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## Comments

Here is the PowerPoint slideshow http://umn.edu/~puk/published/AnalyticalFormulationOfEquatorialStandingWavePhenomena2.ppsx

`Here is the PowerPoint slideshow http://umn.edu/~puk/published/AnalyticalFormulationOfEquatorialStandingWavePhenomena2.ppsx`

nad said:

I find it odd to talk about strange behavior in QBO considering that I still haven't seen a valid physical model of QBO that explains the periodic behavior from 1953 on up. Unless you want to believe in the AGW denier Lindzen's QBO model -- but that model can't even predict the period!

A still unanswered question from the consensus science is why does the QBO flip direction every ~28 months? If it is because of the lunisolar nodal alignment that happens to occur with that exact same period, then you have to start with that premise. And then you should note that within the past year we have had a perigee of the moon with the closest distance to the earth simultaneous with a very full moon since 1948 (i.e. the most super of the supermoons). And 1948 was before the QBO measurements began, so that we have no data to compare against. So saying it's "strange" is simply a value judgement.

I have seen some indication from the online meteorologists that the QBO is back in sync, so the better assumption is that any transient may be due to a stronger perigee forcing. And once that excess forcing disappears, it goes back to aligning with the temporal boundary conditions.

I have run my model of QBO that I presented at AGU this past week and compared it against recent QBO data and can tell you that there is indeed some type of hiccup for the recent 3o hPa data .

Note that this was based on training data from 1953 to 1985 against the major lunar tidal periods, and you can note that most of the peaks sharply align in the extrapolation post 1985 -- until we get to ~2014, where you can see the peaks go in opposite directions. It could also be that this past year's El Nino was strong enough to perturb the lunar forcing enough to transiently force it out of its path.

But to have a real discussion on this means that it has to be a two-way street. I am putting effort into this lunisolar forcing model so that is what I will be talking about. I am not going to go on tangent concerning methane, because methane isn't what is causing the fundamental 28 month period.

`nad said: > ". So with strange QBO behaviour I mean the last strange QBO in 2015/2016 which we discussed [here this August](https://forum.azimuthproject.org/discussion/comment/15460/#Comment_15460). All this looks very very bad." I find it odd to talk about strange behavior in QBO considering that I still haven't seen a valid physical model of QBO that explains the periodic behavior from 1953 on up. Unless you want to believe in the AGW denier Lindzen's QBO model -- but that model can't even predict the period! A still unanswered question from the consensus science is why does the QBO flip direction every ~28 months? If it is because of the lunisolar nodal alignment that happens to occur with that exact same period, then you have to start with that premise. And then you should note that within the past year we have had a perigee of the moon with the closest distance to the earth simultaneous with a very full moon since 1948 (i.e. the most super of the supermoons). And 1948 was before the QBO measurements began, so that we have no data to compare against. So saying it's "strange" is simply a value judgement. I have seen some indication from the online meteorologists that the QBO is back in sync, so the better assumption is that any transient may be due to a stronger perigee forcing. And once that excess forcing disappears, it goes back to aligning with the temporal boundary conditions. I have run my model of QBO that I presented at AGU this past week and compared it against recent QBO data and can tell you that there is indeed some type of hiccup for the recent 3o hPa data . ![](http://imageshack.com/a/img923/4926/de30ud.png) Note that this was based on training data from 1953 to 1985 against the major lunar tidal periods, and you can note that most of the peaks sharply align in the extrapolation post 1985 -- until we get to ~2014, where you can see the peaks go in opposite directions. It could also be that this past year's El Nino was strong enough to perturb the lunar forcing enough to transiently force it out of its path. But to have a real discussion on this means that it has to be a two-way street. I am putting effort into this lunisolar forcing model so that is what I will be talking about. I am not going to go on tangent concerning methane, because methane isn't what is causing the fundamental 28 month period.`

the supermoon happened more or less after the strange QBO pattern started. That is the strange pattern is this dog shaped head looking to the left, where the nose is at the

beginning of 2016.The supermoon was in November 2016 i.e. at the

end of 2016Actually I find measurements often way more trustworthy than models. But sure its bad that we have here not more comparable measurements to look at, that holds by the way also for the sun radiation measurements. I find it quite a scandal that there are so few of those essential environmental measurements. We fly on this earth basically with all instruments turned off. And if I believe the current concerns regarding the US climate science politics we are soon on an almost complete blind flight.

As already said I haven't sofar understood those QBO models, including your model, but since Lindzen spoke about lunar forcing I could imagine that it is somewhere indirectly included.

By the way if you know a visualization where one could have a look on the path of the moon as projected down to earth let me know. I would also be interesting to hear from you if you are being paid by BAE Systems for your QBO research.

`>note that within the past year we have had a perigee of the moon with the closest distance to the earth simultaneous with a very full moon since 1948 (i.e. the most super of the supermoons) the supermoon happened more or less after the strange QBO pattern started. That is the strange pattern is this dog shaped head looking to the left, where the nose is at the <strong>beginning of 2016</strong>. ![QBO](http://www.geo.fu-berlin.de/met/ag/strat/produkte/qbo/qbo_wind.jpg) The supermoon was in November 2016 i.e. at the <strong>end of 2016</strong> >I find it odd to talk about strange behavior in QBO considering that I still haven't seen a valid physical model of QBO that explains the periodic behavior from 1953 on up. Unless you want to believe in the AGW denier Lindzen's QBO model -- but that model can't even predict the period! Actually I find measurements often way more trustworthy than models. But sure its bad that we have here not more comparable measurements to look at, that holds by the way also for the <a href="https://forum.azimuthproject.org/discussion/comment/15673/#Comment_15673">sun radiation measurements</a>. I find it quite a scandal that there are so few of those essential environmental measurements. We fly on this earth basically with all instruments turned off. And if I believe the current concerns regarding the US climate science politics we are soon on an almost complete blind flight. As already said I haven't sofar understood those QBO models, including your model, but since Lindzen spoke about lunar forcing I could imagine that it is somewhere indirectly included. >Note that this was based on training data from 1953 to 1985 against the major lunar tidal periods By the way if you know a visualization where one could have a look on the path of the moon as projected down to earth let me know. I would also be interesting to hear from you if you are being paid by BAE Systems for your QBO research.`

=)) =)) =)) =)) =))

`> "I would also be interesting to hear from you if you are being paid by BAE Systems for your QBO research." =)) =)) =)) =)) =))`

whats so funny about that question?

`whats so funny about that question?`

There is absolutely no doubt that QBO is forced by the lunisolar periods. By inspection and a first-order fit, the majority of the forcing is due to the draconic (aka nodal) lunar tide aliased with the seasonal stimulus. But if we then look at the residual, we see that a majority of the variability is due to the anomalistic lunar tide (i.e. the perigee/apogee lunar effect).

In the figure below, the upper panel is a power spectrum of the residual signal

afterincorporating the draconic tide. The ORANGE dots indicate where the power from the anomalistic aliased cycles should occur. After applying that EXACTLY KNOWN period, the lower panel shows the next level of residual. As you can see, the overall power is greatly reduced by simply adding this one period.The new set of dots shows where a possible nonlinear interaction between the lunar tides would occur. I didn't fit to that yet as it gets close to overfitting at that point. Classical tidal analysis proceeds in this manner, with literally hundreds of possible lunisolar (and planetary) periods contributing at lower significance levels.

The likely reason that this QBO deconstruction has never occurred is that scientists have looked at the power spectrum and couldn't make any sense of it. These aren't classical harmonics in the sense of a fundamental frequency and harmonics of that fundamental, but are in fact

aliasedharmonics that obey a different arithmetic progression.That's why this AGW denier guy Richard Lindzen was completely stumped by the nature of the QBO for over 40 years. He couldn't figure it out even though he knew deep down that it could occur. These are the quotes by Lindzen that I presented at my AGU talk.

These are quotes from Lindzen's papers on QBO and atmospheric science topics from at least 30 years ago!

And consider that these are also statements from the guy that doesn't believe in AGW, and who currently sits on the GWPF board which has a charter to discredit the current climate science consensus. So this is the guy who will likely advise the WH in the next few years. :-B Time to drain the swamp and archive the data, because what is in store is not pretty unless we get in gear.

`There is absolutely no doubt that QBO is forced by the lunisolar periods. By inspection and a first-order fit, the majority of the forcing is due to the draconic (aka nodal) lunar tide aliased with the seasonal stimulus. But if we then look at the residual, we see that a majority of the variability is due to the anomalistic lunar tide (i.e. the perigee/apogee lunar effect). In the figure below, the upper panel is a power spectrum of the residual signal *after* incorporating the draconic tide. The <font color=darkorange>ORANGE</font> dots indicate where the power from the anomalistic aliased cycles should occur. After applying that EXACTLY KNOWN period, the lower panel shows the next level of residual. As you can see, the overall power is greatly reduced by simply adding this one period. ![](http://imageshack.com/a/img924/3836/UQV7Rm.png) The new set of dots shows where a possible nonlinear interaction between the lunar tides would occur. I didn't fit to that yet as it gets close to overfitting at that point. Classical tidal analysis proceeds in this manner, with literally hundreds of possible lunisolar (and planetary) periods contributing at lower significance levels. The likely reason that this QBO deconstruction has never occurred is that scientists have looked at the power spectrum and couldn't make any sense of it. These aren't classical harmonics in the sense of a fundamental frequency and harmonics of that fundamental, but are in fact *aliased* harmonics that obey a different arithmetic progression. That's why this AGW denier guy Richard Lindzen was completely stumped by the nature of the QBO for over 40 years. He couldn't figure it out even though he knew deep down that it could occur. These are the quotes by Lindzen that I presented at my AGU talk. ![](http://imageshack.com/a/img923/2756/rq6xwM.png) ![](http://imageshack.com/a/img921/5338/wmEyEU.png) These are quotes from Lindzen's papers on QBO and atmospheric science topics from at least 30 years ago! And consider that these are also statements from the guy that doesn't believe in AGW, and who currently sits on the GWPF board which has a charter to discredit the current climate science consensus. So this is the guy who will likely advise the WH in the next few years. :-B Time to drain the swamp and archive the data, because what is in store is not pretty unless we get in gear.`

I still haven't gotten an answer to this question. Finally BAE Systems Applied Intelligence Operations include e.g. communications and so climate and metrological findings might be important for that.

`>I would also be interesting to hear from you if you are being paid by BAE Systems for your QBO research. I still haven't gotten an answer to this question. Finally <a href="https://en.wikipedia.org/wiki/BAE_Systems_Applied_Intelligence#Operations">BAE Systems Applied Intelligence Operations</a> include e.g. communications and so <a href="https://en.wikipedia.org/wiki/Space_weather#Long-distance_radio_signals">climate and metrological findings</a> might be important for that.`

I-)

`> "I still haven't gotten an answer to this question" I-)`

As with ENSO, we can train QBO on separate intervals and compare the fit on each interval. The QBO 30 hPa data runs from 1953 to the present. So we take a pair of intervals — one from 1953-1983 (i.e. lower) and one from 1983-2013 (i.e. higher) — and compare the two.

The primary forcing factor is the seasonally aliased nodal or Draconic tide which is shown in the upper left on the figure. The lower interval fit in

BLUEmatches extremely well to the higher interval fit inRED, with a correlation coefficient above 0.8.These two intervals have no inherent correlation other than what can be deduced from the physical behavior generating the time-series. The other factors are the most common long-period tidal cycles, along with the seasonal factor. All have good correlations — even the aliased anomalistic tide (lower left), which features a pair of closely separated harmonics, clearly shows strong phase coherence over the two intervals.

The two intervals used for the fit.

The training region has a correlation coefficient above 0.8 while the validation interval is around 0.6, which indicates that there is likely some overfitting to noise within the training fit.

That's what my AGU presentation was about — demonstrating how QBO and ENSO are simply derived from known geophysical forcing mechanisms applied to the fundamental mathematical geophysical fluid dynamics models. Anybody can reproduce the model fit with nothing more than an Excel spreadsheet and a Solver plugin.

`As with <a href="http://contextearth.com/2016/11/21/presentation-at-agu-2016-on-december-12/">ENSO</a>, we can train QBO on separate intervals and compare the fit on each interval. The QBO 30 hPa data runs from 1953 to the present. So we take a pair of intervals — one from 1953-1983 (i.e. lower) and one from 1983-2013 (i.e. higher) — and compare the two.</p> ![](http://imageshack.com/a/img921/695/J0jL3q.png) <p>The primary forcing factor is the seasonally aliased nodal or Draconic tide which is shown in the upper left on the figure. The lower interval fit in <strong><font color="BLUE">BLUE</font></strong> matches extremely well to the higher interval fit in <strong><font color="RED">RED</font></strong>, with a correlation coefficient above 0.8.</p> <p>These two intervals have no inherent correlation other than what can be deduced from the physical behavior generating the time-series. The other factors are the most common long-period tidal cycles, along with the seasonal factor. All have good correlations — even the aliased anomalistic tide (lower left), which features a pair of closely separated harmonics, clearly shows strong phase coherence over the two intervals.</p> <p>The two intervals used for the fit.</p> <p><img src="http://imageshack.com/a/img921/8046/MLWYom.png"></p> <p><img src="http://imageshack.com/a/img921/1927/VWRMbI.png"></p> <p>The training region has a correlation coefficient above 0.8 while the validation interval is around 0.6, which indicates that there is likely some overfitting to noise within the training fit.</p> <p>That's what my AGU presentation was about — demonstrating how QBO and ENSO are simply derived from known geophysical forcing mechanisms applied to the fundamental mathematical <a href="https://t.co/Cwuoo4Xd5M">geophysical fluid dynamics</a> models. Anybody can reproduce the model fit with nothing more than an Excel spreadsheet and a Solver plugin.</p>`

Tidal gauge sea-level height (SLH) readings can reveal the impact of ENSO if analyzed properly.

If the two are compared directly, there is a faster cycle in the SLH readings (taken from Sydney harbor) than in the ENSO SOI measure:

If we apply an optimal Finite Impulse Response (FIR) filter to the SLH then we get a better fit:

The FIR is shown in the upper left inset, which has units of lagged

month.From that, one can see that harmonics of ~1/4 year combined as a lagged FIR window generate a much better approximation to the ENSO time-series.

But even more interesting, is that this very intriguing FIR of a 2-year lagged differential impulse window gives an equivalent fit!

This is predicted based on the biennial modulation model of ENSO that I presented at AGU. The nonlinear sloshing interaction of external forcing (lunar and annual) with the Pacific ocean leads to this subharmonic. Intriguingly, the 2-year lag tells us that ENSO can be predicted effectively 2 years in advance just from SLH readings!

This is not the first time I have observed this effect, but I will likely explore this further because it gives an alternative perspective to the biennial and lunisolar contributions to ENSO.

`Tidal gauge sea-level height (SLH) readings can reveal the impact of ENSO if analyzed properly. If the two are compared directly, there is a faster cycle in the SLH readings (taken from [Sydney harbor](http://www.psmsl.org/data/obtaining/stations/196.php)) than in the ENSO SOI measure: ![un](http://imageshack.com/a/img922/2552/m2mRSm.png) If we apply an optimal Finite Impulse Response ([FIR](https://en.wikipedia.org/wiki/Finite_impulse_response)) filter to the SLH then we get a better fit: ![third](http://imageshack.com/a/img922/1765/wiIjUE.png) The FIR is shown in the upper left inset, which has units of lagged **month**. From that, one can see that harmonics of ~1/4 year combined as a lagged FIR window generate a much better approximation to the ENSO time-series. But even more interesting, is that this very intriguing FIR of a 2-year lagged differential impulse window gives an equivalent fit! ![two](http://imageshack.com/a/img922/5033/PFmH57.png) This is predicted based on the biennial modulation model of ENSO that I presented at AGU. The nonlinear sloshing interaction of external forcing (lunar and annual) with the Pacific ocean leads to this subharmonic. Intriguingly, the 2-year lag tells us that ENSO can be predicted effectively 2 years in advance just from SLH readings! This is not the first time [I have observed this effect](http://contextearth.com/2016/04/13/seasonal-aliasing-of-tidal-forcing-in-mean-sea-level-height/), but I will likely explore this further because it gives an alternative perspective to the biennial and lunisolar contributions to ENSO.`

Interesting new research on the concept of "time crystals". This appears to be an oscillation in a lattice structure, perhaps akin to phonons, but I can't tell from this paper. What is interesting is this statement:

They find it unusual that a period doubling in jiggled Jell-O occurs? I suppose that would happen if one doesn't know the scientific literature. This is from Ibrahim's book on sloshing

Note that both Faraday and Rayleigh observed period doubling. Rayleigh's original paper On Maintained Vibrations is an interesting read.

A liquid is not a crystal, yet phonons jiggling a crystal is the acoustic wave equivalent of a liquid sloshing.

And the reason I commented in this thread is because I believe the same period doubling behavior features in the sloshing dynamics of ENSO. This is the same Mathieu equation formulation that I apply to refine models of the behavior.

http://contextearth.com/2016/11/21/presentation-at-agu-2016-on-december-12/

`Interesting new research on the concept of ["time crystals"](https://www.sott.net/article/341047-Time-crystals-Scientists-have-confirmed-a-brand-new-form-of-matter). This appears to be an oscillation in a lattice structure, perhaps akin to phonons, but I can't tell from this paper. What is interesting is this statement: >The two lasers that were periodically nudging the ytterbium atoms were producing a repetition in the system at twice the period of the nudges, something that couldn't occur in a normal system. > "Wouldn't it be super weird if you jiggled the Jell-O and found that somehow it responded at a different period?" said Yao. > "But that is the essence of the time crystal. You have some periodic driver that has a period 'T', but the system somehow synchronises so that you observe the system oscillating with a period that is larger than 'T'." They find it unusual that a period doubling in jiggled Jell-O occurs? I suppose that would happen if one doesn't know the scientific literature. This is from [Ibrahim's book on sloshing](https://books.google.com/books/about/Liquid_Sloshing_Dynamics.html?id=ctvhvH74ZzEC) ![](http://imageshack.com/a/img922/1158/qat6lK.gif) Note that both Faraday and Rayleigh observed period doubling. Rayleigh's original paper [On Maintained Vibrations](http://www.tandfonline.com/doi/abs/10.1080/14786448308627342?journalCode=tphm16) is an interesting read. A liquid is not a crystal, yet phonons jiggling a crystal is the acoustic wave equivalent of a liquid sloshing. And the reason I commented in this thread is because I believe the same period doubling behavior features in the sloshing dynamics of ENSO. This is the same Mathieu equation formulation that I apply to refine models of the behavior. http://contextearth.com/2016/11/21/presentation-at-agu-2016-on-december-12/`

I found a reference to a technique called Slow Feature Analysis

http://www.scholarpedia.org/article/Slow_feature_analysis

It falls under the category of an unsupervised learning tool and essentially pulls out patterns from what looks like either noisy or highly erratic signals.

I have a feeling this is related to what I am doing with my ENSO analysis. The key step I find in fitting the ENSO model is to modulate the original signal with a biennially peaked periodic function. This emphasizes the forcing function in a way that is compatible with the way one would formulate the sloshing physics. In other words, sloshing requires a Mathieu equation formulation -- and that includes a time-varying modulation implicit in the DiffEq.

As it happens, the QBO analysis requires a similar approach. With QBO, the key is to model the acceleration of the wind, and not the velocity. The acceleration is important because that is what the atmospheric flow physics uses!

No one in climate science is doing these kinds of transformations -- they all seem to do exactly what everyone else is doing and thus getting stuck in a lock-step dead-end.

I recently watched the congressional EPA hearings -- "Make the EPA great again". The one witness who was essentially schooling the Republican-know-nothings was Rush Holt PhD, who is now CEO of the AAAS but at one time was a physicist congressman from New Jersey. My favorite bit of wisdom he imparted was that science isn't going to make any progress by looking at the same data over and over again the same way, but by

"approaching the problem with a new perspective"Watch it here, set to 101 minutes (1:41 mark) into the hearing

We have to look at the data in new ways -- that's science.

`I found a reference to a technique called Slow Feature Analysis http://www.scholarpedia.org/article/Slow_feature_analysis > ![wiki](http://www.scholarpedia.org/w/images/thumb/4/41/SlowFeatureAnalysis-Algorithm-InputSignal.png/400px-SlowFeatureAnalysis-Algorithm-InputSignal.png) >Input signal of the simple example described in the text. The panels on the left and center show the two individual input components, x1(t) and x2(t) . On the right, the joint 2D trajectory x(t)=(x1(t),x2(t)) is shown. It falls under the category of an unsupervised learning tool and essentially pulls out patterns from what looks like either noisy or highly erratic signals. I have a feeling this is related to what I am doing with my ENSO analysis. The key step I find in fitting the ENSO model is to modulate the original signal with a biennially peaked periodic function. This emphasizes the forcing function in a way that is compatible with the way one would formulate the sloshing physics. In other words, sloshing requires a Mathieu equation formulation -- and that includes a time-varying modulation implicit in the DiffEq. As it happens, the QBO analysis requires a similar approach. With QBO, the key is to model the acceleration of the wind, and not the velocity. The acceleration is important because that is what the atmospheric flow physics uses! No one in climate science is doing these kinds of transformations -- they all seem to do exactly what everyone else is doing and thus getting stuck in a lock-step dead-end. --- I recently watched the congressional EPA hearings -- "Make the EPA great again". The one witness who was essentially schooling the Republican-know-nothings was Rush Holt PhD, who is now CEO of the AAAS but at one time was a physicist congressman from New Jersey. My favorite bit of wisdom he imparted was that science isn't going to make any progress by looking at the same data over and over again the same way, but by *"approaching the problem with a new perspective"* Watch it here, set to 101 minutes (1:41 mark) into the hearing https://youtu.be/v7krqZxXu94?t=6160 --- We have to look at the data in new ways -- that's science.`

This is how to apply a slow feature analysis to a sloshing model.

Start with the Mathieu equation and keep it in its differential form

$\frac{d^2x(t)}{dt^2}+[a-2q\cos(2\omega t)]x(t)=F(t)$

The time-modulating parameter multiplying

x(t)is replaced with a good guess -- which is that it is either an annual or biennial modulation and with a peak near the end of the year. For a biennial modulation, the peak will appear on either an odd or even year. In the modulation below it is on an even year:The RHS

F(t)is essentially the same modulation but with a multiplicative forcing corresponding to the known angular momentum and tidal variations. The strongest known is at the Chandler Wobble frequency of ~432 days. There is another wobble predicted at ~14 years and a nutation at 18.6 years due to the lunar nodal variation and a heavily aliased anomalistic period at 27.54 days. Those are the strongest known forcings and are input with unknown amplitude and phase.The search solver tries to find the best fit by varying these parameters over a "training" window of the ENSO time series. I first take a split window that takes an older time interval and matches with a more recent time interval. The "out-of-band" interval is then used to test the fit.

Then I reverse the fit by using the out-of-band interval as the fitting interval and testing against the former.

The reason that the fit is stationary across the time intervals is because the strength and the phase of the wobble terms remains pretty much constant. The dense chart below is a comparison of the factors used. Note that some of the lesser tidal forcing factors are included as well and they do not fare quite as well.

Here is an animated GIF of how the two fitting intervals compare:

The sloshing model is parsimonious with the data and what remains to be done is to establish the plausibility of the model. In that regard, is there enough of an angular momentum change or torque in the earth's rotation to cause the ocean to slosh? Or is there enough to cause

the thermoclineto slosh, which is actually what's happening? The difference in density of water above and below the thermocline is enough to create a reduced effective gravity that can plausibly be extremely sensitive to momentum changes. The sensitivity is equivalent to an oil/water wave machineYet the current thinking in the consensus ENSO research is that prevailing east-west winds are what causes a sloshing buildup. But what causes those winds? Could those be a

resultof ENSO and not acause? They seem to be highly correlated,but what causes a wind other than a pressure differential? And that pressure differential arises from the pressure dipole measured by the ENSO SOI index. Whenever the pressure is low in the west Pacific, it is high in the east and vice versa.

So the plausibility of this model of torque-assisted sloshing is essentially wrapped around finding out whether this effect is of similar magnitude of any wind-forced mechanism. Perhaps the question to ask -- is it easier to cause a wave machine to slosh by blowing on the surface or to gently rock it?

`This is how to apply a slow feature analysis to a sloshing model. Start with the Mathieu equation and keep it in its differential form $\frac{d^2x(t)}{dt^2}+[a-2q\cos(2\omega t)]x(t)=F(t)$ The time-modulating parameter multiplying *x(t)* is replaced with a good guess -- which is that it is either an annual or biennial modulation and with a peak near the end of the year. For a biennial modulation, the peak will appear on either an odd or even year. In the modulation below it is on an even year: ![1](http://imageshack.com/a/img922/1014/b5bOdp.png) The RHS *F(t)* is essentially the same modulation but with a multiplicative forcing corresponding to the known angular momentum and tidal variations. The strongest known is at the Chandler Wobble frequency of ~432 days. There is another wobble predicted at ~14 years and a nutation at 18.6 years due to the lunar nodal variation and a heavily aliased anomalistic period at 27.54 days. Those are the strongest known forcings and are input with unknown amplitude and phase. The search solver tries to find the best fit by varying these parameters over a "training" window of the ENSO time series. I first take a split window that takes an older time interval and matches with a more recent time interval. The "out-of-band" interval is then used to test the fit. ![4](http://imageshack.com/a/img924/4640/zvC8eb.png) Then I reverse the fit by using the out-of-band interval as the fitting interval and testing against the former. ![2](http://imageshack.com/a/img922/8146/MRiRzi.png) The reason that the fit is stationary across the time intervals is because the strength and the phase of the wobble terms remains pretty much constant. The dense chart below is a comparison of the factors used. Note that some of the lesser tidal forcing factors are included as well and they do not fare quite as well. ![3](http://imageshack.com/a/img924/6414/JkdcEB.png) Here is an animated GIF of how the two fitting intervals compare: ![5](http://imageshack.com/a/img923/7651/vaNakb.gif) The sloshing model is parsimonious with the data and what remains to be done is to establish the plausibility of the model. In that regard, is there enough of an angular momentum change or torque in the earth's rotation to cause the ocean to slosh? Or is there enough to cause **the thermocline** to slosh, which is actually what's happening? The difference in density of water above and below the thermocline is enough to create a reduced effective gravity that can plausibly be extremely sensitive to momentum changes. The sensitivity is equivalent to an oil/water wave machine https://youtu.be/UUZ8vrj-qzM Yet the current thinking in the consensus ENSO research is that prevailing east-west winds are what causes a sloshing buildup. But what causes those winds? Could those be a **result** of ENSO and not a **cause**? They seem to be highly correlated, ![wind](http://imageshack.com/a/img923/2606/PVVxAx.png) but what causes a wind other than a pressure differential? And that pressure differential arises from the pressure dipole measured by the ENSO SOI index. Whenever the pressure is low in the west Pacific, it is high in the east and vice versa. So the plausibility of this model of torque-assisted sloshing is essentially wrapped around finding out whether this effect is of similar magnitude of any wind-forced mechanism. Perhaps the question to ask -- is it easier to cause a wave machine to slosh by blowing on the surface or to gently rock it?`

I mentioned:

This would actually be a very easy experiment to set up. Build a home-made wave machine out of an old aquarium and then compare with two different forcing mechanisms -- (1) with an oscillating translational platform driven by a servo (2) with an oscillating speed fan blowing air over the surface.

It would be easy to measure the average power consumed by each mechanism and see which one takes the least effort to start the wave machine sloshing.

The other interesting idea I have is in the analytical realm. I think I have figured out how to automatically extract the principal factors of forcing from a certain class of Mathieu modulated DiffEq's. The key is to create a modulation that is a delta-function train of spikes at a periodic interval. That is an easy convolution to construct as a matrix eigenvalue problem in frequency space. Essentially for every harmonic created one can introduce an additional periodic forcing factor. Solve the matrix of a chosen size and the factors should pop out from the roots. I should get the same answer as the solver finds in the comment above, since the modulation there looks like a train of spikes in the limiting case:

If that works, we can get an answer back instantaneously instead of letting the solver grind away finding a miminum error solution. Even if it's not the actual physics involved, it certainly qualifies as an interesting applied math solution.

Here is a paper that exactly derives the Mathieu equation for Faraday waves on a sphere. https://www.pmmh.espci.fr/~laurette/papers/FIS_FA_pub.pdf "Faraday instability on a sphere: Floquet analysis"

I recall seeing a paper from years ago that indicated that the Mathieu equation was inapplicable for a spherical geometry, which may have hindered further research along this path. Interesting that this particular paper is also in the highly regarded Journal of Fluid Mechanics, which recently published a review article from 2016 explaining how to best approach the characterization of wave behavior Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited

This paper includes this jarring statement:

This hasn't yet completely sunk in but I still find it strange that at this late date scientists are apparently still struggling to figure out forced wave action in a liquid volume.

The lead author of that paper elsewhere said this:

`I mentioned: > "Perhaps the question to ask -- is it easier to cause a wave machine to slosh by blowing on the surface or to gently rock it?" This would actually be a very easy experiment to set up. Build a home-made wave machine out of an old aquarium and then compare with two different forcing mechanisms -- (1) with an oscillating translational platform driven by a servo (2) with an oscillating speed fan blowing air over the surface. It would be easy to measure the average power consumed by each mechanism and see which one takes the least effort to start the wave machine sloshing. --- The other interesting idea I have is in the analytical realm. I think I have figured out how to automatically extract the principal factors of forcing from a certain class of Mathieu modulated DiffEq's. The key is to create a modulation that is a delta-function train of spikes at a periodic interval. That is an easy convolution to construct as a matrix eigenvalue problem in frequency space. Essentially for every harmonic created one can introduce an additional periodic forcing factor. Solve the matrix of a chosen size and the factors should pop out from the roots. I should get the same answer as the solver finds in the comment above, since the modulation there looks like a train of spikes in the limiting case: ![](http://imageshack.com/a/img922/1014/b5bOdp.png) If that works, we can get an answer back instantaneously instead of letting the solver grind away finding a miminum error solution. Even if it's not the actual physics involved, it certainly qualifies as an interesting applied math solution. --- Here is a paper that exactly derives the Mathieu equation for Faraday waves on a sphere. https://www.pmmh.espci.fr/~laurette/papers/FIS_FA_pub.pdf "Faraday instability on a sphere: Floquet analysis" I recall seeing a paper from years ago that indicated that the Mathieu equation was inapplicable for a spherical geometry, which may have hindered further research along this path. Interesting that this particular paper is also in the highly regarded Journal of Fluid Mechanics, which recently published a review article from 2016 explaining how to best approach the characterization of wave behavior [Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited](http://www.unice.fr/rajchenbach/JFM2015.pdf) This paper includes this jarring statement: > "For instance, to the best of our knowledge, the dispersion relation (relating angular frequency ω and wavenumber k) of parametrically forced water waves has astonishingly not been explicitly established hitherto. " This hasn't yet completely sunk in but I still find it strange that at this late date scientists are apparently still struggling to figure out forced wave action in a liquid volume. The lead author of that paper elsewhere said this: > The prominent physicist Richard P. Feynman wrote in his cerebrated lectures [10]: *“Water waves that are easily seen by everyone, and which are usually used as an example of waves in elementary courses, are the worst possible example; they have all the complications that waves can have.”* This is precisely these complications that make the richness and interest of water waves. Indeed, despite numerous studies, new waves and new wave behaviors are still discovered (e.g. , [26, 27]) and wave dynamics is still far from being fully understood.`

Retired atmospheric sciences professor Judith Curry has a discussion paper out called "Climate Models for the Layman" written for Trump followers apparently: http://www.thegwpf.org/content/uploads/2017/02/Curry-2017.pdf

The paper is essentially Curry whining that climate science is too difficult, instead of getting to work and figuring out the physics and math, like the rest of us try to do.

The GWPF is the Global Warming Policy Foundation, which is one of those misnamed organizations -- the people running it do not actually believe in the science behind AGW. It's based in the UK which you can tell from the trustee list

Notice that the two scientists largely responsible for the primitive state of QBO and ENSO models -- Lindzen and frequent Curry collaborator Tsonis -- are on the GWPF academic advisory board. Since they don't seem to believe in AGW, I don't trust their understanding of QBO and ENSO; which is one of the reasons that I am working on these models. The idea is to work on research where the understanding is weak.

`Retired atmospheric sciences professor Judith Curry has a discussion paper out called "Climate Models for the Layman" written for Trump followers apparently: http://www.thegwpf.org/content/uploads/2017/02/Curry-2017.pdf The paper is essentially Curry whining that climate science is too difficult, instead of getting to work and figuring out the physics and math, like the rest of us try to do. The GWPF is the Global Warming Policy Foundation, which is one of those misnamed organizations -- the people running it do not actually believe in the science behind AGW. It's based in the UK which you can tell from the trustee list ![](http://imageshack.com/a/img922/7752/2QeAFZ.png) Notice that the two scientists largely responsible for the primitive state of QBO and ENSO models -- Lindzen and frequent Curry collaborator Tsonis -- are on the GWPF academic advisory board. Since they don't seem to believe in AGW, I don't trust their understanding of QBO and ENSO; which is one of the reasons that I am working on these models. The idea is to work on research where the understanding is weak.`

Where is she whining ? I mean she is critizising certain features of GCM's, like in particular that

As far as I understood, this critique is based on observations (p.7):

It is clear that predicting climate is shit difficult and at this infancy stage very likely quite error-prone. As she rightly pointed out alone on the math side there are major difficulties, like:

so on page (vii) she writes:

The question here seems what do you make out of this sentence? What are the consequences?

from the last page:

Yes, following the climate models in particular GCMs seem to be quite faulty. But they can be faulty into two directions, i.e. things can be less worse or way worse. If the thermometers in your living room gets suddenly hot (at least at some places) do you keep sitting and pretend nothing has happened? Or do you keep trying to check as best as possible and eventually precautiously switch down the thermostat even if you don't really know wether the heating is the source of the problem?

I think that there had been deficiences especially with respect to the point "check as best as possible." That is as already Judith Curry pointed out, if all models (in the example the "hypothesis why it is so hot in the living room") appear rather bad in explaining the phenomena then nobody really knows what the best model is:

and so the choice for approaches to find solutions was partially less based on purely scientific choices, but also on who is able to secure which funds with which effort.

The fact that people like Tim van Beek is now busy producing cars for the "economic elite" instead of e.g. working on fluid dynamics is saying something.

`>The paper is essentially Curry whining that climate science is too difficult, instead of getting to work and figuring out the physics and math, like the rest of us try to do. Where is she whining ? I mean she is critizising certain features of GCM's, like in particular that >There is growing evidence that climate models are running too hot and that climate sensitivity to carbon dioxide is at the lower end of the range provided by the IPCC. Nevertheless, these lower values of climate sensitivity are not accounted for in IPCC climate model projections of temperature at the end of the 21st century or in esti- mates of the impact on temperatures of reducing carbon dioxide emissions. As far as I understood, this critique is based on observations (p.7): >Lewis and Curry (2014) used an observation-based energy balance approach to estimate ECS. Their calculations used the same values (including uncertainties) for changes in greenhouse gases and other drivers of climate change as given in the Fifth Assessment. However, their range of values for ECS were approximately half those de- termined from the CMIP5 climate models. It is clear that predicting climate is shit difficult and at this infancy stage very likely quite error-prone. As she rightly pointed out alone on the math side there are major difficulties, like: >The solution of Navier–Stokes equations is one of the most vexing problems in all of mathematics: the Clay Mathematics Institute has declared this to be one of the top seven problems in all of mathematics and is offering a $1 million prize for its solution. so on page (vii) she writes: >By extension, GCMs are not fit for the purpose of justifying political policies to fundamentally alter world social, economic and energy systems. The question here seems what do you make out of this sentence? What are the consequences? from the last page: >The Global Warming Policy Foundation is an all-party and non-party think tank and a registered educational charity which, while openminded on the contested science of global warming, is deeply concerned about the costs and other implications of many of the policies currently being advo- cated Yes, following the climate models in particular GCMs seem to be quite faulty. But they can be faulty into two directions, i.e. things can be less worse or way worse. If the thermometers in your living room gets suddenly hot (at least at some places) do you keep sitting and pretend nothing has happened? Or do you keep trying to check as best as possible and eventually precautiously switch down the thermostat even if you don't really know wether the heating is the source of the problem? I think that there had been deficiences especially with respect to the point "check as best as possible." That is as already Judith Curry pointed out, if all models (in the example the "hypothesis why it is so hot in the living room") appear rather bad in explaining the phenomena then nobody really knows what the best model is: >Is it possible to select a ‘best’ model? Well, several models generally show a poorer performance overall when compared with observations. However, the best model depends on how you define ‘best’, and no single model is the best at everything. and so the choice for approaches to find solutions was partially less based on purely scientific choices, but also on who is able to secure which funds with which effort. The fact that people like <a href="https://johncarlosbaez.wordpress.com/2012/05/30/fluid-flows-and-infinite-dimensional-manifolds-part-3/">Tim van Beek</a> is now busy producing cars for the "economic elite" instead of e.g. working on fluid dynamics is saying something.`

Obviously they are unsolvable in general because the equations are largely under-determined. Boundary and initial conditions alone do not provide enough constraints to solve the set of equations in 3-dimensions. No one complains that two equations with three unknowns is unsolvable.

Compare that to Maxwell's equations which tend to be more determined because of the interactions between the B and E fields and how boundary conditions are applied.

Here's an example of what I am talking about. Consider my solution to the QBO. This is essentially solving Navier-Stokes for atmospheric flow. When I worked out the primitive equations, I knew that I would have to be ruthless in reducing the dimensionality from the start. The first simplification was working at the equator, which eliminated a few of the terms arising from the Coriolis effect. QBO is also a stratified system, so the vertical cross-terms are inconsequential. Of course the time-space part of Navier-Stokes was separated by noticing the standing-wave nature of the phenomenon has a wavenumber of zero, which obviously helped quite a bit.

I eventually ran into the last remaining under-determined constraint dealing with a transverse spatial term; I eliminated this by cleverly associating the latitudinal equatorial flow line with a nodal lunisolar forcing. This provide a running boundary-slash-initial condition and thus reduced the initially "unsolvable" Navier-Stokes equations to a Sturm-Liouville formulation -- which fortunately had a remarkable closed-form analytical solution.

So not only did I solve a variant of Navier-Stokes -- i.e. the primitive equations of Laplace's tidal equations -- but it didn't even require an iterative numeric solution. I do have to admit that I used many of the ideas from understanding how to solve the 2-dimensional Hall effect via Maxwell's equations in coming up with the answer.

The other example is ENSO. This is also a standing wave equation which has a rather straightforward solution but needs a numerical computation assist to iterate the solution, unless one can leave it as a convolution of a Mathieu function with sinusoidal forcing terms. Of course I partially adapted this idea from sloshing research in the hydrodynamics literature.

Perhaps I should be the one whining to Professor Curry ...

WHERE IS MY $1,000,00.00 PRIZE AWARD !!!!!:-B :)`>>The paper is essentially Curry whining that climate science is too difficult, instead of getting to work and figuring out the physics and math, like the rest of us try to do. > Where is she whining ? I mean she is critizising certain features of GCM's, like in particular that >> The solution of Navier–Stokes equations is one of the most vexing problems in all of mathematics: the Clay Mathematics Institute has declared this to be one of the top seven problems in all of mathematics and is offering a **$1 million prize** for its solution. Obviously they are unsolvable in general because the equations are largely under-determined. Boundary and initial conditions alone do not provide enough constraints to solve the set of equations in 3-dimensions. No one complains that two equations with three unknowns is unsolvable. Compare that to Maxwell's equations which tend to be more determined because of the interactions between the B and E fields and how boundary conditions are applied. Here's an example of what I am talking about. Consider my solution to the QBO. This is essentially solving Navier-Stokes for atmospheric flow. When I worked out the primitive equations, I knew that I would have to be ruthless in reducing the dimensionality from the start. The first simplification was working at the equator, which eliminated a few of the terms arising from the Coriolis effect. QBO is also a stratified system, so the vertical cross-terms are inconsequential. Of course the time-space part of Navier-Stokes was separated by noticing the standing-wave nature of the phenomenon has a wavenumber of zero, which obviously helped quite a bit. I eventually ran into the last remaining under-determined constraint dealing with a transverse spatial term; I eliminated this by cleverly associating the latitudinal equatorial flow line with a nodal lunisolar forcing. This provide a running boundary-slash-initial condition and thus reduced the initially "unsolvable" Navier-Stokes equations to a Sturm-Liouville formulation -- which fortunately had a remarkable closed-form analytical solution. So not only did I solve a variant of Navier-Stokes -- i.e. the primitive equations of Laplace's tidal equations -- but it didn't even require an iterative numeric solution. I do have to admit that I used many of the ideas from understanding how to solve the 2-dimensional Hall effect via Maxwell's equations in coming up with the answer. The other example is ENSO. This is also a standing wave equation which has a rather straightforward solution but needs a numerical computation assist to iterate the solution, unless one can leave it as a convolution of a Mathieu function with sinusoidal forcing terms. Of course I partially adapted this idea from sloshing research in the hydrodynamics literature. > Where is she whining ? Perhaps I should be the one whining to Professor Curry ... **WHERE IS MY $1,000,00.00 PRIZE AWARD !!!!!** :-B :)`

It's been quite a challenge decoding the physics of ENSO. Anything that makes the model more complex and with more degrees of freedom needs to be treated carefully. The period doubling bifurcation properties of wave sloshing has been an eye-opener for me. I experimented with adding a sub-harmonic period of 4 years to the 2-year Mathieu modulation and see if that improves the fit. By simply masking the odd behavior around 1981-1983, I came up with this breakdown of the RHS/LHS comparison.

This is an iterative solver applied to

two completely orthogonalintervals of the ENSO time series leading to largely identical solutions given the fixed tidal factors. The animated gif flips between the fit on one training interval to the orthogonal interval. All that was allowed to change was the amplitude and phase, as shown in the following phasor diagram.Each of the pairs of sinusoidal factors would line up exactly on a phasor diagram if the analyzed process was perfectly stationary. If they line up closely, then there is good agreement -- subject to the possibility of overfitting.

So to check for overfitting, we take this same formulation and extend it to regions outside of the ENSO interval from 1880-present. We can't look to the future, but we can look into the past via the UEP coral proxy records. This is what it looks like.

From 1650 to about 1800, the correlation is quite good considering that we only have yearly-averaged values . Over the calibrated proxy interval post 1900 the agreement is as before. However, the 1800's are out-of-phase (is that due to the amount of volcanic activity during that century Tambora in 1815 plus Krakatoa in 1883?).

This is not conclusive proof but neither does it contradict the model. To achieve such a high correlation between time series separated by at least 200 years is only remotely possible to achieve via random chance.

So is there something fundamental to the 2-year and 4-year period sub-harmonics? I can understand the two year period as being the first doubling of the annual cycle. But the 4 year period would come about from the doubling of the 2-year cycle. This could be recursively applied to 8-year and 16-year periods, but likely not if there was another resonant period close to 4 years.

Is it possible that the 27.55455 day anomalistic lunar tidal period fits into a four year cycle as 13 + 13 + 13 + 13 + 1 = 53 anomalistic tidal periods nearly exactly? 27.55455*53 = 4 * (365.x) where x = 0.098 instead of x=0.242. This is the difference between adding a leap day every fourth year versus adding one every tenth year in terms of alignment. So what the perigee-apogee lunar cycles could do is reinforce this 4-year period by pumping the gravitational cycle in unison with the ocean at the same seasonal reference point.

I presented the following chart at the AGU, which showed how machine learning picked out the same aliased anomalistic period from the UEP coral proxy records with no human direction:

An angular frequency of 7.821 radians with 12 2$\pi$ added = 83.219 is close to the 2 $\pi$ (365.242/27.55455) = 83.285 expected for an anomalistic forcing period.

`It's been quite a challenge decoding the physics of ENSO. Anything that makes the model more complex and with more degrees of freedom needs to be treated carefully. The period doubling bifurcation properties of wave sloshing has been an eye-opener for me. I experimented with adding a sub-harmonic period of 4 years to the 2-year Mathieu modulation and see if that improves the fit. By simply masking the odd behavior around 1981-1983, I came up with this breakdown of the RHS/LHS comparison. ![angif](http://imageshack.com/a/img923/7456/xA3ZKy.gif) This is an iterative solver applied to *two completely orthogonal* intervals of the ENSO time series leading to largely identical solutions given the fixed tidal factors. The animated gif flips between the fit on one training interval to the orthogonal interval. All that was allowed to change was the amplitude and phase, as shown in the following [phasor diagram](https://en.wikipedia.org/wiki/Phasor). ![factors](http://imageshack.com/a/img924/4075/Q6NDni.png) Each of the pairs of sinusoidal factors would line up exactly on a phasor diagram if the analyzed process was perfectly stationary. If they line up closely, then there is good agreement -- subject to the possibility of overfitting. So to check for overfitting, we take this same formulation and extend it to regions outside of the ENSO interval from 1880-present. We can't look to the future, but we can look into the past via the [UEP coral proxy records](http://contextearth.com/2016/09/27/enso-proxy-revisited/). This is what it looks like. ![proxy](http://imageshack.com/a/img922/8082/blVeqV.png) From 1650 to about 1800, the correlation is quite good considering that we only have yearly-averaged values . Over the calibrated proxy interval post 1900 the agreement is as before. However, the 1800's are out-of-phase (is that due to the amount of volcanic activity during that century Tambora in 1815 plus Krakatoa in 1883?). This is not conclusive proof but neither does it contradict the model. To achieve such a high correlation between time series separated by at least 200 years is only remotely possible to achieve via random chance. So is there something fundamental to the 2-year and 4-year period sub-harmonics? I can understand the two year period as being the first doubling of the annual cycle. But the 4 year period would come about from the doubling of the 2-year cycle. This could be recursively applied to 8-year and 16-year periods, but likely not if there was another resonant period close to 4 years. Is it possible that the 27.55455 day anomalistic lunar tidal period fits into a four year cycle as 13 + 13 + 13 + 13 + 1 = 53 anomalistic tidal periods nearly exactly? 27.55455*53 = 4 * (365.x) where x = 0.098 instead of x=0.242. This is the difference between adding a leap day every fourth year versus adding one every tenth year in terms of alignment. So what the perigee-apogee lunar cycles could do is reinforce this 4-year period by pumping the gravitational cycle in unison with the ocean at the same seasonal reference point. ![perigee](https://smd-prod.s3.amazonaws.com/science-red/s3fs-public/atoms/files/diagram_0.gif) I presented the following chart at the AGU, which showed how machine learning picked out the same aliased anomalistic period from the UEP coral proxy records with no human direction: ![uep](http://imagizer.imageshack.us/a/img539/73/13y2CN.gif) An angular frequency of 7.821 radians with 12 2$\pi$ added = 83.219 is close to the 2 $\pi$ (365.242/27.55455) = 83.285 expected for an anomalistic forcing period.`

Last comment I mentioned I was trying to simplify the ENSO model. Right now the forcing is a mix of angular momentum variations related to Chandler wobble and lunisolar tidal pull. This is more complex than I would like to see. So what happens if the Chandler wobble is directly tied to the draconic/nodal cycles in the lunar tide? There is empirical evidence for this even though it is not acknowledged in the consensus geophysics literature.

The figure below is my fit to the Chandler wobble, seemingly matching the

aliasedlunar draconic cycle rather precisely:http://contextearth.com/2016/01/27/possible-luni-solar-tidal-mechanism-for-the-chandler-wobble/

The consensus is that it is impossible for the moon to induce a nutation in the earth's rotation to match the Chandler wobble. Yet, the seasonally reinforced draconic pull leads to an aliasing that is precisely the same value as the Chandler wobble period over the span of many years. Is this just coincidence or is there something that the geophysicists are missing?

It's kind of hard to believe that this would be overlooked, and I have avoided discussing the correlation out of deference to the research literature. Yet the simplification to the ENSO model that a uniform lunisolar forcing would result in shouldn't be dismissed. To quote Clinton, what if

`Last comment I mentioned I was trying to simplify the ENSO model. Right now the forcing is a mix of angular momentum variations related to Chandler wobble and lunisolar tidal pull. This is more complex than I would like to see. So what happens if the Chandler wobble is directly tied to the draconic/nodal cycles in the lunar tide? There is empirical evidence for this even though it is not acknowledged in the consensus geophysics literature. The figure below is my fit to the Chandler wobble, seemingly matching the *aliased* lunar draconic cycle rather precisely: ![cw](http://imagizer.imageshack.us/a/img922/9128/U1BKZz.png) http://contextearth.com/2016/01/27/possible-luni-solar-tidal-mechanism-for-the-chandler-wobble/ The consensus is that it is impossible for the moon to induce a nutation in the earth's rotation to match the Chandler wobble. Yet, the seasonally reinforced draconic pull leads to an aliasing that is precisely the same value as the Chandler wobble period over the span of many years. Is this just coincidence or is there something that the geophysicists are missing? It's kind of hard to believe that this would be overlooked, and I have avoided discussing the correlation out of deference to the research literature. Yet the simplification to the ENSO model that a uniform lunisolar forcing would result in shouldn't be dismissed. To quote Clinton, what if`

In the current research literature, the Chandler wobble is described as an impulse response with a characteristic frequency determined by the earth's ellipticity.

There is a factor known as the Q-value which describes the resonant "quality" of the impulse response, classically defined as the solution to a 2nd-order DiffEq. The higher the Q, the longer the oscillating response. The following figure shows the impulse and response for a fairly low Q-value. It's thought that the Chandler wobble Q-value is very high, as it doesn't seem to damp quickly.

In contrast, ocean tides are not described as a characteristic frequency but instead as a transfer function and a "steady-state" response due to the forcing frequency. The forcing frequency is in fact

carried throughfrom the input stimulus to the output response. In other words, the tidal frequency matches the rhythm of the lunar (and solar) orbital frequency. There may be a transient associated with the natural response but this eventually transitions into the steady-state through the ocean's damping filter as shown below:This behavior is well known in engineering and science circles and explains why the recorded music you listen to is not a resonant squeal but an amplified (and phase-delayed) replica of the input bits.

So why does the Chandler wobble appear close to 433 days instead of the 305 days that Euler predicted? If there was a resonance close to 305 days, any forcing frequency would be amplified in proportion to how close it was to 305 (or larger in Newcomb's non-rigid earth model). Therefore, why can't the aliased draconic lunar forcing cycle of 432.76 days be responsible for the widely accepted Chandler wobble of 433 days?

This is the biannual geometry giving the driving conditions.

And this is the strength of the draconic lunar pull at a sample of two times a year, computed according to the formula cos(2$\pi$/(13.6061/365.242)*t), where 13.6061 days is the lunar draconic fortnight or half the lunar draconic month.

Can count ~127 cycles in 150 years, which places it between 432 and 433 days, which is the Chandler wobble period.

Yet again Wikipedia explains it this way:

Like ENSO and QBO, there is actually no truly accepted model for the Chandler wobble behavior. The one I give here appears just as valid as any of the others. One can't definitely discount it because the lunar draconic period precisely matches the CW period. If it did't match then the hypothesis could be roundly rejected.

And the same goes for the QBO and ENSO models described herein. The aliased lunisolar models match the data nicely in each of those cases as well and so can't easily be rejected. That's why I have been hammering at these models for so long, as a unified theory of lunisolar geophysical forcing is so tantalizingly close -- one for the atmosphere (QBO), the ocean (ENSO), and for the earth itself (Chandler wobble). These three will then unify with the generally accepted theory for ocean tides.

`In the current research literature, the Chandler wobble is described as an impulse response with a characteristic frequency determined by the earth's ellipticity. > https://en.wikipedia.org/wiki/Chandler_wobble "The existence of Earth's free nutation was predicted by Isaac Newton in Corollaries 20 to 22 of Proposition 66, Book 1 of the Philosophiæ Naturalis Principia Mathematica, and by Leonhard Euler in 1765 as part of his studies of the dynamics of rotating bodies. Based on the known ellipticity of the Earth, Euler predicted that it would have a period of 305 days. Several astronomers searched for motions with this period, but none was found. Chandler's contribution was to look for motions at any possible period; once the Chandler wobble was observed, the difference between its period and the one predicted by Euler was explained by Simon Newcomb as being caused by the non-rigidity of the Earth. The full explanation for the period also involves the fluid nature of the Earth's core and oceans .. " There is a factor known as the Q-value which describes the resonant "quality" of the impulse response, classically defined as the solution to a 2nd-order DiffEq. The higher the Q, the longer the oscillating response. The following figure shows the impulse and response for a fairly low Q-value. It's thought that the Chandler wobble Q-value is very high, as it doesn't seem to damp quickly. ![impulse](http://1.bp.blogspot.com/-hBm3S3Gfu3s/T9Ypvnvq_vI/AAAAAAAADxA/awZ12LAIl5Q/s400/singleImpulse-300.png) In contrast, ocean tides are not described as a characteristic frequency but instead as a transfer function and a "steady-state" response due to the forcing frequency. The forcing frequency is in fact *carried through* from the input stimulus to the output response. In other words, the tidal frequency matches the rhythm of the lunar (and solar) orbital frequency. There may be a transient associated with the natural response but this eventually transitions into the steady-state through the ocean's damping filter as shown below: ![ss](http://www.physik.uzh.ch/local/teaching/SPI301/LV-2015-Help/common/GUID-F5E84855-0619-4990-9053-8ADC3DD2A83E-help-web.png) This behavior is well known in engineering and science circles and explains why the recorded music you listen to is not a resonant squeal but an amplified (and phase-delayed) replica of the input bits. So why does the Chandler wobble appear close to 433 days instead of the 305 days that Euler predicted? If there was a resonance close to 305 days, any forcing frequency would be amplified in proportion to how close it was to 305 (or larger in Newcomb's non-rigid earth model). Therefore, why can't the aliased draconic lunar forcing cycle of 432.76 days be responsible for the widely accepted Chandler wobble of 433 days? This is the biannual geometry giving the driving conditions. ![pic](http://imageshack.com/a/img923/9858/ViIcq9.png) And this is the strength of the draconic lunar pull at a sample of two times a year, computed according to the formula cos(2$\pi$/(13.6061/365.242)*t), where 13.6061 days is the lunar draconic fortnight or half the lunar draconic month. ![](http://imageshack.com/a/img922/1644/bI7uGZ.png) Can count ~127 cycles in 150 years, which places it between 432 and 433 days, which is the Chandler wobble period. Yet again Wikipedia explains it this way: > "While it has to be maintained by changes in the mass distribution or angular momentum of the Earth's outer core, atmosphere, oceans, or crust (from earthquakes), for a long time the actual source was unclear, since no available motions seemed to be coherent with what was driving the wobble. One promising theory for the source of the wobble was proposed in 2001 by Richard Gross at the Jet Propulsion Laboratory managed by the California Institute of Technology. He used angular momentum models of the atmosphere and the oceans in computer simulations to show that from 1985 to 1996, the Chandler wobble was excited by a combination of atmospheric and oceanic processes, with the dominant excitation mechanism being ocean‐bottom pressure fluctuations. Gross found that two-thirds of the "wobble" was caused by fluctuating pressure on the seabed, which, in turn, is caused by changes in the circulation of the oceans caused by variations in temperature, salinity, and wind. The remaining third is due to atmospheric fluctuations." Like ENSO and QBO, there is actually no truly accepted model for the Chandler wobble behavior. The one I give here appears just as valid as any of the others. One can't definitely discount it because the lunar draconic period precisely matches the CW period. If it did't match then the hypothesis could be roundly rejected. And the same goes for the QBO and ENSO models described herein. The aliased lunisolar models match the data nicely in each of those cases as well and so can't easily be rejected. That's why I have been hammering at these models for so long, as a unified theory of lunisolar geophysical forcing is so tantalizingly close -- one for the atmosphere (QBO), the ocean (ENSO), and for the earth itself (Chandler wobble). These three will then unify with the generally accepted theory for ocean tides.`

Here is the math on the Chandler wobble. We start with the seasonally-modulated draconic lunar forcing. This has an envelope of a full-wave rectified signal as the moon and sun will show the greatest gravitational pull on the poles during the full northern and southern nodal excursions (i.e. the two solstices). This creates a full period of a 1/2 year.

The effective lunisolar pull is the multiplication of that envelope with the complete cycle draconic month of $2\pi / \omega_0$ =27.2122 days. Because the full-wave rectified signal will create a large number of harmonics, the convolution in the frequency domain of the draconic period with the biannually modulated signal generates spikes at intervals of :

$ 2\omega_0, 2\omega_0-4\pi, 2\omega_0-8\pi, ... 2\omega_0-52\pi $

According to the Fourier series expansion in the figure above, the intensity of the terms will decrease left to right as $1/n^2$, that is with

decreasingfrequency. The last term shown correlates to the Chandler wobble period of 1.185 years = 432.77 days.One would think this decrease in intensity is quite rapid, but because of the resonance condition of the Chandler wobble nutation, a compensating amplification occurs. Here is the frequency response curve of a 2nd-order resonant DiffEq, written in terms of an equivalent electrical RLC circuit.

So, if we choose values for RLC to give a resonance close to 433 days and with a high enough Q-value, then the diminishing amplitude of the Fourier series is amplified by the peak of the nutation response. Note that it doesn't have to match exactly to the peak, but somewhere within the halfwidth, where Q = $\frac{\omega}{\Delta\omega}$

So we see that the original fortnightly period of 13.606 days is retained, but what also emerges is the 13th harmonic of that signal located right at the Chandler wobble period.

That's how a resonance works in the presence of a driving signal. It's not the characteristic frequency that emerges, but the forcing harmonic closest to resonance frequency. And that's how we get the value of 432.77 days for the Chandler wobble. It may not be entirely intuitive but that's the way that the math of the steady-state dynamics works out.

Alas, you won't find this explanation anywhere in the research literature, even though the value of the Chandler wobble has been known since 1891! Apparently no geophysicist will admit that a lunisolar torque can stimulate the wobble in the earth's rotation. I find that mystifying, but maybe I am missing something.

`Here is the math on the Chandler wobble. We start with the seasonally-modulated draconic lunar forcing. This has an envelope of a full-wave rectified signal as the moon and sun will show the greatest gravitational pull on the poles during the full northern and southern nodal excursions (i.e. the two solstices). This creates a full period of a 1/2 year. > ![nodes](http://imageshack.com/a/img924/1315/A00Y47.png) The effective lunisolar pull is the multiplication of that envelope with the complete cycle draconic month of $2\pi / \omega_0$ =27.2122 days. Because the full-wave rectified signal will create a large number of harmonics, the convolution in the frequency domain of the draconic period with the biannually modulated signal generates spikes at intervals of : $ 2\omega_0, 2\omega_0-4\pi, 2\omega_0-8\pi, ... 2\omega_0-52\pi $ According to the Fourier series expansion in the figure above, the intensity of the terms will decrease left to right as $1/n^2$, that is with *decreasing* frequency. The last term shown correlates to the Chandler wobble period of 1.185 years = 432.77 days. One would think this decrease in intensity is quite rapid, but because of the resonance condition of the Chandler wobble nutation, a compensating amplification occurs. Here is the frequency response curve of a 2nd-order resonant DiffEq, written in terms of an equivalent electrical RLC circuit. > ![rlc](http://imageshack.com/a/img922/5107/wwEHLX.png) So, if we choose values for RLC to give a resonance close to 433 days and with a high enough Q-value, then the diminishing amplitude of the Fourier series is amplified by the peak of the nutation response. Note that it doesn't have to match exactly to the peak, but somewhere within the halfwidth, where Q = $\frac{\omega}{\Delta\omega}$ ![rlc_full](http://imageshack.com/a/img922/5184/SazoGh.png) So we see that the original fortnightly period of 13.606 days is retained, but what also emerges is the 13th harmonic of that signal located right at the Chandler wobble period. That's how a resonance works in the presence of a driving signal. It's not the characteristic frequency that emerges, but the forcing harmonic closest to resonance frequency. And that's how we get the value of 432.77 days for the Chandler wobble. It may not be entirely intuitive but that's the way that the math of the steady-state dynamics works out. Alas, you won't find this explanation anywhere in the research literature, even though the value of the Chandler wobble has been known since 1891! Apparently no geophysicist will admit that a lunisolar torque can stimulate the wobble in the earth's rotation. I find that mystifying, but maybe I am missing something.`

Here is an extended twitter thread I had with an expert on climate dynamics concerning QBO. One of my most important arguments is how can a phenomenon with wavenumber = 0 -- i.e. an infinite wavelength -- get started with anything other than a spatially uniform forcing such as provided by the sun or the moon?

`Here is an extended twitter thread I had with an expert on climate dynamics concerning QBO. One of my most important arguments is how can a phenomenon with wavenumber = 0 -- i.e. an infinite wavelength -- get started with anything other than a spatially uniform forcing such as provided by the sun or the moon? ![disc](http://imageshack.com/a/img923/9060/Y68Gok.jpg)`

I've been thinking the history of these behaviors. Pierre-Simon Laplace came up with his tidal equations in 1776. Lord Rayleigh wrote about wave bifurcations around 1880. Ocean tides and their mechanism had been known forever, but only in the 1900's did they have a detailed approach to mathematically define the cycles (i.e. Doodson arguments).

Chandler discovered the earth's polar wobble in 1891. Scientists knew right away that there was a seasonal wobble that was easily explained by a forced factor. But why couldn't they determine the lunar factor?

The QBO was known in the 1950's I think, but only in the 1960's did they have enough data to notice the strong periodicity. If the Chandler wobble mechanism had been known (hypothetically) it would have been simple to adapt that mechanism to QBO through the application of Laplace's equations. The anti-AGW scientist Richard Lindzen spent his career trying to convince everyone of his overly complex model, and now he is left spending his time sending petitions to Trump to have the USA withdraw from the Paris climate accords. Sad.

And ENSO came a little later, with most scientist bewildered by the near chaotic oscillations observed. Yet, if they would have learned from the (hypothetical) models for the Chandler wobble and QBO and the sloshing mechanisms described by Rayleigh, they may have been able to de-convolute the cycles to see once again the lunisolar forcing.

I bring up this history, because I received this comment via Twitter from a well-regarded climate scientist:

The implication is that all current models may be wrong if these simple models of lunar forcing are correct.

`I've been thinking the history of these behaviors. Pierre-Simon Laplace came up with his tidal equations in 1776. Lord Rayleigh wrote about wave bifurcations around 1880. Ocean tides and their mechanism had been known forever, but only in the 1900's did they have a detailed approach to mathematically define the cycles (i.e. Doodson arguments). Chandler discovered the earth's polar wobble in 1891. Scientists knew right away that there was a seasonal wobble that was easily explained by a forced factor. But why couldn't they determine the lunar factor? The QBO was known in the 1950's I think, but only in the 1960's did they have enough data to notice the strong periodicity. If the Chandler wobble mechanism had been known (hypothetically) it would have been simple to adapt that mechanism to QBO through the application of Laplace's equations. The anti-AGW scientist Richard Lindzen spent his career trying to convince everyone of his overly complex model, and now [he is left spending his time sending petitions to Trump to have the USA withdraw from the Paris climate accords](http://business.financialpost.com/fp-comment/lawrence-solomon-scientists-urging-trump-to-embrace-carbon-among-the-biggest-climate-experts-around). Sad. And ENSO came a little later, with most scientist bewildered by the near chaotic oscillations observed. Yet, if they would have learned from the (hypothetical) models for the Chandler wobble and QBO and the sloshing mechanisms described by Rayleigh, they may have been able to de-convolute the cycles to see once again the lunisolar forcing. I bring up this history, because I received this comment via Twitter from a well-regarded climate scientist: ![roundy](http://imageshack.com/a/img922/7651/4pib6v.png) The implication is that all current models may be wrong if these simple models of lunar forcing are correct.`

I don't quite understand why you got the impression that this lunar forcing wasn't taken into account:

From Wikipedia:

`>Chandler discovered the earth's polar wobble in 1891. Scientists knew right away that there was a seasonal wobble that was easily explained by a forced factor. But why couldn't they determine the lunar factor? I don't quite understand why you got the impression that this lunar forcing wasn't taken into account: <a href="https://en.wikipedia.org/wiki/Nutation#Earth">From Wikipedia:</a> >In the case of the Earth, the principal sources of tidal force are the Sun and Moon, which continuously change location relative to each other and thus cause nutation in Earth's axis. The largest component of Earth's nutation has a period of 18.6 years, the same as that of the precession of the Moon's orbital nodes.[1] However, there are other significant periodic terms that must be accounted for depending upon the desired accuracy of the result. A mathematical description (set of equations) that represents nutation is called a "theory of nutation".`

What's "deposition"?

`What's "deposition"?`

Yes, of course the 18.6 year nodal precession is mentioned. But this is in reference to a specific longitude. The Chandler wobble is independent of longitude and should respond to periods at which the moon crosses the equator through a complete ascending/descending cycle.

That cycle is the draconic month, which is 27.2122 days. The tropical month is 27.3216 days, which is the length of time the moon takes to appear at the same longitude. Those two are slightly different and the difference forms a beat cycle that determines how often the maximum declination is reached

for a particular longitude. That is important for ocean tides as tides are really a localized phenomenon.But the Chandler wobble is global and would only have triaxial components to second order. It doesn't really care where the moon is located in longitude when it reaches a maximum in declination excursion. Consider that at the poles, the longitude converges at a singularity. It's the physics of a spinning top under the influence of a gravity vector.

So the important point is that the moon is cycling a significant gravitational torque on the earth's axis every 1/2 a draconic period. When this is reinforced by the sun's biannual cycle, one gets the Chandler wobble period precisely.

Try Googling "Draconic AND Chandler wobble". You won't find anything. Sure you will find the 18.6 year cycle but that's because of an echo chamber of misleading information. This is important to understand, and if my own reasoning is faulty, this model I have been pushing for the Chandler wobble and QBO collapses. Those are both global behaviors. ENSO, on the other hand, has a longitudinally localized forcing and should show strong effects of the 18.6 year cycle ... and I have shown it does!

`Yes, of course the 18.6 year nodal precession is mentioned. But this is in reference to a specific longitude. The Chandler wobble is independent of longitude and should respond to periods at which the moon crosses the equator through a complete ascending/descending cycle. That cycle is the draconic month, which is 27.2122 days. The tropical month is 27.3216 days, which is the length of time the moon takes to appear at the same longitude. Those two are slightly different and the difference forms a beat cycle that determines how often the maximum declination is reached *for a particular longitude*. That is important for ocean tides as tides are really a localized phenomenon. But the Chandler wobble is global and would only have triaxial components to second order. It doesn't really care where the moon is located in longitude when it reaches a maximum in declination excursion. Consider that at the poles, the longitude converges at a singularity. It's the physics of a spinning top under the influence of a gravity vector. So the important point is that the moon is cycling a significant gravitational torque on the earth's axis every 1/2 a draconic period. When this is reinforced by the sun's biannual cycle, one gets the Chandler wobble period precisely. Try Googling "Draconic AND Chandler wobble". You won't find anything. Sure you will find the 18.6 year cycle but that's because of an echo chamber of misleading information. This is important to understand, and if my own reasoning is faulty, this model I have been pushing for the Chandler wobble and QBO collapses. Those are both global behaviors. ENSO, on the other hand, has a longitudinally localized forcing and should show strong effects of the 18.6 year cycle ... and I have shown it does! ![lunar](https://imagizer.imageshack.us/v2/320xq90/r/924/zGZPS8.png)`

Jim asks

In crystal growth, which was my academic specialty, deposition is used in the context of the rate of growth. As in depositing layers of material.

Honestly, I really can't follow much of the jargon of these meteorologists. Either they really have the insight and really know what's going on, or they are blowing smoke and using the jargon to cover up the lack of their own understanding.

The problem with Roundy is that he can't own up to the fact that QBO has a spatial wavenumber of zero, which makes it a global phenomenon. All the points he makes regarding deposition are spatially dependent, which makes them irrelevant to forcing a zero wavenumber. This is just group theory and symmetry arguments I am applying, which he refuses to consider apparently.

That's my take. If I try to understand his deposition arguments, I am certain to go down the rabbit hole and end up following the same route that Lindzen took with QBO -- one of unwarranted complexity.

`Jim asks > What is "deposition"? In crystal growth, which was my academic specialty, deposition is used in the context of the rate of growth. As in depositing layers of material. Honestly, I really can't follow much of the jargon of these meteorologists. Either they really have the insight and really know what's going on, or they are blowing smoke and using the jargon to cover up the lack of their own understanding. The problem with Roundy is that he can't own up to the fact that QBO has a spatial wavenumber of zero, which makes it a global phenomenon. All the points he makes regarding deposition are spatially dependent, which makes them irrelevant to forcing a zero wavenumber. This is just group theory and symmetry arguments I am applying, which he refuses to consider apparently. That's my take. If I try to understand his deposition arguments, I am certain to go down the rabbit hole and end up following the same route that Lindzen took with QBO -- one of unwarranted complexity.`

Here is part 2 of a conversation on QBO with Roundy from yesterday. I tend to interrupt these meteorology Twitter threads by throwing out comments relating to what I am finding. These arguments usually play out by this analogy: Say that someone like Roundy was analyzing an electrical signal. I am simply pointing out that there is an obvious 60 Hz hum that is emerging in the signal, and I am suggesting a possible source of that signal. Then Roundy responds with that can't happen because there are other sources of noise in there as well. I come back and say that those other noise sources aren't going to eliminate the 60 Hz hum. Then he comes back and asserts that it is more complicated than that, because if that was the case, all the models would fail.

`Here is part 2 of a conversation on QBO with Roundy from yesterday. I tend to interrupt these meteorology Twitter threads by throwing out comments relating to what I am finding. These arguments usually play out by this analogy: Say that someone like Roundy was analyzing an electrical signal. I am simply pointing out that there is an obvious 60 Hz hum that is emerging in the signal, and I am suggesting a possible source of that signal. Then Roundy responds with that can't happen because there are other sources of noise in there as well. I come back and say that those other noise sources aren't going to eliminate the 60 Hz hum. Then he comes back and asserts that it is more complicated than that, because if that was the case, all the models would fail. ![ROUNDY](http://imageshack.com/a/img922/6186/xDiJo6.jpg)`

Question deleted. I'd forgotten that you're "whut".

`Question deleted. I'd forgotten that you're "whut".`

Wikipedia writes:

So by this definition thats the longitude with respect to the equinoxes. I don't know how the equinoxes move with respect to geographic longitudes. I had somewhere else seen the definition that "tropical" means "with respect to the earth reference system", but at the moment I don't see quickly that the geographic longitudes should be the same as the longitudes with respect to the equinoxes.

There might be various reasons for that is first -I don't know- but I would think that nutation theory uses coordinate systems and not those rather confusing definitions about sidereal, draconic, tropical etc. secondly you always have to take into account that your google bubble might be ill-adjusted to your purpose.

`>That cycle is the draconic month, which is 27.2122 days. The tropical month is 27.3216 days, which is the length of time the moon takes to appear at the same longitude. <a href="https://en.wikipedia.org/wiki/Lunar_month#Tropical_month">Wikipedia writes:</a> >It is customary to specify positions of celestial bodies with respect to the vernal equinox. Because of Earth's precession of the equinoxes, this point moves back slowly along the ecliptic. Therefore, it takes the Moon less time to return to an ecliptic longitude of zero than to the same point amidst the fixed stars: 27.321582 days (27 d 7 h 43 min 4.7 s). So by this definition thats the longitude with respect to the equinoxes. I don't know how the equinoxes move with respect to geographic longitudes. I had somewhere else seen the definition that "tropical" means "with respect to the earth reference system", but at the moment I don't see quickly that the geographic longitudes should be the same as the longitudes with respect to the equinoxes. >Try Googling "Draconic AND Chandler wobble". You won't find anything. There might be various reasons for that is first -I don't know- but I would think that nutation theory uses coordinate systems and not those rather confusing definitions about sidereal, draconic, tropical etc. secondly you always have to take into account that your google bubble might be ill-adjusted to your purpose.`

You seem to have a mental block. Maybe I can find a diagram.

`You seem to have a mental block. Maybe I can find a diagram.`

+1 I was about to search for a decent animation!

`+1 I was about to search for a decent animation!`

This is a great animation. At the two minute mark the Draconic month is highlighted

`This is a great animation. At the two minute mark the Draconic month is highlighted https://youtu.be/jWCBhVfeAQU`

Maximum combined torque on the axis occurs when the draconic month maximum declination aligns with a solstice. This happens at precisely the same frequency as the Chandler wobble. If this is not an additive stimulus and is simply a coincidence, I don't know what to say.

The Devil's Advocate View: The Chandler wobble, QBO, ENSO periods are all coincidences with respect to lunisolar cycles apparently.`Maximum combined torque on the axis occurs when the draconic month maximum declination aligns with a solstice. This happens at precisely the same frequency as the Chandler wobble. If this is not an additive stimulus and is simply a coincidence, I don't know what to say. *The Devil's Advocate View* : The Chandler wobble, QBO, ENSO periods are all coincidences with respect to lunisolar cycles apparently.`

In trying to isolate the Chandler wobble mechanism, Grumbine seems to agree that it has something to do with external forcing, but rules out lunar because books on the earth's rotation by Munk and Lambeck rules that mechanism out.

I can see this hypothetically if the earth was a perfect sphere, since there is no moment of inertia to torque against. But the earth is not a perfect sphere, so triaxial moments exist. It is also well-known and not contested that the moon will perturb the Earth's rotation rate based on decades worth of LOD (length-of-day) measurements. Changes in LOD perfectly align with tidal periods.

Yet the only mechanism acknowledged for

forcedprecession is this very long period torqueThis is the beat difference between 365.24219 and 365.25636 days, the tropical and sidereal years. Every 26,000 years an extra tropical year is gained.

I think they just overlooked the possibility of the lunar month physically aliasing with the yearly forcing creating the same period as the Chandler wobble. The lunar month was rejected long ago because obviously the periods don't match. But they match if the seasonal aliasing math is done correctly.

`In trying to isolate the Chandler wobble mechanism, Grumbine seems to agree that it has something to do with external forcing, but rules out lunar because books on the earth's rotation by Munk and Lambeck rules that mechanism out. > "Gravitational torques have been examined previously as the main driver of the Chandler Wobble and rejected [**Munk and MacDonald , 1960; Lambeck , 1980**], which means only non-gravitational external forces, such as earth-sun distance, force Chandler Wobble at these periods, if any external sources do." http://moregrumbinescience.blogspot.com/2016/01/earth-sun-distance-and-chandler-wobble.html I can see this hypothetically if the earth was a perfect sphere, since there is no moment of inertia to torque against. But the earth is not a perfect sphere, so triaxial moments exist. It is also well-known and not contested that the moon will perturb the Earth's rotation rate based on decades worth of LOD (length-of-day) measurements. Changes in LOD perfectly align with tidal periods. Yet the only mechanism acknowledged for *forced* precession is this very long period torque > "The free precession of the Earth's symmetry axis in space, which is known as the Chandler wobble--because it was discovered by the American astronomer S.C. Chandler (1846-1913) in 1891--is superimposed on a much slower forced precession, with a period of about 26,000 years, caused by the small gravitational torque exerted on the Earth by the Sun and Moon, as a consequence of the Earth's slight oblateness." https://farside.ph.utexas.edu/teaching/celestial/Celestial/node72.html This is the beat difference between 365.24219 and 365.25636 days, the tropical and sidereal years. Every 26,000 years an extra tropical year is gained. I think they just overlooked the possibility of the lunar month physically aliasing with the yearly forcing creating the same period as the Chandler wobble. The lunar month was rejected long ago because obviously the periods don't match. But they match if the seasonal aliasing math is done correctly.`

There's a report out from NASA today suggesting that the EN piece of ENSO might have some hysteresis: https://www.jpl.nasa.gov/news/news.php?feature=6776

This raises questions of (a) what shuts it down if there's heat left over? and (b) what starts it up again?

`There's a report out from NASA today suggesting that the EN piece of ENSO might have some hysteresis: https://www.jpl.nasa.gov/news/news.php?feature=6776 This raises questions of (a) what shuts it down if there's heat left over? and (b) what starts it up again?`

Richard Lindzen, the "father" of QBO modeling, is at it again with his anti-AGW crusade. He recently sent in a petition to Trump asking to withdraw from the Paris agreement. But now his MIT colleagues are fighting back:

Lindzen seems to be a pariah among his colleagues. This is an interesting piece from years ago

Note the condescending tone of Lindzen in the last sentences. Who is willing to bet that Lindzen's theory of QBO is fundamentally wrong? Lindzen is so sure of himself and his talents that he probably declared victory on his QBO model long before it was ready for prime-time.

Lindzen likely manufactured his own consensus around QBO, and was able to make a convincing case based on his credentials.

`Richard Lindzen, the "father" of QBO modeling, is at it again with his anti-AGW crusade. He recently sent in a petition to Trump asking to withdraw from the Paris agreement. But now his MIT colleagues are fighting back: >"Their letter, sent last week, was drafted in response to a letter that Lindzen sent Trump last month, urging him to withdraw the United States from the international climate accord signed in Paris in 2015. That agreement, signed by nearly 200 nations, seeks to curb the greenhouse gases linked to global warming." https://www.bostonglobe.com/metro/2017/03/08/mit-professors-denounce-their-colleague-letter-trump-for-denying-evidence-climate-change/86K8ur31YIUbMO4SAI7U2N/story.html Lindzen seems to be a pariah among his colleagues. This is an interesting piece from years ago > http://www.washingtonpost.com/wp-dyn/content/article/2006/05/23/AR2006052301305.html > “Of all the skeptics, MIT’s Richard Lindzen probably has the most credibility among mainstream scientists, who acknowledge that he’s doing serious research on the subject. ” >“When I ask (William) Gray who his intellectual soul mates are regarding global warming, he responds, “I have nobody really to talk to about this stuff.” > That’s not entirely true. He has many friends and colleagues, and the meteorologists tend to share his skeptical streak. > I ask if he has ever collaborated on a paper with Richard Lindzen. Gray says he hasn’t. He looks a little pained. “Lindzen, he’s a hard guy to deal with,” Gray says. “He doesn’t think he can learn anything from me.” Which is correct. Lindzen says of Gray: “His knowledge of theory is frustratingly poor, but he knows more about hurricanes than anyone in the world. I regard him in his own peculiar way as a national resource.” Note the condescending tone of Lindzen in the last sentences. Who is willing to bet that Lindzen's theory of QBO is fundamentally wrong? Lindzen is so sure of himself and his talents that he probably declared victory on his QBO model long before it was ready for prime-time. > "There're people like [Lindzen] in every field of science. There are always people in the fringes. They're attracted to the fringe . . . It may be as simple as, how do you prove you're smarter than everyone else? You don't do that by being part of the consensus," (Isaac) Held says. Lindzen likely manufactured his own consensus around QBO, and was able to make a convincing case based on his credentials.`

I will be finishing up the basic research on my ENSO model soon. I don't have many loose ends left and what I presented at the AGU is standing the test of time. The Chandler wobble is tied to the aliased draconic month cycle so I can essentially get perfect agreement by applying Laplace's tidal equations with the known seasonally reinforced lunar forcing. The ENSO fit then has only amplitude and phase unknowns with regard to the 3 lunar monthly cycles, making it conceptually identical to an ocean tidal model fit. The complexity is similar to the ocean tidal setup with the 3 monthly cycles combining with the 2 seasonal cycles (yearly and biannual) to create 18 linear and nonlinear interactions.

I performed a long running fit to the ENSO time series by allowing the lunar cycle periods to vary and then waiting for it to converge to steady state values.

Draconic month (strongest) should be 27.2122 days Anomalistic month should be 27.5545 Sidereal month is 27.3216

What the model converges to is 27.2120 days for Draconic 27.5580 for Anomalistic 27.3259 for Sidereal

These errors are less than a minute per month in the case of the Draconic and 5 and 6 minutes for the other two. The latter error contributes less than a quarter of a lunar month phase shift over the 130 year ENSO interval.

`I will be finishing up the basic research on my ENSO model soon. I don't have many loose ends left and what I presented at the AGU is standing the test of time. The Chandler wobble is tied to the aliased draconic month cycle so I can essentially get perfect agreement by applying Laplace's tidal equations with the known seasonally reinforced lunar forcing. The ENSO fit then has only amplitude and phase unknowns with regard to the 3 lunar monthly cycles, making it conceptually identical to an ocean tidal model fit. The complexity is similar to the ocean tidal setup with the 3 monthly cycles combining with the 2 seasonal cycles (yearly and biannual) to create 18 linear and nonlinear interactions. I performed a long running fit to the ENSO time series by allowing the lunar cycle periods to vary and then waiting for it to converge to steady state values. Draconic month (strongest) should be 27.2122 days Anomalistic month should be 27.5545 Sidereal month is 27.3216 What the model converges to is 27.2120 days for Draconic 27.5580 for Anomalistic 27.3259 for Sidereal These errors are less than a minute per month in the case of the Draconic and 5 and 6 minutes for the other two. The latter error contributes less than a quarter of a lunar month phase shift over the 130 year ENSO interval.`

Previous comment I wrote:

Here is an example of the detail I can get with the ENSO model.

The ENSO model is able to discern the variation in the length (and phase) of the lunar Draconic month (see here https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html#draconic). This is a detailed second-order effect that would only be possible to measure if the model of the first-order effect is correct.

So the AGU-2016 ENSO model is based completely on the long-period lunar tidal cycles reinforced by seasonal cycle impulses. The same numerical techniques used for modeling ocean tides are applied, but with a different concept for seasonal reinforcing. The model therefore has gone from being an explanation of the ENSO behavior to a sensitive metrology technique -- specifically for measuring lunar and solar cycles based on the sloshing sensitivity of a layered fluid medium to forcing changes.

That's one way you substantiate a model's veracity -- flip it from a question of over-fitting to one of precisely identifying physical constants. That's one way to get buy-in.

`Previous comment I wrote: > "The ENSO fit then has only amplitude and phase unknowns with regard to the 3 lunar monthly cycles, making it conceptually identical to an ocean tidal model fit. " Here is an example of the detail I can get with the ENSO model. ![drac](http://imageshack.com/a/img923/1269/GhdGtU.png) The ENSO model is able to discern the variation in the length (and phase) of the lunar Draconic month (see here https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html#draconic). This is a detailed second-order effect that would only be possible to measure if the model of the first-order effect is correct. So the AGU-2016 ENSO model is based completely on the long-period lunar tidal cycles reinforced by seasonal cycle impulses. The same numerical techniques used for modeling ocean tides are applied, but with a different concept for seasonal reinforcing. The model therefore has gone from being an explanation of the ENSO behavior to a sensitive [metrology](https://en.wikipedia.org/wiki/Metrology) technique -- specifically for measuring lunar and solar cycles based on the sloshing sensitivity of a layered fluid medium to forcing changes. That's one way you substantiate a model's veracity -- flip it from a question of over-fitting to one of precisely identifying physical constants. That's one way to get buy-in.`

This is my description for a canonical model for ENSO http://contextEarth.com/2017/04/10/tidal-model-of-enso/

`This is my description for a canonical model for ENSO http://contextEarth.com/2017/04/10/tidal-model-of-enso/`

In the last post, I added a link to a downloadable spreadsheet where one can play with the ENSO solver.

Interesting what it can do with very short training intervals. This is one from 1908 to 1920, a short training interval of only 12 years. You would think that it would overfit, yet the extrapolated out-of-band model looks reasonable and not at all chaotic

`In the last post, I added a link to a downloadable spreadsheet where one can play with the ENSO solver. Interesting what it can do with very short training intervals. This is one from 1908 to 1920, a short training interval of only 12 years. You would think that it would overfit, yet the extrapolated out-of-band model looks reasonable and not at all chaotic ![train](http://imageshack.com/a/img924/6987/URH1w1.png)`

Also added the QBO model to the spreadsheet. Essentially the same model fitting algorithm is used for both ENSO and QBO, but using a different set of forcing functions. For ENSO, the draconic and anomalistic tide is modulated with a biennial signal, while QBO modulates only the draconic with an annual signal.

`Also added the QBO model to the spreadsheet. Essentially the same model fitting algorithm is used for both ENSO and QBO, but using a different set of forcing functions. For ENSO, the draconic and anomalistic tide is modulated with a biennial signal, while QBO modulates only the draconic with an annual signal. ![qbo](http://imageshack.com/a/img923/8453/z8RdQL.png)`

Update to #141. Forcing with a 1st-order lag and an opposing feedback from one year prior. This serves to smooth the ENSO model fit (in contrast to the blocky one-year window integration) and to reinforce the biennial modulation in the lunar tidal forcing.

Like the QBO model, this is a remarkable result showing the ability of a very short interval of the time series to recover the rest of the behavior. And like the QBO model, there are so few degrees of freedom in the fitting parameters that over-fitting is not an issue. No remaining kludges in the model and the only ansatz is the assumption of a metastable biennial modulation. Yet, this is by all accounts a result of a year-to-year compensating feedback leading to a period doubling (and which Rayleigh discovered in his maintained vibrations paper )

http://contextearth.com/2017/04/18/shortest-training-fit-for-enso/

`Update to #141. Forcing with a 1st-order lag and an opposing feedback from one year prior. This serves to smooth the ENSO model fit (in contrast to the blocky one-year window integration) and to reinforce the biennial modulation in the lunar tidal forcing. ![enso](http://imagizer.imageshack.us/a/img923/8048/jKp07B.png) Like the QBO model, this is a remarkable result showing the ability of a very short interval of the time series to recover the rest of the behavior. And like the QBO model, there are so few degrees of freedom in the fitting parameters that over-fitting is not an issue. No remaining kludges in the model and the only ansatz is the assumption of a metastable biennial modulation. Yet, this is by all accounts a result of a year-to-year compensating feedback leading to a period doubling (and which Rayleigh discovered in his [maintained vibrations paper](http://www.tandfonline.com/doi/abs/10.1080/14786448308627342?journalCode=tphm16) ) http://contextearth.com/2017/04/18/shortest-training-fit-for-enso/`

On this Earth Day, I present the final touches on the ENSO model, which uses only the primary lunar cycles and the yearly period. The training goes from 1880 to 1950, and everything after that is projected:

More info here http://contextEarth.com/2017/04/21/canonical-solution-of-mathieu-equation-for-enso/.

Didn't go to a #MarchForScience, but instead am doing this kind of stuff.

Whut's more important?

`On this Earth Day, I present the final touches on the ENSO model, which uses only the primary lunar cycles and the yearly period. The training goes from 1880 to 1950, and everything after that is projected: ![enso](http://imageshack.com/a/img924/9420/2go1p5.png) More info here http://contextEarth.com/2017/04/21/canonical-solution-of-mathieu-equation-for-enso/. Didn't go to a #MarchForScience, but instead am doing this kind of stuff. Whut's more important?`

Given the two fitting intervals, a low range from 1880 to 1950 and a high range 1950 to 2016, we can compare the resultant parameters.

The strongest lunar parameters are the D=Draconic and A=Anomalistic periods. The higher-order parameters $D^n$ and $A^n$ also align as does a cross-term $D\cdot A^2$. The $D\cdot A$ cross-term is negligible.

One thing you will notice is that the overall amplitude is different on the two axis. That has to do with the error metric used - optimizing WRT a correlation coefficient does not preserve the absolute scale. It does take a few minutes to do the optimizing fit on each range but the resultant alignment is correlated above 0.99.

`Given the two fitting intervals, a low range from 1880 to 1950 and a high range 1950 to 2016, we can compare the resultant parameters. ![compare](http://imageshack.com/a/img923/905/cWg8dM.png) The strongest lunar parameters are the D=Draconic and A=Anomalistic periods. The higher-order parameters $D^n$ and $A^n$ also align as does a cross-term $D\cdot A^2$. The $D\cdot A$ cross-term is negligible. One thing you will notice is that the overall amplitude is different on the two axis. That has to do with the error metric used - optimizing WRT a correlation coefficient does not preserve the absolute scale. It does take a few minutes to do the optimizing fit on each range but the resultant alignment is correlated above 0.99.`

This is as good as any tidal analysis I have ever seen. Notice the perturbation right around 1980 is now gone. It turns out that may have been a red herring, as the past fitting routines were missing the secret ingredient.

This is a short summary I wrote to respond to someone:

In fact there is a magical differential equation that is part Mathieu equation and part delay-differential that captures the sloshing dynamics perfectly. Mathieu equations are favored by hydrodynamics engineers that study sloshing of volumes of water, while delay-differential equations are favored by ocean climatologists studying ENSO. So I figured why not put the two together and try to solve it on a spreadsheet?

Of course a magical equation needs magical ingredients, so as stimulus I provided the only known forcing that could plausibly cause the thermocline sloshing -- the lunar long-period tidal pull. These have to be exact in period and phase or else the signals will destructively interfere over many years.

It looks as if the noise in the ENSO signal is minimal. Just about any interval of the time-series can reconstruct any other interval. That's the hallmark of an ergodic stationary process with strong deterministic properties. It's essentially what oceanographers do with conventional tidal analysis -- train an interval of measured sea-level height gauge measurements against the known lunar periods and out pops an extrapolated tidal prediction algorithm. Basically what this infers is that Curry and Tsonis and the other deniers are flat wrong when they say that climate change on this multidecadal scale is chaotic.

Perhaps the climate is only complex when dealing with vortices, i.e. hurricanes, etc. A standing wave behavior like ENSO is not a vortex and it has a chance to be simplified. Anyone that has done any physics has learned this from their undergrad classes. And the QBO behavior may be an anti-vortex type of standing wave, which also can be simplified.

Yet, who knows if various other vortex patterns can't at least be partially simplified. I spent some time discussing with a poster presenter at last year's AGU why he was looking at analyzing jet-stream patterns at higher latitudes while the behavior at the equator (i.e. the QBO) has a much better chance of being simplified. And then that could be used for evaluating higher latitude behavior as a stimulus. Recall that these vortices peel off the equator before developing into larger patterns. He didn't have a good answer.

`This is as good as any tidal analysis I have ever seen. Notice the perturbation right around 1980 is now gone. It turns out that may have been a red herring, as the past fitting routines were missing the secret ingredient. This is a short summary I wrote to respond to someone: In fact there is a magical differential equation that is part <a href="http://mathworld.wolfram.com/MathieuDifferentialEquation.html" rel="nofollow">Mathieu equation</a> and part <a href="https://en.wikipedia.org/wiki/Delay_differential_equation" rel="nofollow">delay-differential</a> that captures the sloshing dynamics perfectly. Mathieu equations are favored by hydrodynamics engineers that study sloshing of volumes of water, while delay-differential equations are favored by ocean climatologists studying ENSO. So I figured why not put the two together and try to solve it on a spreadsheet? <br><br>Of course a magical equation needs magical ingredients, so as stimulus I provided the only known forcing that could plausibly cause the thermocline sloshing -- the lunar long-period tidal pull. These have to be exact in period and phase or else the signals will destructively interfere over many years. ![enso](http://imageshack.com/a/img924/9420/2go1p5.png) It looks as if the noise in the ENSO signal is minimal. Just about any interval of the time-series can reconstruct any other interval. That's the hallmark of an ergodic stationary process with strong deterministic properties. It's essentially what oceanographers do with conventional tidal analysis -- train an interval of measured sea-level height gauge measurements against the known lunar periods and out pops an extrapolated tidal prediction algorithm. Basically what this infers is that Curry and Tsonis and the other deniers are flat wrong when they say that climate change on this multidecadal scale is chaotic. Perhaps the climate is only complex when dealing with vortices, i.e. hurricanes, etc. A standing wave behavior like ENSO is not a vortex and it has a chance to be simplified. Anyone that has done any physics has learned this from their undergrad classes. And the QBO behavior may be an anti-vortex type of standing wave, which also can be simplified. Yet, who knows if various other vortex patterns can't at least be partially simplified. I spent some time discussing with a poster presenter at last year's AGU why he was looking at analyzing jet-stream patterns at higher latitudes while the behavior at the equator (i.e. the QBO) has a much better chance of being simplified. And then that could be used for evaluating higher latitude behavior as a stimulus. Recall that these vortices peel off the equator before developing into larger patterns. He didn't have a good answer.`

Incredibly, the ENSO model is able to detect the Draconic and Anomalistic lunar month values to within ~30 seconds out of ~27 days.

http://contextEarth.com/2017/05/01/the-enso-model-turns-into-a-metrology-tool/

The huge lever arm of a 130+ year ENSO record helps enormously to be able to resolve the value that precisely. Any other values would cause destructive interference over that interval.

`Incredibly, the ENSO model is able to detect the Draconic and Anomalistic lunar month values to within ~30 seconds out of ~27 days. http://contextEarth.com/2017/05/01/the-enso-model-turns-into-a-metrology-tool/ The huge lever arm of a 130+ year ENSO record helps enormously to be able to resolve the value that precisely. Any other values would cause destructive interference over that interval.`

The ENSO model is essentially two known (monthly) tidal parameters and one (annual) solar parameter provided as a RHS forcing to a primitive differential equation with LHS parameters that are roughly estimated.

Once these 5 LHS parameters are estimated from trial-and-error experiments shat show the greatest stationarity over the entire interval, the magnitudes and phases of the RHS parameters are discovered by fitting to any interval within the ENSO time series

That is a essentially an extreme over-fit to the interval covering the range 1900 to 1920. Normally this amount of over-fitting will not extrapolate well, yet this does. The model probably will end up working as well as any conventional tidal analysis, especially considering the maturity of this model.

`The ENSO model is essentially two known (monthly) tidal parameters and one (annual) solar parameter provided as a RHS forcing to a primitive differential equation with LHS parameters that are roughly estimated. Once these 5 LHS parameters are estimated from trial-and-error experiments shat show the greatest stationarity over the entire interval, the magnitudes and phases of the RHS parameters are discovered by fitting to any interval within the ENSO time series ![1900](http://imageshack.com/a/img924/7921/4KVw6j.png) That is a essentially an extreme over-fit to the interval covering the range 1900 to 1920. Normally this amount of over-fitting will not extrapolate well, yet this does. The model probably will end up working as well as any conventional tidal analysis, especially considering the maturity of this model.`

This is a straightforward validation of the ENSO model presented at last December's AGU.

What I did was use the modern instrumental record of ENSO — the NINO34 data set — as a training interval, and then tested across the historical coral proxy record — the UEP data set.

The correlation coefficient in the out-of-band region of 1650 to 1880 is excellent, considering that only two RHS lunar periods (draconic and anomalistic month) are used for forcing. As a matter of fact, trying to get any kind of agreement with the UEP using an arbitrary set of sine waves is problematic as the time-series appears nearly chaotic and thus requires may Fourier components to fit. With the ENSO model in place, the alignment with the data is automatic. It predicts the strong El Nino in 1877-1878 and then nearly everything before that.

http://contextearth.com/2017/05/12/enso-proxy-validation/

This is an expanded view of the proxy agreement. Now the ENSO proxy is in red with squares showing the yearly readings. It would be nice to get sub-year resolution but that will never happen with yearly-growth-ring data.

`This is a straightforward validation of the ENSO model presented at last December's AGU. What I did was use the modern instrumental record of ENSO — the NINO34 data set — as a training interval, and then tested across the historical coral proxy record — the UEP data set. ![proxy](http://imagizer.imageshack.us/a/img923/3825/bhFYEn.png) The correlation coefficient in the out-of-band region of 1650 to 1880 is excellent, considering that only two RHS lunar periods (draconic and anomalistic month) are used for forcing. As a matter of fact, trying to get any kind of agreement with the UEP using an arbitrary set of sine waves is problematic as the time-series appears nearly chaotic and thus requires may Fourier components to fit. With the ENSO model in place, the alignment with the data is automatic. It predicts the strong El Nino in 1877-1878 and then nearly everything before that. http://contextearth.com/2017/05/12/enso-proxy-validation/ This is an expanded view of the proxy agreement. Now the ENSO proxy is in red with squares showing the yearly readings. It would be nice to get sub-year resolution but that will never happen with yearly-growth-ring data. ![exp](http://imagizer.imageshack.us/a/img924/8849/loYuQ4.png)`

http://contextearth.com/2017/05/15/enso-and-noise/

This is a great validation of the ENSO model.

`http://contextearth.com/2017/05/15/enso-and-noise/ This is a great validation of the ENSO model.`