the same set of parameters as a starting point is *squared* as unlikely if each process was based on random chance.
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309.

Using the same approach on the other major climate index, the PDO, a pattern is starting to emerge. The PDO has a significant LTE sin() modulation that is the same as ENSO, but also has a strong factor with a wavenumber that is 5 times as rapid. In contrast the AMO wavenumber modulation is 3 times as fast as ENSO (with a much weaker modulation that's the same as ENSO).

The odd-number multiplicative scaling may be related to properties of inversion symmetry in the medium. So if there was an even harmonic among the standing waves, it would be imbalanced with respect to positive and negative excursions.

Again, the tidal forcing matches to a correlation above 0.99 for years greater than 1920. The AMO fit extends back to 1856 so the mean forcing appears to stay closer to zero for those years, so is flatter.

Comment Source:Using the same approach on the other major climate index, the PDO, a pattern is starting to emerge. The PDO has a significant LTE sin() modulation that is the same as ENSO, but also has a strong factor with a wavenumber that is 5 times as rapid. In contrast the AMO wavenumber modulation is 3 times as fast as ENSO (with a much weaker modulation that's the same as ENSO). ![lte](https://imagizer.imageshack.com/img923/9738/pUxrf1.png) The odd-number multiplicative scaling may be related to properties of [inversion symmetry in the medium](https://physics.stackexchange.com/questions/127531/lack-of-inversion-symmetry-in-crystal). So if there was an even harmonic among the standing waves, it would be imbalanced with respect to positive and negative excursions. Again, the tidal forcing matches to a correlation above 0.99 for years greater than 1920. The AMO fit extends back to 1856 so the mean forcing appears to stay closer to zero for those years, so is flatter. ![tidal](https://imagizer.imageshack.com/img924/3595/GPUERT.png) 
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310.
edited August 2019

I'm working a model of one of the fastest ocean oscillations, the North Atlantic Oscillation (NAO). The NAO can show up to 2 strong cycles per year in comparison to ENSO, which has El Nino peaks every 2 to 7 years

It appears that the model for NAO is not as sensitive to annual or semi-annual impulses like all the other behaviors (ENSO, QBO, PDO, AMO, IOP) require for modeling. So instead of an impulse, it appears to more directly correspond to monthly tidal variations.

This is promising. The time-series is dense enough that cross-validation may work well here.

New/Edited follows

This is a cross-validation, trying to overfit an interval from 1980 to 2000 and observing how the out-of-band intervals respond -- quite stable.

The non-impulsed lunisolar tidal forcing is identical for ENSO and NAO, which removes many degrees of freedom from the fitting process.

The LTE modulation for NAO is quite strong, approximately that of used in PDO in the prior comment #309. Perhaps this is expected as both NAO and PDO are northern/higher latitude behaviors

Comment Source:I'm working a model of one of the fastest ocean oscillations, the North Atlantic Oscillation (NAO). The NAO can show up to 2 strong cycles per year in comparison to ENSO, which has El Nino peaks every 2 to 7 years It appears that the model for NAO is not as sensitive to annual or semi-annual impulses like all the other behaviors (ENSO, QBO, PDO, AMO, IOP) require for modeling. So instead of an impulse, it appears to more directly correspond to monthly tidal variations. ![](https://imagizer.imageshack.com/img923/8181/xjA7Ww.png) This is promising. The time-series is dense enough that cross-validation may work well here. --- *New/Edited follows* This is a cross-validation, trying to overfit an interval from 1980 to 2000 and observing how the out-of-band intervals respond -- quite stable. ![fit](https://imagizer.imageshack.com/img922/1354/NppJ53.png) The non-impulsed lunisolar tidal forcing is identical for ENSO and NAO, which removes many degrees of freedom from the fitting process. ![forcing](https://imagizer.imageshack.com/img923/1213/gPGBfx.png) The LTE modulation for NAO is quite strong, approximately that of used in PDO in the prior comment #309. Perhaps this is expected as both NAO and PDO are northern/higher latitude behaviors ![mod](https://imagizer.imageshack.com/img923/1534/aGC6Y4.png) 
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311.

interesting!thanks

Comment Source:interesting!thanks
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312.
edited August 2019

It's the two-layers-of-complexity category. Impossible to figure out unless one layer reveals itself.

Longer post here: https://geoenergymath.com/2019/08/04/north-atlantic-oscillation/

Comment Source:It's the two-layers-of-complexity category. Impossible to figure out unless one layer reveals itself. Longer post here: https://geoenergymath.com/2019/08/04/north-atlantic-oscillation/
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313.

Random observation, or an aside: Tierney, Haywood, Feng, Bhattacharya, and Otto-Bliesner have a paper accepted in GRL, which addresses, in the negative, the idea from Fedorov, et al (2006, 2010) of "permanent El Nino in the early Pliocene epoch". The paper is:

J. E. Tierney, A. M. Haywood, R. Feng, T. Bhattacharya, B. L. Otto-Bliesner, Pliocene warmth consistent with greenhouse gas forcing, Geophysical Research Letters (2019).

Comment Source:Random observation, or an aside: Tierney, Haywood, Feng, Bhattacharya, and Otto-Bliesner have a paper accepted in _GRL_, which addresses, in the negative, the idea from Fedorov, _et_ _al_ (2006, 2010) of "permanent El Nino in the early Pliocene epoch". The paper is: J. E. Tierney, A. M. Haywood, R. Feng, T. Bhattacharya, B. L. Otto-Bliesner, [Pliocene warmth consistent with greenhouse gas forcing](https://sci-hub.tw/10.1029/2019GL083802), _[Geophysical Research Letters](https://agupubs.onlinelibrary.wiley.com/journal/19448007)_ (2019).
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314.

Interesting paper in the sense of I don't understand what a permanent El Nino even means. Since ENSO is conceptually a dipole and always reverts to a mean value of zero, how can it ever balance out to being predominately El Nino over La Nina?

Unless what they are saying is that warmer El Nino peaks are the quiescent state of the system and that the cold La Nina valleys were thus less severe. That is, the sloshing gradient was lower, resulting in an overall greater distribution of warmer surface water along the equator and thus less cold water brought up from the depths from a steeper thermocline gradient.

From the paper:

"Previous work suggested a low zonal sea-surface temperature (SST) gradient in the tropical Pacific during the Pliocene, the so-called “permanent El Ni˜no.”

Comment Source:Interesting paper in the sense of I don't understand what a permanent El Nino even means. Since ENSO is conceptually a dipole and always reverts to a mean value of zero, how can it ever balance out to being predominately El Nino over La Nina? Unless what they are saying is that warmer El Nino peaks are the quiescent state of the system and that the cold La Nina valleys were thus less severe. That is, the sloshing gradient was lower, resulting in an overall greater distribution of warmer surface water along the equator and thus less cold water brought up from the depths from a steeper thermocline gradient. From the paper: > "Previous work suggested a low zonal sea-surface temperature (SST) gradient in the tropical Pacific during the Pliocene, the so-called “permanent El Ni˜no.”
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315.

One index that I haven't looked at is the Indian Ocean Dipole and its gradient measure the Dipole Mode Index. This is important because it is correlated with India subcontinent monsoons. It also shows a correlation to ENSO, which is quite apparent by comparing specific peak positions, with a correlation coefficient of 0.2.

I think the reason the correlation isn't higher is that there is likely another standing wave solution that complements the major standing wave that stretches across the equatorial Pacific. The latter contributes the majority of ENSO but only a portion of IOD, so the mystery standing wave is what generates the busier cyclic behavior o IOD.

Like with the other oceanic indices, it has a similar tidal forcing to ENSO, with R^2>0.95.

What differs from ENSO is the LTE modulation, of which IOD consists of a background similar to ENSO, but also a faster modulation that is 3 to 4 that of the background.

The fit over the entire time span is good, with the Fourier spectrum in the lower panel.

So far, the following indices highlighted in yellow have been modeled. The QBO is the only atmospheric behavior, and it has a distinct tidal forcing. Each of ENSO, PDO, AMO, NAO, and now IOD have a nearly identical set of fundamental forced tidal cycles but distinct standing mode modulations.

Comment Source:One index that I haven't looked at is the [Indian Ocean Dipole](https://www.esrl.noaa.gov/psd/gcos_wgsp/Timeseries/DMI/) and its gradient measure the [Dipole Mode Index](http://www.jamstec.go.jp/frsgc/research/d1/iod/dmi.html). This is important because it is correlated with India subcontinent monsoons. It also shows a correlation to ENSO, which is quite apparent by comparing specific peak positions, with a correlation coefficient of 0.2. I think the reason the correlation isn't higher is that there is likely another standing wave solution that complements the major standing wave that stretches across the equatorial Pacific. The latter contributes the majority of ENSO but only a portion of IOD, so the mystery standing wave is what generates the busier cyclic behavior o IOD. Like with the other oceanic indices, it has a similar tidal forcing to ENSO, with R^2>0.95. ![Forcing](https://imagizer.imageshack.com/img922/1511/GHgwci.png) What differs from ENSO is the LTE modulation, of which IOD consists of a background similar to ENSO, but also a faster modulation that is 3 to 4 that of the background. ![LTE](https://imagizer.imageshack.com/img922/523/y1BfTi.png) The fit over the entire time span is good, with the Fourier spectrum in the lower panel. ![Model](https://imagizer.imageshack.com/img922/8189/npvxsW.png) So far, the following indices highlighted in yellow have been modeled. The QBO is the only atmospheric behavior, and it has a distinct tidal forcing. Each of ENSO, PDO, AMO, NAO, and now IOD have a nearly identical set of fundamental forced tidal cycles but distinct standing mode modulations. ![map](https://imagizer.imageshack.com/img923/5279/oTlfQk.gif) 
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316.

Very, very cool!

Comment Source:Very, very cool!
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317.
edited August 2019

I wonder if the 3-4x frequency modulation of the IOD relative to ENSO might be related to some sort of "shudder" due to the various chokepoints around the straights of Malacca which iirc were adduced as contributing the slowdown of the increased SST in the last decade?

Comment Source:I wonder if the 3-4x frequency modulation of the IOD relative to ENSO might be related to some sort of "shudder" due to the various chokepoints around the straights of Malacca which iirc were adduced as contributing the slowdown of the increased SST in the last decade?
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318.

Thanks, I'm next working on the PNA, which is a Pacific North America oscillation which stretches across the continent

The results are looking good but the tidal equation (LTE) modulation is extremely large

Comment Source:Thanks, I'm next working on the PNA, which is a [Pacific North America oscillation](https://climate.ncsu.edu/climate/patterns/pna) which stretches across the continent ![](https://climate.ncsu.edu/images/climate/enso/PNA_POSITIVE_1981_TEMPS.gif) The results are looking good but the tidal equation (LTE) modulation is extremely large 
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"I wonder if the 3-4x frequency modulation of the IOD relative to ENSO might be related to some sort of "shudder" due to the various chokepoints around the straights of Malacca which iirc were adduced as contributing the slowdown of the increased SST in the last decade?"

Th concept of LTE modulation is somewhat counter-intuitive. It's not really a modulation in frequency but a modulation of amplitude level, which is an indirect frequency multiplier in that it introduces harmonics. So when the multiplier is 3 to 4, harmonics are definitely introduced but I am not certain if these change the frequency by 3-4.

In this case, it appears as if there is a North/South aspect to the IOD definition and that may have an impact on the higher multiplying factor. This would contribute likely a shorter bounding waveguide in that dimension, which would create a tighter standing wave and thus higher frequency standing wave harmonics. The definition states: "Intensity of the IOD is represented by anomalous SST gradient between the western equatorial Indian Ocean (50oE-70oE and 10oS-10oN) and the south eastern equatorial Indian Ocean (90oE-110oE and 10oS-0oN). This gradient is named as Dipole Mode Index (DMI)."

I am thinking along those lines, because the multiplier is also very high for the NAO, and that has a significant North/South aspect to the dipole definition.

In general, what I am characterizing as the LTE multiplier may require a different vocabulary to describe the resultant behavior. Since there is nothing in the research literature that is even close to this solution, there is no lingo or common understanding to draw from.

The PNA also has a large LTE multiplier as I mentioned in the last comment. From Wikipedia: "The positive phase of the PNA pattern features above-average barometric pressure heights in the vicinity of Hawaii and over the inter-mountain region of North America, and below-average heights located south of the Aleutian Islands and over the southeastern United States. The PNA pattern is associated with strong fluctuations in the strength and location of the East Asian jet stream."

This is the tidal forcing for the PNA model and once again it matches that to ENSO

This is the fit on a relatively short training interval, which shows good cross-validation out-of-band. What the strong LTE modulation does is bring the peaks into strong relief

The dipoles that are left to do are the arctic (AO) and antarctic (SAM). I looked at the AO before and that has very high frequency content, to the point it almost looks like white noise. I am not sure what kind of success I will have in characterizing it via the model.

Comment Source:Jim asked: > "I wonder if the 3-4x frequency modulation of the IOD relative to ENSO might be related to some sort of "shudder" due to the various chokepoints around the straights of Malacca which iirc were adduced as contributing the slowdown of the increased SST in the last decade?" Th concept of LTE modulation is somewhat counter-intuitive. It's not really a modulation in frequency but a modulation of amplitude level, which is an indirect frequency multiplier in that it introduces harmonics. So when the multiplier is 3 to 4, harmonics are definitely introduced but I am not certain if these change the frequency by 3-4. In this case, it appears as if there is a North/South aspect to the IOD definition and that may have an impact on the higher multiplying factor. This would contribute likely a shorter bounding waveguide in that dimension, which would create a tighter standing wave and thus higher frequency standing wave harmonics. The definition states: *"Intensity of the IOD is represented by anomalous SST gradient between the western equatorial Indian Ocean (50oE-70oE and 10oS-10oN) and the south eastern equatorial Indian Ocean (90oE-110oE and 10oS-0oN). This gradient is named as Dipole Mode Index (DMI)."* ![iod](http://www.bom.gov.au/climate/influences/images/map-indices.png) I am thinking along those lines, because the multiplier is also very high for the NAO, and that has a significant North/South aspect to the dipole definition. ![nao](https://www.air-worldwide.com/uploadedImages/Blog/europe_winterstorms_fig1.jpg) In general, what I am characterizing as the LTE multiplier may require a different vocabulary to describe the resultant behavior. Since there is nothing in the research literature that is even close to this solution, there is no lingo or common understanding to draw from. --- The PNA also has a large LTE multiplier as I mentioned in the last comment. From Wikipedia: *"The positive phase of the PNA pattern features above-average barometric pressure heights in the vicinity of Hawaii and over the inter-mountain region of North America, and below-average heights located south of the Aleutian Islands and over the southeastern United States. The PNA pattern is associated with strong fluctuations in the strength and location of the East Asian jet stream."* ![LTM](https://imagizer.imageshack.com/img923/2324/S4N6mq.png) This is the tidal forcing for the PNA model and once again it matches that to ENSO ![Forcing](https://imagizer.imageshack.com/img924/4748/ynaTV2.png) This is the fit on a relatively short training interval, which shows good cross-validation out-of-band. What the strong LTE modulation does is bring the peaks into strong relief ![PNA](https://imagizer.imageshack.com/img921/7110/m6zFyT.png) The dipoles that are left to do are the arctic (AO) and antarctic (SAM). I looked at the AO before and that has very high frequency content, to the point it almost looks like white noise. I am not sure what kind of success I will have in characterizing it via the model. 
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What's also interesting is that the PNA (Pacific-North America) and now AO (Arctic Oscillation) can be easily fit from a perturbation of the NAO model. These are all northern latitude behaviors as highlighted in orange below.

This is the common tidal forcing for each of these models, with the LTE modulation in the lower panel. The tidal forcing has a strong semi-annual factor, as with the QBO.

The LTE modulation differs subtly between the three, as the multipliers are slightly different for NAO and AO and within ~15% for PNA. They are in sync at the yellow arrows shown in the lower panel. The LTE modulation is dependent on the fundamental spatial wavenumber defining the dipole, which should be different for each of the regions.

These are the fits for each of the time-series

You can see how the NAO and AO are vaguely similar and the the PNA is similar but flipped in polarity. It is known that the QBO has a connection to the polar vortex, so the semi-annual commonality between QBO and AO makes some sense.

The only major index left is the Southern Annular Mode (SAM) index associated with the Antarctic Oscillation.

Comment Source:What's also interesting is that the PNA (Pacific-North America) and now [AO (Arctic Oscillation)](https://www.ncdc.noaa.gov/teleconnections/ao/) can be easily fit from a perturbation of the NAO model. These are all northern latitude behaviors as highlighted in orange below. ![map](https://imagizer.imageshack.com/img922/285/4ibPDd.gif) This is the common tidal forcing for each of these models, with the LTE modulation in the lower panel. The tidal forcing has a strong semi-annual factor, as with the QBO. ![forcing](https://imagizer.imageshack.com/img921/9856/JRPwqM.png) The LTE modulation differs subtly between the three, as the multipliers are slightly different for NAO and AO and within ~15% for PNA. They are in sync at the yellow arrows shown in the lower panel. The LTE modulation is dependent on the fundamental spatial wavenumber defining the dipole, which should be different for each of the regions. These are the fits for each of the time-series ![fit](https://imagizer.imageshack.com/img923/9754/xFg3hd.png) You can see how the NAO and AO are vaguely similar and the the PNA is similar but flipped in polarity. It is known that the QBO has a connection to the polar vortex, so the semi-annual commonality between QBO and AO makes some sense. The only major index left is the Southern Annular Mode (SAM) index associated with the [Antarctic Oscillation](https://en.wikipedia.org/wiki/Antarctic_oscillation). 
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edited August 2019

The SAM index (data source) is the last to do on the map

Based of the complexity of these waveforms, this should have taken a long time to adequately fit a model if starting from scratch. Yet, since the tidal forcing is nearly identical for each, the computation took no time at all.

The LTE modulation was close to that of the complementary AO, as it retains the same phase over a greater range of forcing levels (indicated by the yellow arrow):

The fit is very good (Fourier spectrum comparison in lower panel)

As a bottom-line, these climate indices are likely not related as teleconnections (which is the current consensus idea), but more likely by a common-mode forcing . The set is synchronized by the common lunisolar tidal forces operating across the earth and individually distinguished by the standing wave constraints of each region.

Moreover, it's highly unlikely that the quality of these model fits is due to overfitting as there are very few DOF available given the common-mode forcing constraint shared by each model.

Comment Source:The [SAM](https://en.wikipedia.org/wiki/Antarctic_oscillation) index ([data source](https://climatedataguide.ucar.edu/climate-data/marshall-southern-annular-mode-sam-index-station-based)) is the last to do on the map ![map](https://imagizer.imageshack.com/img923/774/e15Iic.gif) Based of the complexity of these waveforms, this should have taken a long time to adequately fit a model if starting from scratch. Yet, since the tidal forcing is nearly identical for each, the computation took no time at all. ![forcing](https://imagizer.imageshack.com/img924/2990/7IYddU.png) The LTE modulation was close to that of the complementary AO, as it retains the same phase over a greater range of forcing levels (indicated by the yellow arrow): ![lte](https://imagizer.imageshack.com/img924/43/dCwZBX.png) The fit is very good (Fourier spectrum comparison in lower panel) ![model](https://imagizer.imageshack.com/img923/9608/hvwZQj.png) As a bottom-line, these climate indices are likely not related as teleconnections (which is the current consensus idea), but more likely by a common-mode forcing . The set is synchronized by the common lunisolar tidal forces operating across the earth and individually distinguished by the standing wave constraints of each region. Moreover, it's highly unlikely that the quality of these model fits is due to overfitting as there are very few DOF available given the common-mode forcing constraint shared by each model. 
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Just wow! Heading for the denoument I'm looking forward to trying to explain this in a suitable pop version for my ecomath mates. :)

Comment Source:Just wow! Heading for the denoument I'm looking forward to trying to explain this in a suitable pop version for my ecomath mates. :) 
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edited August 2019

Jim, Appreciate the interest as always.

I'm trying to relate the LTE modulation factor to something akin to a Reynolds number (Re) or a Richardson number (Ri), which makes it a single scalar that describes the breaking or folding of the waves (like a turbulence factor but not chaotic) and to the primary wavenumber.

Right now the trend of the LTE value is closer to zero if the climate index is measured close to the equator (QBO is the lowest) and it tends to increase as the index moves away from the equator. The ordering is about like this:

QBO < ENSO < (AMO ~ IOD) < PDO < ( NAO ~ AO ~ SAM ~ PNA)

The wavenumber of QBO approaches zero because the standing wave encircles the equator and cycles in unison. Correspondingly the wavenumber values may be required to increase away from the equator -- which is forced to be smaller closer to the poles -- but it also may be due to the specific waveguide bounding box of the index. For example, the equatorial Pacific is widest and thus ENSO has the lowest primary wavenumber next to QBO.

Added for entertainment. This is a typical YouTube search for "fluid motion in glycerine"

https://youtu.be/Yy0-1nWVgls

Notice how the fluid flow is complete reversible in the sense that all the dispersion observed "undisperses" on reversing direction. This is a consequence of the low Reynolds number limit -- via the highly viscous glycerine media -- of the Navier-Stokes equation.

Comment Source:Jim, Appreciate the interest as always. I'm trying to relate the LTE modulation factor to something akin to a Reynolds number (Re) or a Richardson number (Ri), which makes it a single scalar that describes the breaking or folding of the waves (like a turbulence factor but not chaotic) and to the primary wavenumber. Right now the trend of the LTE value is closer to zero if the climate index is measured close to the equator (QBO is the lowest) and it tends to increase as the index moves away from the equator. The ordering is about like this: QBO < ENSO < (AMO ~ IOD) < PDO < ( NAO ~ AO ~ SAM ~ PNA) The wavenumber of QBO approaches zero because the standing wave encircles the equator and cycles in unison. Correspondingly the wavenumber values may be required to increase away from the equator -- which is forced to be smaller closer to the poles -- but it also may be due to the specific waveguide bounding box of the index. For example, the equatorial Pacific is widest and thus ENSO has the lowest primary wavenumber next to QBO. --- Added for entertainment. This is a typical YouTube search for "fluid motion in glycerine" https://youtu.be/Yy0-1nWVgls Notice how the fluid flow is complete reversible in the sense that all the dispersion observed "undisperses" on reversing direction. This is a consequence of the low Reynolds number limit -- via the highly viscous glycerine media -- of the Navier-Stokes equation. 
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324.

What's interesting about the common tidal forcing of (AO, NAO, PNA, SAM) is that there is a distinct visible period in the time-series which is the lunar tropical month (27.321582 days) aliased against the annual signal.

1/(365.242/(27.321582)-13) = 2.72 years

For the QBO, the forcing and response are very close to each other (due to the low LTE factor) and the tidal forcing is the lunar draconic month (27.2122 days) aliased against the annual signal. This gives the measured QBO periodicity of:

1/(365.242/(27.2122)-13) = 2.37 years

There are many papers suggesting that there is a connection between QBO and polar behavior, but it is not always there. The wavenumber 0 symmetry of the QBO precludes any tropical (synodic) dependence so the cycle is draconic while the the other indices require a tropical dependence as they are geospatially specific. The two distinct cycles will go in and out of sync gradually with an 18.6 year cycle.

One such paper from earlier this year claiming the teleconnection: Observed and Simulated Teleconnections Between the Stratospheric Quasi‐Biennial Oscillation and Northern Hemisphere Winter Atmospheric Circulation

Comment Source:What's interesting about the common tidal forcing of (AO, NAO, PNA, SAM) is that there is a distinct visible period in the time-series which is the lunar *tropical* month (27.321582 days) aliased against the annual signal. ![forcing](https://imagizer.imageshack.com/img924/2990/7IYddU.png) 1/(365.242/(27.321582)-13) = 2.72 years For the QBO, the forcing and response are very close to each other (due to the low LTE factor) and the tidal forcing is the lunar *draconic* month (27.2122 days) aliased against the annual signal. This gives the measured QBO periodicity of: 1/(365.242/(27.2122)-13) = 2.37 years There are many papers suggesting that there is a connection between QBO and polar behavior, but it is not always there. The wavenumber 0 symmetry of the QBO precludes any tropical (synodic) dependence so the cycle is draconic while the the other indices require a tropical dependence as they are geospatially specific. The two distinct cycles will go in and out of sync gradually with an 18.6 year cycle. One such paper from earlier this year claiming the teleconnection: [Observed and Simulated Teleconnections Between the Stratospheric Quasi‐Biennial Oscillation and Northern Hemisphere Winter Atmospheric Circulation](https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2018JD029368) 
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Comment Source:Interesting reading :) 
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326.

The bizarreness of the solution can perhaps be rationalized when considered in the context of solving equations derived from the full Navier-Stokes. If Navier-Stokes was straightforward, the solution would likely be a familiar formulation, but since N-S is challenging you might imagine that the solution would be odd-looking.

Comment Source:The bizarreness of the solution can perhaps be rationalized when considered in the context of solving equations derived from the full Navier-Stokes. If Navier-Stokes was straightforward, the solution would likely be a familiar formulation, but since N-S is challenging you might imagine that the solution would be odd-looking.
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327.

Jan, Saw your comment attached to the Antarctic Ice sheet melting post over at RC and responded, waiting for it to come out of moderation.

I summarized the findings of common-mode forcing here: https://geoenergymath.com/2019/08/12/ao-pna-and-sam-models/

The RC post is saying that variability in ice sheet melting is linked to ENSO. SAM is located right off Antarctica, and has that connection to ENSO via the common-mode tidal forcing.

Comment Source:Jan, Saw your [comment attached to the Antarctic Ice sheet melting post over at RC](http://www.realclimate.org/index.php/archives/2019/08/the-antarctic-ice-sheet-is-melting-and-yeah-its-probably-our-fault) and responded, waiting for it to come out of moderation. ![](https://imagizer.imageshack.com/img922/9829/0QZBS4.png) I summarized the findings of common-mode forcing here: https://geoenergymath.com/2019/08/12/ao-pna-and-sam-models/ The RC post is saying that variability in ice sheet melting is linked to ENSO. SAM is located right off Antarctica, and has that connection to ENSO via the common-mode tidal forcing.
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328.

Thanks, Paul!

Comment Source:Thanks, Paul!
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329.

There was a paper titled "Numerical Bifurcation Methods applied to Climate Models: Analysis beyond Simulation" which was open review

I asked a question based on an excerpt in the paper:

"All of the results of continuation methods described above were obtained under stationary forcing and for many in the field this seems disjoint from the real climate system, which is obviously forced by a non-stationary insolation component (on diurnal, seasonal and orbital time scales). "

Are tidal forcing factors considered on orbital time scales? According to Munk and Wunsch, tidal factors are a factor in overturning circulation. -- Munk, W. & Wunsch, C. Abyssal recipes II: energetics of tidal and wind mixing. Deep Sea Research Part I: Oceanographic Research Papers 45, 1977–2010 (1998).

Obviously I was trying to provoke the author into addressing what I think are the real drivers of the system, that is the tidal+seasonal forcing.

The author responded:

"Tidal factors are certainly important the maintain the mean state ocean circulation on long time scales, but they are usually not considered when looking at orbital variations, where changes in this mean state are considered. Effectively, they are represented at a high aggregate level by the vertical mixing coefficients in the ocean model component."

This is an inadequate response in that admitting that tidal forcing works on long scales with the knowledge that it also works on short time scales (i.e. ocean tides) misses the obvious intermediate level that ENSO and the other oceanic dipoles work on. From the second sentence, I am assuming any contributions of tides are essentially introduced only by parameterizing the mean-value fluid coefficients so the forcing will never pass through and show up in the output of the model as a response.

ps. Cant comment further on the article as the review period is now closed.

Comment Source:There was a paper titled ["Numerical Bifurcation Methods applied to Climate Models: Analysis beyond Simulation"](https://www.nonlin-processes-geophys-discuss.net/npg-2019-29/#discussion) which was open review I asked a question based on an excerpt in the paper: > "*All of the results of continuation methods described above were obtained under stationary forcing and for many in the field this seems disjoint from the real climate system, which is obviously forced by a non-stationary insolation component (on diurnal, seasonal and orbital time scales).* " >Are tidal forcing factors considered on orbital time scales? According to Munk and Wunsch, tidal factors are a factor in overturning circulation. -- Munk, W. & Wunsch, C. Abyssal recipes II: energetics of tidal and wind mixing. Deep Sea Research Part I: Oceanographic Research Papers 45, 1977–2010 (1998). Obviously I was trying to provoke the author into addressing what I think are the real drivers of the system, that is the tidal+seasonal forcing. The author responded: > "Tidal factors are certainly important the maintain the mean state ocean circulation on long time scales, but they are usually not considered when looking at orbital variations, where changes in this mean state are considered. Effectively, they are represented at a high aggregate level by the vertical mixing coefficients in the ocean model component." This is an inadequate response in that admitting that tidal forcing works on long scales with the knowledge that it also works on short time scales (i.e. ocean tides) misses the obvious intermediate level that ENSO and the other oceanic dipoles work on. From the second sentence, I am assuming any contributions of tides are essentially introduced only by parameterizing the mean-value fluid coefficients so the forcing will never pass through and show up in the output of the model as a response. ps. Cant comment further on the article as the review period is now closed. 
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"Prominent precession‐band variance in ENSO intensity over the last 300,000 years"

"The simulated ENSO and AC amplitudes change in‐phase, and both have pronounced precession‐band variance (~21,000 years). The precession‐modulated slow (orbital time scales) ENSO evolution is dominated linearly by the change of the coupled ocean‐atmosphere instability, notably the Ekman upwelling feedback and thermocline feedback." in Geophysical Research Letters

How can they say this about orbital factors that impact ENSO via the thermocline and upwelling over thousands of years and yet neglect the orbital factors that clearly occur over the monthly cycle?

Part of the reason is that the orbital factors are all solar related, and so they preclude lunar forcing

"First, we want to explore the ENSO response to the orbital forcing that includes full cycles of eccentricity (~100 ka), obliquity (~41 ka) and precession (~21 ka) (Berger and Loutre, 1991), with more extreme precessional forcing effects (modulated by a larger eccentricity compared to the last 21 ka). "

And this is how they can rationalize ignoring the short term scale

It looks as if they simplify that substantially differs from the shallow-water wave equation (i.e. Laplace's tidal equation) ansatz that I apply. They lose track of the non-linear terms and leave it in a form that essentially models a linearly sloshing thermocline see-saw. Moreover, they then conclude that it's too complex due to stratification. Yet if it's too complicated, how can they understand it enough to make extrapolations over a much longer time-scale?

Comment Source:["Prominent precession‐band variance in ENSO intensity over the last 300,000 years"](https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2019GL083410) > "The simulated ENSO and AC amplitudes change in‐phase, and both have pronounced precession‐band variance (~21,000 years). The precession‐modulated slow (orbital time scales) ENSO evolution is dominated linearly by the change of the coupled ocean‐atmosphere instability, notably the Ekman upwelling feedback and thermocline feedback." in **Geophysical Research Letters** How can they say this about orbital factors that impact ENSO via the thermocline and upwelling over thousands of years and yet neglect the orbital factors that clearly occur over the monthly cycle? Part of the reason is that the orbital factors are all solar related, and so they preclude lunar forcing > "First, we want to explore the ENSO response to the orbital forcing that includes full cycles of eccentricity (~100 ka), obliquity (~41 ka) and precession (~21 ka) (Berger and Loutre, 1991), with more extreme precessional forcing effects (modulated by a larger eccentricity compared to the last 21 ka). " And this is how they can rationalize ignoring the short term scale ![](https://imagizer.imageshack.com/img921/7459/yRxe3s.png) It looks as if they simplify that substantially differs from the shallow-water wave equation (i.e. Laplace's tidal equation) ansatz that I apply. They lose track of the non-linear terms and leave it in a form that essentially models a linearly sloshing thermocline see-saw. Moreover, they then conclude that it's too complex due to stratification. Yet if it's too complicated, how can they understand it enough to make extrapolations over a much longer time-scale? 
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edited August 2019

Terry Tao proposes ideas for solving Navier-Stokes. May have to set up a new discussion thread for this -- will edit this if it goes anywhere. https://www.quantamagazine.org/terence-tao-proposes-fluid-new-path-in-navier-stokes-problem-20140224/

"... the Navier-Stokes equations of fluid flow, which physicists use to model ocean currents, weather patterns and other phenomena. "

His latest Navier-Stokes post was last week: https://terrytao.wordpress.com/2019/08/15/quantitative-bounds-for-critically-bounded-solutions-to-the-navier-stokes-equations/

Comment Source:Terry Tao proposes ideas for solving Navier-Stokes. May have to set up a new discussion thread for this -- will edit this if it goes anywhere. https://www.quantamagazine.org/terence-tao-proposes-fluid-new-path-in-navier-stokes-problem-20140224/ > "... the Navier-Stokes equations of fluid flow, which physicists use to model ocean currents, weather patterns and other phenomena. " His latest Navier-Stokes post was last week: https://terrytao.wordpress.com/2019/08/15/quantitative-bounds-for-critically-bounded-solutions-to-the-navier-stokes-equations/ 
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332.

New paper at Journal of Climate : https://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-18-0681.1, Xian Wu, Yuko M. Okumura, and Pedro N. DiNezio, "What Controls the Duration of El Niño and La Niña Events?", 2019.

Comment Source:New paper at _Journal_ _of_ _Climate_ : https://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-18-0681.1, Xian Wu, Yuko M. Okumura, and Pedro N. DiNezio, "What Controls the Duration of El Niño and La Niña Events?", 2019.
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edited April 2020

The inability of forecasters to track hurricanes accurately is fundamentally related to the inability to model tropical behavior such as ENSO

Professor Phil Klotzbach of Colorado State University’s Department of Atmospheric Science told Express.co.uk: “The primary reason for NOAA’s increase in their forecast was due to the weakening of El Niño.

“El Niño is warmer than normal water in the central and eastern tropical Pacific.

“Typically, when you have El Niño conditions, it increases vertical wind shear in the Caribbean into the tropical Atlantic, tearing apart hurricanes.

“With El Niño going away, they anticipated less vertical wind shear and consequently more conducive conditions for hurricanes.”

Consider the simplicity and the symmetry of the forcing model I am using for ENSO. The essential tidal forcing is the sidereal (tropical) and synodic lunar cycles. The combination of this pair of fortnightly cycles leads to a semi-annual symmetry in the time-series. Even though the analysis started with a comprehensive forcing model, just a couple of the main factors provide the basic pattern.

From the amplitude spectrum, one can see how these are the strongest tidal factors, labelled Mf (fortnightly cycle doubled from the monthly sidereal) and Msf (from the synodic), with the Mf+Ssa (solar semi-annual) peak generating the semi-annual beat pattern. (The inset is an earlier model that gave a hint of the beat pattern)

The tricky fluid dynamics aspect is in determining the LTE modulation, both for the major ENSO standing wave and the higher harmonics. The major cycle represents the primary dipole, while the higher frequency noise is actually constructive in that it provides harmonics that shape the peaks in the time series.

Considering the simplicity and low dimensionality of the model, the good fit suggests that it's a very plausible mechanism.

The LTE modulation derives from an assumed wiggle in the equatorial latitude. Imagine that the cyclic forcing impacts the location of the equatorial latitude, pushing it north & south in a deterministic pattern. This is the ansatz that leads to the solution to Laplace's Tidal Equations, and thus providing a means to create a non-linear modulation of the tidal forcing.

Comment Source:The inability of forecasters to track hurricanes accurately is fundamentally related to the inability to model tropical behavior such as ENSO https://theusposts.com/hurricane-tracker-what-causes-deadly-hurricanes-what-is-el-nino/ > Professor Phil Klotzbach of Colorado State University’s Department of Atmospheric Science told Express.co.uk: “The primary reason for NOAA’s increase in their forecast was due to the weakening of El Niño. > “El Niño is warmer than normal water in the central and eastern tropical Pacific. > “Typically, when you have El Niño conditions, it increases vertical wind shear in the Caribbean into the tropical Atlantic, tearing apart hurricanes. > “With El Niño going away, they anticipated less vertical wind shear and consequently more conducive conditions for hurricanes.” --- Consider the simplicity and the symmetry of the forcing model I am using for ENSO. The essential tidal forcing is the sidereal (tropical) and synodic lunar cycles. The combination of this pair of fortnightly cycles leads to a semi-annual symmetry in the time-series. Even though the analysis started with a comprehensive forcing model, just a couple of the main factors provide the basic pattern. ![forcing](https://imagizer.imageshack.com/img923/7954/W6WLAC.png) From the amplitude spectrum, one can see how these are the strongest tidal factors, labelled Mf (fortnightly cycle doubled from the monthly sidereal) and Msf (from the synodic), with the Mf+Ssa (solar semi-annual) peak generating the semi-annual beat pattern. (The inset is an earlier model that gave a hint of the beat pattern) ![forcing_spectrum](https://geoenergymath.files.wordpress.com/2019/02/rkzhjm.png) The tricky fluid dynamics aspect is in determining the LTE modulation, both for the major ENSO standing wave and the higher harmonics. The major cycle represents the primary dipole, while the higher frequency noise is actually constructive in that it provides harmonics that shape the peaks in the time series. ![lte](https://imagizer.imageshack.com/img923/6237/LBYTVq.png) Considering the simplicity and low dimensionality of the model, the good fit suggests that it's a very plausible mechanism. ![fit](https://imagizer.imageshack.com/img921/6009/uejZOH.png) --- The LTE modulation derives from an assumed wiggle in the equatorial latitude. Imagine that the cyclic forcing impacts the location of the equatorial latitude, pushing it north & south in a deterministic pattern. This is the ansatz that leads to the solution to Laplace's Tidal Equations, and thus providing a means to create a non-linear modulation of the tidal forcing. ![](https://imagizer.imageshack.com/img923/3812/9U1Dki.gif) ![](https://imagizer.imageshack.com/img921/748/gJdY7Y.gif) ![](https://imagizer.imageshack.com/img924/2450/TVnE17.gif) ![](https://imagizer.imageshack.com/img924/4990/DGK8Kv.gif) 
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334.

One of the tricky parts of dealing with an uncommon mathematical formulation is in finding appropriate analysis techniques. The Laplace's Tidal Equation solution is essentially a "sine of sine" formulation, which is rare to come across. Yet it is quite important in the realm of Mach-Zehnder modulation (MZM), which is applied in interferometry-based measurement applications and in fiber optic communication devices.

This is from a book called Fiber Optics: Physics and Technology by Mitschke, where the sine of sine modulation is defined and often applied as a means to finely control the phase of a signal.

In a specific optical application, MZM can be used to encrypt comms such that enough phase shifts are applied as to make the signal nearly impossible to decode. This paper describes a method operating in the Fresnel domain where $$2 \pi$$ phase shifts are the norm.

https://www.ncbi.nlm.nih.gov/pubmed/24663801

Abstract: A method for optical image hiding and for optical image encryption and hiding in the Fresnel domain via completely optical means is proposed, which encodes original object image into the encrypted image and then embeds it into host image in our modified Mach-Zehnder interferometer architecture. The modified Mach-Zehnder interferometer not only provides phase shifts to record complex amplitude of final encrypted object image on CCD plane but also introduces host image into reference path of the interferometer to hide it. The final encrypted object image is registered as interference patterns, which resemble a Fresnel diffraction pattern of the host image, and thus the secure information is imperceptible to unauthorized receivers. The method can simultaneously realize image encryption and image hiding at a high speed in pure optical system. The validity of the method and its robustness against some common attacks are investigated by numerical simulations and experiments.

So this is essentially what we are up against when trying to invert an LTE modulation that follows a $$sin(A sin(t))$$ behavior. If the value of $$A$$ is small, there's no issue with doing a Taylor's series expansion, but if $$A$$ gets too large, we have no way of extracting the number of $$2 \pi$$ phase shifts that are occurring, particularly if the internal $$sin(t)$$ is an unknown Fourier series representation of the embedded signal. That's part of the reason it works effectively as an encryption approach, as only the sender and receiver know the key = number of phase shifts involved.

The best I have been able to do so far with an $$arcsin()$$ inversion is to start with a candidate solution for LTE (e.g. a tropical lunar sinewave mixed with an annual impulse) and then estimate the $$2 \pi$$ phase shifts required to match the ENSO signal. This works fairly effectively but suffers from a biased coupling so as to render any correlation coefficient meaningless. The approach is essentially working from both ends of the following figure towards the middle.

The other aspect of MZM that I need to ponder is whether there is some commonality in the topological physics between the LTE formulation for constrained fluid dynamics and the wave modulation created via a MZ device. They are both waves, but there is likely something more fundamental involving how the "modulation of a modulation" arises.

I have been toying with the idea that it may have something to do with the physics of topological insulators from the cite below. This group is already trying to understand equatorial indices such as ENSO by analogizing from the fractional quantum Hall effect world, so perhaps there is a happy medium.

Topological Origin of Equatorial Waves : Pierre Delplace, J. B. Marston, Antoine Venaille

"The first Chern number is an integer that quantifies the number of phase singularities in a bundle of eigenmodes parameterized on a closed manifold. These singularities are somewhat analogous to amphidromic points (±2π phase vortices of tidal modes), but they occur in parameter space rather than in physical space"

Famous last words "defined only up to a phase"

Comment Source:One of the tricky parts of dealing with an uncommon mathematical formulation is in finding appropriate analysis techniques. The Laplace's Tidal Equation solution is essentially a "sine of sine" formulation, which is rare to come across. Yet it is quite important in the realm of [Mach-Zehnder modulation](https://en.wikipedia.org/wiki/Electro-optic_modulator) (MZM), which is applied in interferometry-based measurement applications and in fiber optic communication devices. This is from a book called Fiber Optics: Physics and Technology by Mitschke, where the sine of sine modulation is defined and often applied as a means to finely control the phase of a signal. ![sinofsin](https://imagizer.imageshack.com/img922/5844/tHuQUv.png) In a specific optical application, MZM can be used to encrypt comms such that enough phase shifts are applied as to make the signal nearly impossible to decode. This paper describes a method operating in the Fresnel domain where \$$2 \pi\$$ phase shifts are the norm. https://www.ncbi.nlm.nih.gov/pubmed/24663801 > Abstract: A method for optical image hiding and for optical image encryption and hiding in the Fresnel domain via completely optical means is proposed, which encodes original object image into the encrypted image and then embeds it into host image in our modified Mach-Zehnder interferometer architecture. The modified Mach-Zehnder interferometer not only provides phase shifts to record complex amplitude of final encrypted object image on CCD plane but also introduces host image into reference path of the interferometer to hide it. The final encrypted object image is registered as interference patterns, which resemble a Fresnel diffraction pattern of the host image, and thus the secure information is imperceptible to unauthorized receivers. The method can simultaneously realize image encryption and image hiding at a high speed in pure optical system. The validity of the method and its robustness against some common attacks are investigated by numerical simulations and experiments. So this is essentially what we are up against when trying to invert an LTE modulation that follows a \$$sin(A sin(t)) \$$ behavior. If the value of \$$A\$$ is small, there's no issue with doing a Taylor's series expansion, but if \$$A\$$ gets too large, we have no way of extracting the number of \$$2 \pi\$$ phase shifts that are occurring, particularly if the internal \$$sin(t) \$$ is an unknown Fourier series representation of the embedded signal. That's part of the reason it works effectively as an encryption approach, as only the sender and receiver know the key = number of phase shifts involved. The best I have been able to do so far with an \$$arcsin()\$$ inversion is to start with a candidate solution for LTE (e.g. a tropical lunar sinewave mixed with an annual impulse) and then estimate the \$$2 \pi\$$ phase shifts required to match the ENSO signal. This works fairly effectively but suffers from a biased coupling so as to render any correlation coefficient meaningless. The approach is essentially working from both ends of the following figure towards the middle. ![](https://imagizer.imageshack.com/img924/8215/ycXSL8.png) The other aspect of MZM that I need to ponder is whether there is some commonality in the topological physics between the LTE formulation for constrained fluid dynamics and the wave modulation created via a MZ device. They are both waves, but there is likely something more fundamental involving how the "modulation of a modulation" arises. I have been toying with the idea that it may have something to do with the physics of topological insulators from the cite below. This group is already trying to understand equatorial indices such as ENSO by analogizing from the fractional quantum Hall effect world, so perhaps there is a happy medium. [Topological Origin of Equatorial Waves : Pierre Delplace, J. B. Marston, Antoine Venaille](https://arxiv.org/pdf/1702.07583.pdf) > "The first Chern number is an integer that quantifies the number of phase singularities in a bundle of eigenmodes parameterized on a closed manifold. These singularities are somewhat analogous to amphidromic points (±2π phase vortices of tidal modes), but they occur in parameter space rather than in physical space" Famous last words *"defined only up to a phase"* 
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335.

# #ClimateStrike

This was apparently about a year ago

I am sure this forum is receiving the same interest several years in

Comment Source:#\#ClimateStrike This was apparently about a year ago ![](https://pbs.twimg.com/media/EE75By8XsAA4VTX.jpg) I am sure this forum is receiving the same interest several years in
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336.
edited September 2019

Why I didn't do this bit of signal processing earlier, I don't know.

Take the Fourier spectrum of the ENSO time-series and shift and reverse the order of frequencies between 0.5 to 1/year and by 0.5/year (i.e. fold it back) to quantify the underlying forcing symmetry.

The correlation coefficient is ~0.6 which is impossible to attain with a stochastic auto-regressive spectrum.

Compare this to the model tidal forcing where the spectrum is not folded as above but the symmetry centered at 0.5 /year (and 1.5 /year) is clearly apparent.

The folded cross-correlation of this ideal model forcing is > 0.9, so likely the LTE modulation is responsible for the multiple splitting of the peaks in the ENSO forcing response, thus degrading the symmetry to some degree.

Analysts may overlook this kind of approach because they will simply take the spectrum, not see any annual spectral peak, and so just assume that an annual signal doesn't exist. Yet, taking an annual impulse and convolving that with any kind of sinusoidal waveform with a mean of zero will create only satellite peaks in the response, with the annual spectral peak completely missing. Then with any modulation, such as via the LTE tidal response, the spectral response gets reshaped -- much like the Mach-Zehnder modulation creates a "comb" response (from the post from last week).

The insight is that LTE and MZM is essentially the same analysis. After the MZ modulation is applied to the input signal, it combs out as in the rightmost column below. The comb is essentially a large number of precisely spaced spectral lines that are comprised of harmonics of the original signal.

"Characterization of dual-electrode Mach-Zehnder modulator based optical frequency comb generator in two regimes"

The reason that the ENSO is difficult to decode is that we are only guessing what the LTE modulation is. In contrast, with Mach-Zehnder, that is part of the device design and one calibrates the modulation beforehand. As I said earlier, this is why MZM is hard to decode when used for crypto applications. With a strong modulation, the time domain trace is shown below, creating erratic harmonics of the fundamental frequency.

Consider that there may be a number of varying input signals and if the comb is in 2D, the output will obviously become quite scrambled, making it impossible to decode without knowing the MTM key.

But here is the next mathematical idea. The MZM modulation is typically expanded as a Taylor's series with the coefficients of the harmonics given by a Bessel function, see this

https://en.wikipedia.org/wiki/Electro-optic_modulator#Phase_modulation

So with enough data points and enough coefficients in this so-called Jacobi-Anger expansion, we should be able to do a multiple linear regression to optimize for a best fit of the amplitude (z below) and phase (weighting of sin vs cos below) of the modulation (or more than a single modulation if that works) https://en.wikipedia.org/wiki/Jacobi–Anger_expansion

The reason that this may be possible without iteration and just by applying a regression algorithm is that the input frequencies (inner terms above) will be restricted to the primary tidal periods, each one of these will have an amplitude and phase. Since we will have ~1700 data points over 140 years worth of monthly ENSO data, there will be enough spectral resolution between 0 an 0.5 /year frequency to do a decent regression with a couple dozen parameters -- two modulation and perhaps 10 tidal cycles (amplitude + phase). The tricky part is the Bessel function on z which is non-linear, so perhaps these will need to be expanded and another regression applied.

Comment Source:Why I didn't do this bit of signal processing earlier, I don't know. Take the Fourier spectrum of the ENSO time-series and shift and reverse the order of frequencies between 0.5 to 1/year and by 0.5/year (i.e. fold it back) to quantify the underlying forcing symmetry. ![](https://imagizer.imageshack.com/img922/2671/YmyFPN.png) The correlation coefficient is ~0.6 which is impossible to attain with a stochastic auto-regressive spectrum. Compare this to the model tidal forcing where the spectrum is not folded as above but the symmetry centered at 0.5 /year (and 1.5 /year) is clearly apparent. ![](https://imagizer.imageshack.com/img923/559/8zIrn1.png) The folded cross-correlation of this ideal model forcing is > 0.9, so likely the LTE modulation is responsible for the multiple splitting of the peaks in the ENSO forcing response, thus degrading the symmetry to some degree. Analysts may overlook this kind of approach because they will simply take the spectrum, not see any annual spectral peak, and so just assume that an annual signal doesn't exist. Yet, taking an annual impulse and **convolving** that with any kind of sinusoidal waveform with a mean of zero will create **only** satellite peaks in the response, with the annual spectral peak completely missing. Then with any modulation, such as via the LTE tidal response, the spectral response gets reshaped -- much like the Mach-Zehnder modulation creates a "comb" response (from the post from last week). The insight is that LTE and MZM is essentially the same analysis. After the MZ modulation is applied to the input signal, it combs out as in the rightmost column below. The comb is essentially a large number of precisely spaced spectral lines that are comprised of harmonics of the original signal. ["Characterization of dual-electrode Mach-Zehnder modulator based optical frequency comb generator in two regimes" ](https://www.semanticscholar.org/paper/Characterization-of-dual-electrode-Mach-Zehnder-in-Fontaine-Scott/640e962d202e714aa225887ad33fb45d7f6d77a4) ![](https://imagizer.imageshack.com/img922/7568/IdxxmI.gif) The reason that the ENSO is difficult to decode is that we are only guessing what the LTE modulation is. In contrast, with Mach-Zehnder, that is part of the device design and one calibrates the modulation beforehand. As I said earlier, this is why MZM is hard to decode when used for crypto applications. With a strong modulation, the time domain trace is shown below, creating erratic harmonics of the fundamental frequency. ![](https://imagizer.imageshack.com/img923/3179/cxabdA.gif) Consider that there may be a number of varying input signals and if the comb is in 2D, the output will obviously become quite scrambled, making it impossible to decode without knowing the MTM key. But here is the next mathematical idea. The MZM modulation is typically expanded as a Taylor's series with the coefficients of the harmonics given by a Bessel function, see this https://en.wikipedia.org/wiki/Electro-optic_modulator#Phase_modulation <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7ff9546c46824262b3d20397e9c50112d2b5a3b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:72.43ex; height:7.509ex;" alt="{\displaystyle Ae^{i\omega t+i\beta \sin(\Omega t)}=Ae^{i\omega t}\left(J_{0}(\beta )+\sum _{k=1}^{\infty }J_{k}(\beta )e^{ik\Omega t}+\sum _{k=1}^{\infty }(-1)^{k}J_{k}(\beta )e^{-ik\Omega t}\right),}"> So with enough data points and enough coefficients in this so-called Jacobi-Anger expansion, we should be able to do a multiple linear regression to optimize for a best fit of the amplitude (z below) and phase (weighting of sin vs cos below) of the modulation (or more than a single modulation if that works) https://en.wikipedia.org/wiki/Jacobi%E2%80%93Anger_expansion <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/479f84bed8d8c7faf30d65a780f65362e242bf92" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -13.505ex; width:51.159ex; height:28.176ex;" alt="{\displaystyle {\begin{aligned}\cos(z\cos \theta )&amp;\equiv J_{0}(z)+2\sum _{n=1}^{\infty }(-1)^{n}J_{2n}(z)\cos(2n\theta ),\\\sin(z\cos \theta )&amp;\equiv -2\sum _{n=1}^{\infty }(-1)^{n}J_{2n-1}(z)\cos \left[\left(2n-1\right)\theta \right],\\\cos(z\sin \theta )&amp;\equiv J_{0}(z)+2\sum _{n=1}^{\infty }J_{2n}(z)\cos(2n\theta ),\\\sin(z\sin \theta )&amp;\equiv 2\sum _{n=1}^{\infty }J_{2n-1}(z)\sin \left[\left(2n-1\right)\theta \right].\end{aligned}}}"> The reason that this may be possible without iteration and just by applying a regression algorithm is that the input frequencies (inner terms above) will be restricted to the primary tidal periods, each one of these will have an amplitude and phase. Since we will have ~1700 data points over 140 years worth of monthly ENSO data, there will be enough spectral resolution between 0 an 0.5 /year frequency to do a decent regression with a couple dozen parameters -- two modulation and perhaps 10 tidal cycles (amplitude + phase). The tricky part is the Bessel function on z which is non-linear, so perhaps these will need to be expanded and another regression applied. 
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337.

More on deep learning for multi-year ENSO forecasts: https://www.nature.com/articles/s41586-019-1559-7

Comment Source:More on deep learning for multi-year ENSO forecasts: https://www.nature.com/articles/s41586-019-1559-7
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338.

Saw that deep-learning paper, thanks. As far as I can tell, they haven't been able to reverse engineer any of the connections in their model yet. So it may work but they don't understand why. If you have something I missed that would be interesting to hear about.

It points out how connections are developed. For example, in the finding I made in the previous comment, the neural net would need to have picked out correlations in frequency space by swapping out various frequencies and doing comparisons. So, in this case, the net would need to have been designed to accommodate that.

Comment Source:Saw that deep-learning paper, thanks. As far as I can tell, they haven't been able to reverse engineer any of the connections in their model yet. So it may work but they don't understand why. If you have something I missed that would be interesting to hear about. It points out how connections are developed. For example, in the finding I made <a href="#Comment_336">in the previous comment</a>, the neural net would need to have picked out correlations in frequency space by swapping out various frequencies and doing comparisons. So, in this case, the net would need to have been designed to accommodate that. 
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339.

A few comments up I said:

"The folded cross-correlation of this ideal model forcing is > 0.9, so likely the LTE modulation is responsible for the multiple splitting of the peaks in the ENSO forcing response, thus degrading the symmetry to some degree."

I initially thought the Laplace's Tidal Equation (LTE) modulation may remove the spectral symmetry around the 0.5/year position, but this is definitely not the case. The symmetry is perfectly preserved independent of the amount of modulation, with more satellite peaks added the greater the modulation. This is easy to understand as only harmonics of the no-LTE fundamental forcing frequency are generated and these will preserve the +/- satellite symmetry, no matter how strong the modulation.

This brings up an interesting possibility in terms of classical tidal analysis. Early on it was realized that there were quite a few tidal constituents to deal with:

"Darwin's harmonic developments of the tide-generating forces were later improved when A T Doodson, applying the lunar theory of E W Brown, developed the tide-generating potential (TGP) in harmonic form, distinguishing 388 tidal frequencies. Doodson's work was carried out and published in 1921."

Even though there are even more than that now, only a few are needed for short-term forecasts. Yet, the reason that so many exist is because these are all generated by combinations of all the possible products of the constituent factors. The interesting finding is that the LTE modulation will also generate all the harmonic combinations, which may explain how the harmonics come about. The LTE modulation is likely very weak for conventional tides, but if it is there at all, it will generate all the tidal harmonics tabulated.

For 4 primary tidal constituents (solar, synodic, draconic, anomalistic) and a 4 harmonic depth on each (i.e. monthly, fortnightly, 9 day, 6.5 day for lunar, and 1 year, 1/2 year, 1/3 year, etc for solar) this will generate ~4^4 = 256 additional harmonics.

Comment Source:A few comments up [I said](#Comment_21195): > "The folded cross-correlation of this ideal model forcing is > 0.9, so likely the LTE modulation is responsible for the multiple splitting of the peaks in the ENSO forcing response, thus degrading the symmetry to some degree." I initially thought the Laplace's Tidal Equation (LTE) modulation may remove the spectral symmetry around the 0.5/year position, but this is definitely not the case. The symmetry is perfectly preserved independent of the amount of modulation, with more satellite peaks added the greater the modulation. This is easy to understand as only harmonics of the no-LTE fundamental forcing frequency are generated and these will preserve the +/- satellite symmetry, no matter how strong the modulation. This brings up an interesting possibility in terms of classical tidal analysis. Early on it was realized that there were quite a few tidal constituents to deal with: > "Darwin's harmonic developments of the tide-generating forces were later improved when A T Doodson, applying the lunar theory of E W Brown, developed the tide-generating potential (TGP) in harmonic form, distinguishing **388** tidal frequencies. Doodson's work was carried out and published in 1921." Even though there are even more than that now, only a few are needed for short-term forecasts. Yet, the reason that so many exist is because these are all generated by combinations of all the possible products of the constituent factors. The interesting finding is that the LTE modulation will also generate all the harmonic combinations, which may explain how the harmonics come about. The LTE modulation is likely very weak for conventional tides, but if it is there at all, it will generate all the tidal harmonics tabulated. For 4 primary tidal constituents (solar, synodic, draconic, anomalistic) and a 4 harmonic depth on each (i.e. monthly, fortnightly, 9 day, 6.5 day for lunar, and 1 year, 1/2 year, 1/3 year, etc for solar) this will generate ~4^4 = 256 additional harmonics. 
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340.
Comment Source:Tibetan Singing Bowl https://youtu.be/oob8zENYt0g The orbiting perturbation causes wild fluctuations https://www.iop.org/news/11/july/page_51352.html https://iopscience.iop.org/article/10.1088/0951-7715/24/8/R01 ![](https://imagizer.imageshack.com/img924/7735/QDk7RF.gif) 
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341.

very interesting

Comment Source:very interesting
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342.

This book on the Physics of Waves by H. Georgi (online PDF) provides some insight.

Incompressibility is an aspect of the ansatz I am using in generating the solution to Laplace's Tidal Equations (LTE) along the equator. If you think of the behavior as a liquid string that can't compress but can wiggle along its equatorial track, then this provides the closure necessary for the solution.

The nature of a compressible liquid is that a displacement in one direction must be compensated by a displacement in another direction. This could be just a speed up in flow a la Bernoulli's principle.

https://youtu.be/UJ3-Zm1wbIQ

So the ansatz that I used in the following derivation is to apply a bending wiggle to the latitudinal displacement of the equatorial wave. This conserves matter by slightly displacing the position of the axis, pulled by gravitational/tidal forces -- at the expense of what would have to be a compensating change in the speed of the fluid.

Without this ansatz in place, the latitudinal and longitudinal behaviors of LTE are uncoupled, yet we know they are physically coupled, otherwise the topological feature of the ENSO equatorial wave would not exist.

Comment Source:This book on the Physics of Waves by H. Georgi *([online PDF](http://www.people.fas.harvard.edu/~hgeorgi/onenew.pdf))* provides some insight. ![](https://imagizer.imageshack.com/img923/528/DyLLf4.gif) Incompressibility is an aspect of the ansatz I am using in generating the solution to Laplace's Tidal Equations (LTE) along the equator. If you think of the behavior as a liquid string that can't compress but can wiggle along its equatorial track, then this provides the closure necessary for the solution. ![](https://imagizer.imageshack.com/img922/8370/IKPfCA.gif) ![](https://imagizer.imageshack.com/img923/4687/aitHVq.gif) The nature of a compressible liquid is that a displacement in one direction must be compensated by a displacement in another direction. This could be just a speed up in flow a la Bernoulli's principle. https://youtu.be/UJ3-Zm1wbIQ So the ansatz that I used in the following derivation is to apply a bending wiggle to the latitudinal displacement of the equatorial wave. This conserves matter by slightly displacing the position of the axis, pulled by gravitational/tidal forces -- at the expense of what would have to be a compensating change in the speed of the fluid. ![](https://imagizer.imageshack.com/img922/2638/PCuY7c.gif) Without this ansatz in place, the latitudinal and longitudinal behaviors of LTE are uncoupled, yet we know they are physically coupled, otherwise the topological feature of the ENSO equatorial wave would not exist. 
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343.
edited October 2019

A recent paper that claims that tides have a greater influence than surface wind on the forcing dynamics of ENSO.

Lin, J. & Qian, T. Switch Between El Nino and La Nina is Caused by Subsurface Ocean Waves Likely Driven by Lunar Tidal Forcing. Sci Rep 9, 1–10 (2019). "Here, we demonstrate that the switch between El Nino and La Nina is caused by a subsurface ocean wave propagating from western Pacifc to central and eastern Pacifc and then triggering development of SST anomaly. This is based on analysis of all ENSO events in the past 136 years using multiple long-term observational datasets. The wave’s slow phase speed and decoupling from atmosphere indicate that it is a forced wave. Further analysis of Earth’s angular momentum budget and NASA’s Apollo Landing Mirror Experiment suggests that the subsurface wave is likely driven by lunar tidal gravitational force."

This is essentially the same thing I have been suggesting -- it just doesn't make sense that a surface wind can drive the sub-surface water sloshing at the thermocline. Take a look at this post from a few years ago from NOAA. They show the following chart of deeper water temperature anomaly (300 feet deep!) while saying with a straight face that surface wind is causing it.

Slow slosh of warm water across Pacific hints El Niño is brewing "As the warm surface water is pushed westward by the prevailing winds, cool water from deeper in the ocean rises to the surface near South America. This temperature gradient—warm waters around Indonesia and cooler waters off South America—lasts only as long as the easterly winds are blowing."

Give credit to Lin & Qian for pointing out how implausible it is to suggest that wind alone can push water around to that extent. Obviously, tidal forces are the one agent that can influence the reduce gravity imbalance on the deeper thermocline (as hypothesized long ago by the legendary physical oceanographers Munk and Wunsch). Yet no real response from anyone else on a paper that directly contradicts the research consensus.

 Just to indicate how conflicting the research is on the topic of atmosphere versus ocean causality, there's this paper

"The ability to anticipate changes to the ocean or atmosphere based on information from the other system provides society with the opportunity to prepare for future impacts, such as to agriculture and fisheries," said Wills. "This is a very important paper in the history of predictability research," said Shukla, "It will surely inspire further research by the predictability research community. In particular, this paper identifies geographical regions on the globe over which there exists potential predictability which can be harvested for improving operational predictions."

Comment Source:A recent paper that claims that tides have a greater influence than surface wind on the forcing dynamics of ENSO. > Lin, J. & Qian, T. [Switch Between El Nino and La Nina is Caused by Subsurface Ocean Waves Likely Driven by Lunar Tidal Forcing](https://www.nature.com/articles/s41598-019-49678-w.pdf). Sci Rep 9, 1–10 (2019). > "Here, we demonstrate that the switch between El Nino and La Nina is caused by a subsurface ocean wave propagating from western Pacifc to central and eastern Pacifc and then triggering development of SST anomaly. This is based on analysis of all ENSO events in the past 136 years using multiple long-term observational datasets. The wave’s slow phase speed and decoupling from atmosphere indicate that it is a forced wave. Further analysis of Earth’s angular momentum budget and NASA’s Apollo Landing Mirror Experiment suggests that the subsurface wave is likely driven by lunar tidal gravitational force." This is essentially the same thing I have been suggesting -- it just doesn't make sense that a surface wind can drive the sub-surface water sloshing at the thermocline. Take a look at this post from a few years ago from NOAA. They show the following chart of deeper water temperature anomaly (300 feet deep!) while saying with a straight face that surface wind is causing it. > [Slow slosh of warm water across Pacific hints El Niño is brewing](https://www.climate.gov/news-features/featured-images/slow-slosh-warm-water-across-pacific-hints-el-ni%C3%B1o-brewing) > "As the warm surface water is pushed westward by the prevailing winds, cool water from deeper in the ocean rises to the surface near South America. This temperature gradient—warm waters around Indonesia and cooler waters off South America—lasts only as long as the easterly winds are blowing." > ![](https://pbs.twimg.com/media/BmFyl9-CAAAGLJ4.jpg) Give credit to Lin & Qian for pointing out how implausible it is to suggest that wind alone can push water around to that extent. Obviously, tidal forces are the one agent that can influence the reduce gravity imbalance on the deeper thermocline (as hypothesized long ago by the legendary physical oceanographers [Munk and Wunsch](https://www.sciencedirect.com/science/article/pii/S0967063798000703)). Yet no real response from anyone else on a paper that directly contradicts the research consensus. ---  Just to indicate how conflicting the research is on the topic of atmosphere versus ocean causality, there's [this paper](https://phys.org/news/2019-10-method-global-picture-mutual-atmosphere.html) >"The ability to anticipate changes to the ocean or atmosphere based on information from the other system provides society with the opportunity to prepare for future impacts, such as to agriculture and fisheries," said Wills. >"This is a very important paper in the history of predictability research," said Shukla, "It will surely inspire further research by the predictability research community. In particular, this paper identifies geographical regions on the globe over which there exists potential predictability which can be harvested for improving operational predictions." 
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344.

One way to prevent overfitting of a model is to use parameters that are fixed by an independent measurement. In the case of forcing for ENSO, and according to the LTE model, this is all ultimately related to angular momentum changes acting on the fluid. The angular momentum changes are reflected in measurements of the dLOD of the earth's length of day anomaly. The same lunisolar gravitational force that slows and speeds up the earth's rotation rate also impacts the sloshing of the ocean.

This is a model of ENSO with the top panel fitted according to the dLOD in the scans below

To extrapolate the LOD data before 1962 (which is when the high-resolution measurements started), the tidal pattern was decoded as a Fourier series.

This allowed the LOD data to be extended to the years before 1962.

Comment Source:One way to prevent overfitting of a model is to use parameters that are fixed by an independent measurement. In the case of forcing for ENSO, and according to the LTE model, this is all ultimately related to angular momentum changes acting on the fluid. The angular momentum changes are reflected in measurements of the dLOD of the earth's length of day anomaly. The same lunisolar gravitational force that slows and speeds up the earth's rotation rate also impacts the sloshing of the ocean. This is a model of ENSO with the top panel fitted according to the dLOD in the scans below ![](https://imagizer.imageshack.com/img921/1227/yTjzaJ.png) To extrapolate the LOD data before 1962 (which is when the high-resolution measurements started), the tidal pattern was decoded as a Fourier series. ![ft](https://imagizer.imageshack.com/img924/3396/oT68Vm.png) This allowed the LOD data to be extended to the years before 1962. 
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345.

An experiment with visual math/symbology

I still think it will take a lot of effort to make the model prediction-ready. It occurred to me that the annual impulse is equivalent to a day's worth of tidal information, so the equivalent of a tidal forecast based on 100 days worth of tidal gauge info requires 100 years of ENSO data.

Sean Carroll's 30-second take on predictions: https://youtu.be/Jqu7McECcgk

Comment Source:An experiment with visual math/symbology ![](https://imagizer.imageshack.com/img924/382/xHPJon.png) I still think it will take a lot of effort to make the model prediction-ready. It occurred to me that the annual impulse is equivalent to a day's worth of tidal information, so the equivalent of a tidal forecast based on 100 days worth of tidal gauge info requires 100 years of ENSO data. Sean Carroll's 30-second take on predictions: https://youtu.be/Jqu7McECcgk 
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346.

Haven't read it since I do not have a subscription.

"Ocean dynamics shapes the structure and timing of Atlantic Equatorial Modes" Marta Martín‐Rey Irene Polo Belén Rodríguez‐Fonseca Alban Lazar Teresa Losada

Abstract

A recent study has brought to light the co‐existence of two distinct Atlantic Equatorial Modes during negative phases of the Atlantic Multidecadal Variability: the Atlantic Niño and Horse‐Shoe (HS) mode. Nevertheless, the associated air‐sea interactions for HS mode have not been explored so far and the prevailing dynamic view of the Atlantic Niño has been questioned. Here, using forced ocean model simulations, we find that for both modes, ocean dynamics is essential to explain the equatorial SST variations, while air‐sea fluxes control the off‐equatorial SST anomalies. Moreover, we demonstrate the key role played by ocean waves in shaping their distinct structure and timing. For the positive phase of both Atlantic Niño and HS, anomalous westerly winds trigger a set of equatorial downwelling Kelvin waves (KW) during spring‐summer. These dKWs deepen the thermocline, favouring the equatorial warming through vertical diffusion and horizontal advection. Remarkably, for the HS, an anomalous north‐equatorial wind stress curl excites an upwelling Rossby wave (RW), which propagates westward and is reflected at the western boundary becoming an equatorial upwelling KW. The uKW propagates to the east, activating the thermocline feedbacks responsible to cool the sea surface during summer months. This RW‐reflected mechanism acts as a negative feedback causing the early termination of the HS mode. Our results provide an improvement in the understanding of the TAV modes and emphasize the importance of ocean wave activity to modulate the equatorial SST variability. These findings could be very useful to improve the prediction of the Equatorial Modes.

Comment Source:Fresh meat: https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2019JC015030 Haven't read it since I do not have a subscription. "Ocean dynamics shapes the structure and timing of Atlantic Equatorial Modes" Marta Martín‐Rey Irene Polo Belén Rodríguez‐Fonseca Alban Lazar Teresa Losada Abstract A recent study has brought to light the co‐existence of two distinct Atlantic Equatorial Modes during negative phases of the Atlantic Multidecadal Variability: the Atlantic Niño and Horse‐Shoe (HS) mode. Nevertheless, the associated air‐sea interactions for HS mode have not been explored so far and the prevailing dynamic view of the Atlantic Niño has been questioned. Here, using forced ocean model simulations, we find that for both modes, ocean dynamics is essential to explain the equatorial SST variations, while air‐sea fluxes control the off‐equatorial SST anomalies. Moreover, we demonstrate the key role played by ocean waves in shaping their distinct structure and timing. For the positive phase of both Atlantic Niño and HS, anomalous westerly winds trigger a set of equatorial downwelling Kelvin waves (KW) during spring‐summer. These dKWs deepen the thermocline, favouring the equatorial warming through vertical diffusion and horizontal advection. Remarkably, for the HS, an anomalous north‐equatorial wind stress curl excites an upwelling Rossby wave (RW), which propagates westward and is reflected at the western boundary becoming an equatorial upwelling KW. The uKW propagates to the east, activating the thermocline feedbacks responsible to cool the sea surface during summer months. This RW‐reflected mechanism acts as a negative feedback causing the early termination of the HS mode. Our results provide an improvement in the understanding of the TAV modes and emphasize the importance of ocean wave activity to modulate the equatorial SST variability. These findings could be very useful to improve the prediction of the Equatorial Modes.
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347.

Advancing the yard-stick on that paper

"ocean dynamics is essential to explain the equatorial SST variations, while air‐sea fluxes control the off‐equatorial SST anomalies. "

Here is a chart supporting an annual impulse driving AMO. An example of double-sideband carrier suppressed modulation. It's not quite as obvious as for ENSO or PNA, but the long-period sidebands are distinct.

The way to read the chart is to fold the frequency spectrum from 0 to 0.5/year onto the reversed spectrum from 1.0 down to 0.5 year. Across the range, the correlation coefficient is above 0.5 which is highly significant.

Comment Source:Advancing the yard-stick on that paper https://www.researchgate.net/profile/Marta_Martin-Rey/publication/336306423_Ocean_dynamics_shapes_the_structure_and_timing_of_Atlantic_Equatorial_Modes/links/5d9b264ba6fdccfd0e7f4148/Ocean-dynamics-shapes-the-structure-and-timing-of-Atlantic-Equatorial-Modes.pdf > "ocean dynamics is essential to explain the equatorial SST variations, while air‐sea fluxes control the off‐equatorial SST anomalies. " Here is a chart supporting an annual impulse driving AMO. An example of [double-sideband carrier suppressed modulation](https://en.wikipedia.org/wiki/Double-sideband_suppressed-carrier_transmission). It's not quite as obvious as for ENSO or PNA, but the long-period sidebands are distinct. ![](https://imagizer.imageshack.com/img922/6499/Slw1rm.png) The way to read the chart is to fold the frequency spectrum from 0 to 0.5/year onto the reversed spectrum from 1.0 down to 0.5 year. Across the range, the correlation coefficient is above 0.5 which is highly significant. 
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348.

This is related to comments #344 & #345. Given that it will take many decades to validate any ENSO model via prediction, the only technique to use in the interim are various flavors of cross-validation using existing data. The following chart uses a test interval that is short (~18 years, the nodal repeat cycle) but constrains the fit by applying a stringent calibration to the angular momentum forcing as measured by length-of-day (LOD) changes -- see middle panel.

The correlation coefficient in the training interval and the calibration interval is demanding at 0.95. The calibration interval is fit to long-period tidal constituents so that this can be extended outside the training interval as an applied ENSO forcing -- note that LOD changes are only high precision back to 1962 so this must be extrapolated for the 82 prior years. The completely unknown factor is the LTE modulation (shown in the upper inset) which is kept constant across the entire interval.

The lower panel is an amplitude spectrum across the entire extrapolated model range, which maintains a resemblance to the spectrum of the ENSO data in spite of the short training interval applied. Recall that because of the LTE modulation, the position of the Fourier peaks become scrambled similar to what occurs with Mach-Zehnder modulation (MZM) described in comment #336.

This has been a challenging modeling problem partly because of structural stability in the non-linear aspects of the LTE solution. Here is a recent article titled "Escape from Model Land" which relates structural stability to the Hawkmoth effect and distinguishes it from the (what I consider inapplicable) Butterfly effect.

https://www.econstor.eu/bitstream/10419/194875/1/1662970102.pdf

"It is sometimes thought that if a model is only slightly wrong, then its outputs will correspondingly be only slightly wrong. The Butterfly Effect revealed that in deterministic nonlinear dynamical systems, a “slightly wrong” initial condition can yield wildly wrong outputs. The Hawkmoth Effect implies that when the mathematical structure of the model is only “slightly wrong” then one loses topological conjugacy (with probability one), and even the best formulated probability forecasts will be wildly wrong. This result, due to Smale in the early 1960’s holds consequences not only for the aims of prediction but also for model development and calibration, and of course for the formation of initial condition ensembles. Naïvely, we might hope that by making incremental improvements to the “realism” of a model (more accurate representations, greater details of processes, finer spatial or temporal resolution, etc.) we would also see incremental improvement in the outputs (either qualitative realism or according to some quantitative performance metric). Regarding the realism of short term trajectories, this may well be true! It is not expected to be true in terms of probability forecasts. And it is not always true in terms of short term trajectories; we note that fields of research where models have become dramatically more complex are experiencing exactly this problem: the nonlinear compound effects of any given small tweak or addition to the model structure are so great that calibration becomes a very computationally-intensive task and the marginal performance benefits of additional subroutines or processes may be zero or even negative. In plainer terms, adding detail to the model can make it less accurate."

This is why it is so difficult to decrypt MZM-encoded optical transmissions. Not only is the modulation unknown to a hacker but that slight variations in the modulation can provide completely different results. That helps explains why one needs to spend lots of effort on cross-validating the LTE model for ENSO before it is even close to becoming a practical tool for predictions, so re-read comment #345 in that context.

Comment Source:This is related to comments #344 & #345. Given that it will take many decades to validate any ENSO model via prediction, the only technique to use in the interim are various flavors of cross-validation using existing data. The following chart uses a test interval that is short (~18 years, the nodal repeat cycle) but constrains the fit by applying a stringent calibration to the angular momentum forcing as measured by length-of-day (LOD) changes -- see middle panel. ![](https://imagizer.imageshack.com/img921/9389/7YSRF8.png) The correlation coefficient in the training interval and the calibration interval is demanding at 0.95. The calibration interval is fit to long-period tidal constituents so that this can be extended outside the training interval as an applied ENSO forcing -- note that LOD changes are only high precision back to 1962 so this must be extrapolated for the 82 prior years. The completely unknown factor is the LTE modulation (shown in the upper inset) which is kept constant across the entire interval. The lower panel is an amplitude spectrum across the entire extrapolated model range, which maintains a resemblance to the spectrum of the ENSO data in spite of the short training interval applied. Recall that because of the LTE modulation, the position of the Fourier peaks become scrambled similar to what occurs with Mach-Zehnder modulation (MZM) described in comment #336. --- This has been a challenging modeling problem partly because of structural stability in the non-linear aspects of the LTE solution. Here is a recent article titled "Escape from Model Land" which relates structural stability to the Hawkmoth effect and distinguishes it from the (what I consider inapplicable) Butterfly effect. https://www.econstor.eu/bitstream/10419/194875/1/1662970102.pdf > "It is sometimes thought that if a model is only slightly wrong, then its outputs will correspondingly be only slightly wrong. The Butterfly Effect revealed that in deterministic nonlinear dynamical systems, a “slightly wrong” initial condition can yield wildly wrong outputs. The Hawkmoth Effect implies that when the mathematical structure of the model is only “slightly wrong” then one loses topological conjugacy (with probability one), and even the best formulated probability forecasts will be wildly wrong. This result, due to Smale in the early 1960’s holds consequences not only for the aims of prediction but also for model development and calibration, and of course for the formation of initial condition ensembles. Naïvely, we might hope that by making incremental improvements to the “realism” of a model (more accurate representations, greater details of processes, finer spatial or temporal resolution, etc.) we would also see incremental improvement in the outputs (either qualitative realism or according to some quantitative performance metric). Regarding the realism of short term trajectories, this may well be true! It is not expected to be true in terms of probability forecasts. And it is not always true in terms of short term trajectories; we note that fields of research where models have become dramatically more complex are experiencing exactly this problem: the nonlinear compound effects of any given small tweak or addition to the model structure are so great that calibration becomes a very computationally-intensive task and the marginal performance benefits of additional subroutines or processes may be zero or even negative. In plainer terms, adding detail to the model can make it less accurate." This is why it is so difficult to decrypt MZM-encoded optical transmissions. Not only is the modulation unknown to a hacker but that slight variations in the modulation can provide completely different results. That helps explains why one needs to spend lots of effort on cross-validating the LTE model for ENSO before it is even close to becoming a practical tool for predictions, so re-read comment #345 in that context. 
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349.

Whether an unknown Laplace's Tidal Equation or Mach-Zehnder modulation can be decoded easily is still an open question but here is a trig trick that doesn't require an $$arcsin$$ function. The puzzle is how to recover $$g(t)$$ from $$\sin( k g(t) )$$, where both $$k$$ and $$g(t)$$ are unknown. The hint is that it is easier to recover $$g'(t)$$ than $$g(t)$$. I am certain it's in the literature somewhere on inverting MZ modulation but it will take a while to find.

This is a recovery of $$g'(t)$$ for an LTE/MZ modulation of $$k=3$$ for three mixed frequency sine waves (periods 12, 25, 40)

This is a modulation of $$k=10$$

Note that the only problem is that it can't decide whether the recovered signal is $$g'(t)$$ or $$-g'(t)$$ and switches between the two whenever a $$2 \pi$$ boundary is crossed.

This is a slight modulation of $$k=0.01$$, which essentially recovers the derivative of the original mix of sine waves without switching to the opposite sine since no $$2 \pi$$ boundary crossings occur.

I think the approach should work fine on an ideal case once an algorithm is applied to prevent jump discontinuities when piecing together the intervals (the zero crossing points are the trickiest to handle). Whether it will work on a non-ideal signal such as ENSO is another question.

This is actually an easy formulation to derive if anyone wants to give it a go.

Comment Source:Whether an unknown Laplace's Tidal Equation or Mach-Zehnder modulation can be decoded easily is still an open question but here is a trig trick that doesn't require an \$$arcsin \$$ function. The puzzle is how to recover \$$g(t) \$$ from \$$\sin( k g(t) ) \$$, where both \$$k \$$ and \$$g(t) \$$ are unknown. The hint is that it is easier to recover \$$g'(t) \$$ than \$$g(t) \$$. I am certain it's in the literature somewhere on inverting MZ modulation but it will take a while to find. This is a recovery of \$$g'(t) \$$ for an LTE/MZ modulation of \$$k=3 \$$ for three mixed frequency sine waves (periods 12, 25, 40) ![](https://imagizer.imageshack.com/img923/7536/fGmQ5Y.png) This is a modulation of \$$k=10 \$$ ![](https://imagizer.imageshack.com/img922/6126/iaJXJD.png) Note that the only problem is that it can't decide whether the recovered signal is \$$g'(t) \$$ or \$$-g'(t) \$$ and switches between the two whenever a \$$2 \pi \$$ boundary is crossed. This is a slight modulation of \$$k=0.01 \$$, which essentially recovers the derivative of the original mix of sine waves without switching to the opposite sine since no \$$2 \pi \$$ boundary crossings occur. ![](https://imagizer.imageshack.com/img924/6109/gIfmLU.png) I think the approach should work fine on an ideal case once an algorithm is applied to prevent jump discontinuities when piecing together the intervals (the zero crossing points are the trickiest to handle). Whether it will work on a non-ideal signal such as ENSO is another question. This is actually an easy formulation to derive if anyone wants to give it a go. 
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350.

This is the result of a basic reconstruction on the LTE/MZ modulated waveform in comment #349. The reconstructed curve picked close to the actual except where it flipped polarity as indicated. The problem is where the anticipated continuation of the curve was extrapolated close to a zero crossing (highlighted in yellow) whereafter the algorithm selected the polarity-reversed curve to continue as being an arbitrarily closer fit. This might be corrected by adding a 2nd order factor to the extrapolation.

The following amplitude spectrum is a marvelous illustration of what the LTE demodulation algorithm provides. The dotted spectrum in blue is of the raw unprocessed LTE modulation test waveform. This is a jumble of indistinct peaks spread across the spectrum -- much like an actual ENSO spectrum. In red is the reconstructed spectrum corresponding to the algorithmically demodulated LTE waveform. Here one can distinctly pick out the 3 correctly positioned sine waves that were mixed together by the LTE modulation. Interesting to also note that the smaller satellite peaks are due to the incorrect extrapolation in the previous chart, and that these will disappear with a better optimized reconstruction algorithm.

What's remarkable about this approach is the ability to reconstruct the original signal properties based on only some basic trig identities.

Comment Source:This is the result of a basic reconstruction on the LTE/MZ modulated waveform in comment #349. The reconstructed curve picked close to the actual except where it flipped polarity as indicated. The problem is where the anticipated continuation of the curve was extrapolated close to a zero crossing (highlighted in yellow) whereafter the algorithm selected the polarity-reversed curve to continue as being an arbitrarily closer fit. This might be corrected by adding a 2nd order factor to the extrapolation. ![](https://imagizer.imageshack.com/img924/1406/dbvDwa.png) The following amplitude spectrum is a marvelous illustration of what the LTE demodulation algorithm provides. The dotted spectrum in blue is of the raw unprocessed LTE modulation test waveform. This is a jumble of indistinct peaks spread across the spectrum -- much like an actual ENSO spectrum. In red is the reconstructed spectrum corresponding to the algorithmically demodulated LTE waveform. Here one can distinctly pick out the 3 correctly positioned sine waves that were mixed together by the LTE modulation. Interesting to also note that the smaller satellite peaks are due to the incorrect extrapolation in the previous chart, and that these will disappear with a better optimized reconstruction algorithm. ![](https://imagizer.imageshack.com/img923/5825/yNI6Ml.png) What's remarkable about this approach is the ability to reconstruct the original signal properties based on only some basic trig identities. 
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