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## Comments

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.

`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.`

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.

`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.`

Jim said:

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

`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`

+1

`+1`

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

Athat 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

`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)`

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"

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

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

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 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.

`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.`

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 andsin (A f(t) + φ)for AMO, where theEandAfactors calibrate the distinct standing wave number boundary conditions for the two regions, andf(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

EandAfactors), 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

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

`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)`

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

squaredas unlikely if each process was based on random chance.`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.`

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.

`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)`

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 followsThis 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

`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)`

interesting!thanks

`interesting!thanks`

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/

`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/`

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,etal(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).`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).`

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:

`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.”`

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.

`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)`

Very, very cool!

`Very, very cool!`

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?

`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?`

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

`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`

Jim asked:

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.

`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.`

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.

`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).`

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.

`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.`

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

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

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"

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.

`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.`

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

tropicalmonth (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

draconicmonth (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

`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)`

Interesting reading :)

`Interesting reading :)`

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.

`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.`

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.

`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.`

Thanks, Paul!

`Thanks, Paul!`

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:

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:

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.

`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.`

"Prominent precession‐band variance in ENSO intensity over the last 300,000 years"

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

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?

`["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?`

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/

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/

`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/`

New paper at

JournalofClimate: 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.`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.`

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/

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.

`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)`

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

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

Famous last words

"defined only up to a phase"`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"*`