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## Comments

Abundant data exists for ENSO before 1880 in the form of proxies from coral ring measurements. The data set called the Universal ENSO Proxy (UEP) has been calibrated up to 1978 against the direct ENSO measurements, so there is at least some confidence that data going back to 1650 reflects the historical ENSO behavior.

I can use the proxy data to help with model verification, as the current formulation can be extrapolated in the forward and reverse directions. The only drawback is that the UEP proxy is only a yearly measurement so that biennial oscillations are right at the Nyquist limit of detectability. If a 2 year period sinusoid was detected only at the every-year zero crossings, then a zero amplitude signal would result, even though the actual data may have a large biennial peak-to-peak signal. Further, I can't afford to do any filtering on the data, as the yearly sampling is doing enough filtering-damage as it is.

Given that, the results of back-extrapolating a model fit from 1880-1978 to the year 1650 shows high significance with respect to the data. The in-band fit correlation coefficient is 0.78 and the out-of-band is 0.28.

The key thing to consider with extended fits is that for a given sinusoidal factor, it is very easy for the phase to start slipping unless the frequencies are known with some certainty. A fit that works over a 100 year span with a slip of one radian will grow to 3 radians over 300 years, and 3 radians is enough to cause a phase inversion in the model to data match.

The other factor to consider is the possibility of phase inversions in the data itself, such as the inferred inversion from 1981 to 1996 in the modern-day ENSO records. So it an inversion does happen along some interval, the fact that the resolution is only one year may make it difficult to detect.

`Abundant data exists for ENSO before 1880 in the form of proxies from coral ring measurements. The data set called the Universal ENSO Proxy (UEP) has been calibrated up to 1978 against the direct ENSO measurements, so there is at least some confidence that data going back to 1650 reflects the historical ENSO behavior. I can use the proxy data to help with model verification, as the current formulation can be extrapolated in the forward and reverse directions. The only drawback is that the UEP proxy is only a yearly measurement so that biennial oscillations are right at the Nyquist limit of detectability. If a 2 year period sinusoid was detected only at the every-year zero crossings, then a zero amplitude signal would result, even though the actual data may have a large biennial peak-to-peak signal. Further, I can't afford to do any filtering on the data, as the yearly sampling is doing enough filtering-damage as it is. Given that, the results of back-extrapolating a model fit from 1880-1978 to the year 1650 shows high significance with respect to the data. The in-band fit correlation coefficient is 0.78 and the out-of-band is 0.28. ![UEP](http://imageshack.com/a/img908/7884/pYBhqr.png) The key thing to consider with extended fits is that for a given sinusoidal factor, it is very easy for the phase to start slipping unless the frequencies are known with some certainty. A fit that works over a 100 year span with a slip of one radian will grow to 3 radians over 300 years, and 3 radians is enough to cause a phase inversion in the model to data match. The other factor to consider is the possibility of phase inversions in the data itself, such as the inferred inversion from 1981 to 1996 in the modern-day ENSO records. So it an inversion does happen along some interval, the fact that the resolution is only one year may make it difficult to detect.`

Trying to determine how significant the back-extrapolated fit in #151 is.

If I slide the time series by a few years left and right with respect to the model, the correlation decreases both in the in-band (20th century data) and out-of-band (>16th century data)

If this was a spurious random correlation (given the CC is only 0.28), then I wouldn't expect the peak to align at zero.

`Trying to determine how significant the back-extrapolated fit in [#151](#Comment_15110) is. If I slide the time series by a few years left and right with respect to the model, the correlation decreases both in the in-band (20th century data) and out-of-band (>16th century data) ![fit](http://imageshack.com/a/img907/1770/P3YlyG.png) If this was a spurious random correlation (given the CC is only 0.28), then I wouldn't expect the peak to align at zero.`

I have been on this path to understand ENSO via its relationship to QBO and the Chandler wobble (along with possible TSI contributions, which is fading) for awhile now. Factors such as QBO and CW have all been considered as possible forcing mechanisms or at least correlations to ENSO in the research literature.

I got sidetracked into trying to figure out the causes of QBO and the Chandler wobble hoping that it might shed light into how they could be driving ENSO.

But now that we see how the QBO and the Chandler wobble both derive from the seasonally aliased lunar Draconic cycle, it may not take as long to piece the individual bits of evidence together.

http://contextearth.com/2016/01/27/possible-luni-solar-tidal-mechanism-for-the-chandler-wobble/

I am optimistic based on how simple these precursor models are. As far as both QBO and the Chandler wobble, one can't ask for a simpler explanation than applying the moon's Draconic orbital cycle as a common forcing mechanism.

I think this is the way that scientific research is supposed to proceed. You keep on turning over rocks and watch what scurries out :)

`I have been on this path to understand ENSO via its relationship to QBO and the Chandler wobble (along with possible TSI contributions, which is fading) for awhile now. Factors such as QBO and CW have all been considered as possible forcing mechanisms or at least correlations to ENSO in the research literature. ![ENSO](http://imageshack.com/a/img923/1716/Adfpq8.png) I got sidetracked into trying to figure out the causes of QBO and the Chandler wobble hoping that it might shed light into how they could be driving ENSO. But now that we see how the QBO and the Chandler wobble both derive from the seasonally aliased lunar Draconic cycle, it may not take as long to piece the individual bits of evidence together. http://contextearth.com/2016/01/27/possible-luni-solar-tidal-mechanism-for-the-chandler-wobble/ I am optimistic based on how simple these precursor models are. As far as both QBO and the Chandler wobble, one can't ask for a simpler explanation than applying the moon's Draconic orbital cycle as a common forcing mechanism. I think this is the way that scientific research is supposed to proceed. You keep on turning over rocks and watch what scurries out :)`

Gray, William M, John D Sheaffer, and John A Knaff. 1992. “Inﬂuence of the Stratospheric QBO on ENSO Variability.” J. Meteor: Soc. Japan 70: 975–95. [1992]

This describes the current model fairly well, a modulated strict biennial forcing with one clear phase reversal.

Biennial oscillations are metastable but reinforcing, as a biennial signal modulated by a seasonal signal will retain the biennial and create a higher harmonic. Then that new modulated signal will continue to feedback a reinforcement via further seasonal modulation. What Gray is suggesting is that the QBO may initiate the disturbance and synchronize the biennial until it gets established.

What makes the analysis difficult is the arbitrariness of a phase reversal, which is difficult to detect by spectral methods as Gray suggests. Also, finding the modulation on a biennial signal is difficult in the midst of phase reversals.

Another interesting behavior described in the paper is the max SOI at the beginning of the year, showing how seasonal synchronization is observed.

The bad news is that Gray has now become an AGW contrarian.

`Gray, William M, John D Sheaffer, and John A Knaff. 1992. “Inﬂuence of the Stratospheric QBO on ENSO Variability.” J. Meteor: Soc. Japan 70: 975–95. [1992] ![gray](http://imageshack.com/a/img924/1484/eaDkPL.png) This describes the current [model](/discussion/comment/15091/#Comment_15091) fairly well, a modulated strict biennial forcing with one clear phase reversal. Biennial oscillations are metastable but reinforcing, as a biennial signal modulated by a seasonal signal will retain the biennial and create a higher harmonic. Then that new modulated signal will continue to feedback a reinforcement via further seasonal modulation. What Gray is suggesting is that the QBO may initiate the disturbance and synchronize the biennial until it gets established. What makes the analysis difficult is the arbitrariness of a phase reversal, which is difficult to detect by spectral methods as Gray suggests. Also, finding the modulation on a biennial signal is difficult in the midst of phase reversals. Another interesting behavior described in the paper is the max SOI at the beginning of the year, showing how seasonal synchronization is observed. ![gray2](http://imageshack.com/a/img921/6947/dQoqsv.png) The bad news is that Gray has now become an AGW contrarian.`

The model for QBO is quite effective at capturing the behavior and has a plausible theory behind it. For awhile I thought that the ENSO model was lagging behind in terms of quality, but it is now actually looking quite good.

My latest strategy is to use a minimal training interval for the data and then validate this over the longer test interval. I first chose a very aggressive 30 year training interval and then applied it to the entire 1880 to 2013 ENSO time series.

Validating ENSO cyclostationary deterministic behavior

`The model for QBO is quite effective at capturing the behavior and has a plausible theory behind it. For awhile I thought that the ENSO model was lagging behind in terms of quality, but it is now actually looking quite good. My latest strategy is to use a minimal training interval for the data and then validate this over the longer test interval. I first chose a very aggressive 30 year training interval and then applied it to the entire 1880 to 2013 ENSO time series. [ Validating ENSO cyclostationary deterministic behavior ](http://contextearth.com/2016/03/31/validating-enso-cyclostationary-deterministic-behavior) ![Fit](http://imagizer.imageshack.us/a/img923/219/OyG6rV.png)`

I go back to the ENSO coral proxy data on occasion to crosscheck results. The main problem with the ENSO proxy, even though it is calibrated to current measurements, is that it only has resolution to 1 year. And since the Nyquist limit on yearly sampling is a 2 year period, it doesn't capture the sub-biennial fluctuations that effectively.

So the fitting using the lunar periods is mainly restricted to the lomg-period tidal periods. The 6 year period appears to be modulated by the 133 year period which defines the interval between times at which the sun, moon, and earth are co-linear [1]. Since the proxy data goes back to 1650, there is enough of a run to be able to pick up the longer periods.

As far as sensitivity, only the 18.6 year nodal declination period is off slightly, picking a best fit at 18.4 years. The longer the data time-series, the more important it is to get the alignment exact. The multiple regression also picks up a significant contribution from the 22/11 year TSI sunspot cycle.

[1] Brannen, Noah Samuel. "The sun, the moon, and convexity." College Mathematics Journal 32.4 (2001): 268-272. "The unusual situation where the sun, the moon, and the earth are nearly collinear, while at the same time a full moon occurs at the winter solstice with the moon at perigee, takes place every 7 x 19 = 133 years."

`I go back to the ENSO coral proxy data on occasion to crosscheck results. The main problem with the ENSO proxy, even though it is calibrated to current measurements, is that it only has resolution to 1 year. And since the Nyquist limit on yearly sampling is a 2 year period, it doesn't capture the sub-biennial fluctuations that effectively. So the fitting using the lunar periods is mainly restricted to the lomg-period tidal periods. The 6 year period appears to be modulated by the 133 year period which defines the interval between times at which the sun, moon, and earth are co-linear [1]. Since the proxy data goes back to 1650, there is enough of a run to be able to pick up the longer periods. ![fit](http://imageshack.com/a/img921/7398/y51jAi.png) As far as sensitivity, only the 18.6 year nodal declination period is off slightly, picking a best fit at 18.4 years. The longer the data time-series, the more important it is to get the alignment exact. The multiple regression also picks up a significant contribution from the 22/11 year TSI sunspot cycle. ![sens](http://imageshack.com/a/img924/29/oxtp7y.png) [1] Brannen, Noah Samuel. "The sun, the moon, and convexity." College Mathematics Journal 32.4 (2001): 268-272. "The unusual situation where the sun, the moon, and the earth are nearly collinear, while at the same time a full moon occurs at the winter solstice with the moon at perigee, takes place every 7 x 19 = 133 years."`

My recent blog post on the topic Biennial Mode of SST and ENSO, with analysis pertaining to the latest paper by Kim

Kim is I believe a PhD from the FSU meteorology department, where most of the key ENSO players such as Clarke and O'Brien have hung out. I have gained most of my insight into the behavior of ENSO from that school.

`My recent blog post on the topic [Biennial Mode of SST and ENSO](http://contextearth.com/2016/04/21/biennial-mode-of-sst-and-enso/), with analysis pertaining to the latest paper by Kim > Kim, Jinju, and Kwang-Yul Kim. "The tropospheric biennial oscillation defined by a biennial mode of sea surface temperature and its impact on the atmospheric circulation and precipitation in the tropical eastern Indo-western Pacific region." Climate Dynamics, 2016: 1-15. Kim is I believe a PhD from the FSU meteorology department, where most of the key ENSO players such as Clarke and O'Brien have hung out. I have gained most of my insight into the behavior of ENSO from that school.`

As I have been formulating a model for ENSO, I always try to relate it to a purely physical basis. The premise I have had from the beginning is that some external factor is driving the forcing of the equatorial Pacific thermocline. This forcing stimulus essentially causes a sloshing in the ocean volume due to small changes in the angular momentum of the rotating earth. I keep thinking that the origin is lunar as the success of the QBO model in relating lunisolar forcing to the oscillatory behavior of the QBO winds is enough motivation to keep on a lunar path.

Yet, I am finding that the detailed mechanism for ENSO differs from that of QBO. An interesting correlation I found is in the tidal-locking of the Earth to the moon. I think this is a subset of the more general case of spin-orbit resonance, where the rotation rate of a satellite is an integral ratio of the main body. In the case of the moon and the earth, it explains why the same moon face is always directed at the earth -- as they spin at the same rate during their mutual orbit, thus compensating via a kind of counter-rotation as shown in the left figure below..

The reason that they spin at the same rate is like most physical behaviors explainable via classical mechanics and the principle of least action. In essence, if one body slows down, the other will slow in response.

If the earth and moon spin at a uniform rate, there is no means for changes in angular momentum to get imparted to the Pacific ocean. And so a spinning planet is not enough to induce a physical sloshing. This is probably too obvious an assertion, but if you just hold a glass of water, it won't start sloshing just because the earth is rotating underneath it. So we can make an assumption that the initial momentum changes needed for the earth to get up to speed have long since equilibrated.

The question is whether there is an obvious modulation of the moon's rotation rate that could impart subtle angular momentum shifts in the earth's own spin. This would quickly get absorbed in the sensitive response of ocean sloshing. As described in ref [1] below, the solid moon does show a longitudinal modulation (i.e. libration) at a period of 1095 days (due to the earth-moon-sun interaction of 1/2 of the perigee cycle ~3 years) and a free resonant period of 1056 days, or ~2.89 years.

Does that latter period show up in the ENSO forcing model? Yes, and it is a very strong and sharp forcing factor in the wave-equation ENSO model, see this comment. But what is even more substantiating for the idea is that an induced seasonal harmonic is also seen in the power spectra of the ENSO time series. This value should be at 1/(1+1/2.89) = 0.743 years if the yearly cycle modulates the lunar libration. The other cycle should be at 1/(1-1/2.89) = 1.53 years. In the following figure, which is a power spectra of the residual from the ENSO model, a clear sharp peak can be seen in the 2048 point monthly Fourier Transform at wave number 230. Seen in the figure below, sitting all by its lonesome but rising distinctly from the higher frequency ENSO noise is a peak corresponding to a period of 2048/12/230=0.742 years!

In the figure, the other labeled points each have a meaning. Peak 1 is the residual of the 2.89 year factor, split into two and reduced in magnitude after being included in the model. Peak #2 is likewise the ubiquitous 2.33 year period model reduced. And the peaks at #3 are centered nearly at a period corresponding to 1.53 years, which is the factored complement to the 0.743 year libration modulation period.

The other relation is that 2.89 years is very close to the Chandler wobble of ~6.5 years modulated by the biennial cycle, so that this relation holds. 1/2.89 + 1/6.5 = 1/2

Which means that a period of 1/(1/2+1/6.5) = 1.53 would show up around peak #3. Its possible that the lunar libration of 2.9 years is working with the chandler wobble to induce a metastable biennial locking to ENSO.

This is a segment of a multiple regression fit to the second derivative showing how the libration modulation emerges. The modulation is subtle but significant as a count of 40 cycles of 0.742 year period spans that 30 year interval.

This is a similar plot but the 2nd derivative was applied after a wave-equation DiffEq solution was fitted to the ENSO time series. Since the 1940's the amount of noise in the ENSO measures has been reduced so that correlation is obvious.

[1] Rambaux, N., and J. G. Williams. "The Moon’s physical librations and determination of their free modes." Celestial Mechanics and Dynamical Astronomy 109.1 (2011): 85-100.

`As I have been formulating a model for ENSO, I always try to relate it to a purely physical basis. The premise I have had from the beginning is that some external factor is driving the forcing of the equatorial Pacific thermocline. This forcing stimulus essentially causes a sloshing in the ocean volume due to small changes in the angular momentum of the rotating earth. I keep thinking that the origin is lunar as the success of the QBO model in relating lunisolar forcing to the oscillatory behavior of the QBO winds is enough motivation to keep on a lunar path. Yet, I am finding that the detailed mechanism for ENSO differs from that of QBO. An interesting correlation I found is in the [tidal-locking](https://en.wikipedia.org/wiki/Tidal_locking) of the Earth to the moon. I think this is a subset of the more general case of [spin-orbit resonance](https://en.wikipedia.org/wiki/Tidal_locking#Rotation.E2.80.93orbit_resonance), where the rotation rate of a satellite is an integral ratio of the main body. In the case of the moon and the earth, it explains why the same moon face is always directed at the earth -- as they spin at the same rate during their mutual orbit, thus compensating via a kind of counter-rotation as shown in the left figure below.. ![moon](https://upload.wikimedia.org/wikipedia/commons/thumb/5/56/Tidal_locking_of_the_Moon_with_the_Earth.gif/300px-Tidal_locking_of_the_Moon_with_the_Earth.gif) The reason that they spin at the same rate is like most physical behaviors explainable via classical mechanics and the principle of least action. In essence, if one body slows down, the other will slow in response. If the earth and moon spin at a uniform rate, there is no means for changes in angular momentum to get imparted to the Pacific ocean. And so a spinning planet is not enough to induce a physical sloshing. This is probably too obvious an assertion, but if you just hold a glass of water, it won't start sloshing just because the earth is rotating underneath it. So we can make an assumption that the initial momentum changes needed for the earth to get up to speed have long since equilibrated. The question is whether there is an obvious modulation of the moon's rotation rate that could impart subtle angular momentum shifts in the earth's own spin. This would quickly get absorbed in the sensitive response of ocean sloshing. As described in ref [1] below, the solid moon does show a longitudinal modulation (i.e. libration) at a period of 1095 days (due to the earth-moon-sun interaction of 1/2 of the perigee cycle ~3 years) and a free resonant period of 1056 days, or ~2.89 years. Does that latter period show up in the ENSO forcing model? Yes, and it is a very strong and sharp forcing factor in the wave-equation ENSO model, [see this comment](https://forum.azimuthproject.org/discussion/comment/14853/#Comment_14853). But what is even more substantiating for the idea is that an induced seasonal harmonic is also seen in the power spectra of the ENSO time series. This value should be at 1/(1+1/2.89) = 0.743 years if the yearly cycle modulates the lunar libration. The other cycle should be at 1/(1-1/2.89) = 1.53 years. In the following figure, which is a power spectra of the residual from the ENSO model, a clear sharp peak can be seen in the 2048 point monthly Fourier Transform at wave number 230. Seen in the figure below, sitting all by its lonesome but rising distinctly from the higher frequency ENSO noise is a peak corresponding to a period of 2048/12/230=0.742 years! ![spectra](http://imageshack.com/a/img924/761/2CpWlv.png) In the figure, the other labeled points each have a meaning. Peak 1 is the residual of the 2.89 year factor, split into two and reduced in magnitude after being included in the model. Peak #2 is likewise the ubiquitous 2.33 year period model reduced. And the peaks at #3 are centered nearly at a period corresponding to 1.53 years, which is the factored complement to the 0.743 year libration modulation period. The other relation is that 2.89 years is very close to the Chandler wobble of ~6.5 years [modulated by the biennial cycle](http://contextearth.com/2016/04/21/biennial-mode-of-sst-and-enso/), so that this relation holds. 1/2.89 + 1/6.5 = 1/2 Which means that a period of 1/(1/2+1/6.5) = 1.53 would show up around peak #3. Its possible that the lunar libration of 2.9 years is working with the chandler wobble to induce a metastable biennial locking to ENSO. This is a segment of a multiple regression fit to the second derivative showing how the libration modulation emerges. The modulation is subtle but significant as a count of 40 cycles of 0.742 year period spans that 30 year interval. ![ts](http://imageshack.com/a/img924/4311/HFrhzJ.png) This is a similar plot but the 2nd derivative was applied after a wave-equation DiffEq solution was fitted to the ENSO time series. Since the 1940's the amount of noise in the ENSO measures has been reduced so that correlation is obvious. ![ts2](http://imageshack.com/a/img922/4552/XxOADG.png) [1] Rambaux, N., and J. G. Williams. "The Moon’s physical librations and determination of their free modes." Celestial Mechanics and Dynamical Astronomy 109.1 (2011): 85-100.`

New post up http://contextearth.com/2016/05/07/crucial-recent-citations-for-enso/

`![fit](http://imageshack.com/a/img921/4664/mJFuv4.png) New post up http://contextearth.com/2016/05/07/crucial-recent-citations-for-enso/`

This is the equivalent DiffEq solution applying the R 'deSolve' routine. I applied a 180 degree phase reversal to the biennial forcing between 1980 and 1996 and, sure enough, the response reverses sign. Overall it's difficult to work with this because I need some automation to get the amplitudes and initial phases optimally fit.

The parameters are essentially the demodulated Chandler wobble period of 6.48 years, another triaxial wobble at 14 years, and the nodal and anomalistic lunar periods. Each of these is modulated by the same biennial factor.

Animated GIF of the wave eq transformed ENSO model from 1880 to 2013. I created this on my phone; what it is doing is scanning the cursor along the time axis and so the model in red tries to track the data in blue.

`This is the equivalent DiffEq solution applying the R 'deSolve' routine. I applied a 180 degree phase reversal to the biennial forcing between 1980 and 1996 and, sure enough, the response reverses sign. Overall it's difficult to work with this because I need some automation to get the amplitudes and initial phases optimally fit. ![diffEq](http://imageshack.com/a/img922/3611/x2tMp9.png) The parameters are essentially the demodulated Chandler wobble period of 6.48 years, another triaxial wobble at 14 years, and the nodal and anomalistic lunar periods. Each of these is modulated by the same biennial factor. Animated GIF of the wave eq transformed ENSO model from 1880 to 2013. I created this on my phone; what it is doing is scanning the cursor along the time axis and so the model in <font color=red>red</font> tries to track the data in <font color=blue>blue</font>. ![anim](http://imageshack.com/a/img924/5808/W4s5kw.gif)`

This is the most recent model featuring the 4 significant angular momentum variations, in order of influence:

These are modulated by a strictly biennial forcing, of which undergoes a 180 degree phase reversal after 1980.

I'm kind of marveling at this because the physics is so plausible and the results are so parsimonious with what is observed. Probably the only-leap-of-faith argument is accounting for the phase reversal between 1980 and 1996. Yet, if this glitch wasn't there and if the biennial modulation did not occur, the model would have been discovered long ago. Same goes for the QBO model, as that is also influenced by tides, but the seasonal aliasing made it difficult to unravel.

I would like to see it get to the point whereby the deviations from a good fit open up for other possibilities to explore. For example what is with the deviation at 1936? The US midwest was hit by a record cold winter to start the year and that was followed by a record hot summer. The model says that 1936 was a cool la Nina cycle, but the data says it was neutral.

`This is the most recent model featuring the 4 significant angular momentum variations, in order of influence: * 6.48 year Chandler wobble angular momentum variation period * 14 year additional triaxial wobble period * 18.6 year lunar nodal forcing period * Aliased 27.55 day anomalistic forcing period These are modulated by a strictly biennial forcing, of which undergoes a 180 degree phase reversal after 1980. ![enso](http://imageshack.com/a/img922/2900/V6do4t.png) I'm kind of marveling at this because the physics is so plausible and the results are so parsimonious with what is observed. Probably the only-leap-of-faith argument is accounting for the phase reversal between 1980 and 1996. Yet, if this glitch wasn't there and if the biennial modulation did not occur, the model would have been discovered long ago. Same goes for the QBO model, as that is also influenced by tides, but the seasonal aliasing made it difficult to unravel. I would like to see it get to the point whereby the deviations from a good fit open up for other possibilities to explore. For example what is with the deviation at 1936? The US midwest was hit by a record cold winter to start the year and that was followed by a record hot summer. The model says that 1936 was a cool la Nina cycle, but the data says it was neutral.`

Here's an example of how you can gain confidence that you are on the right track. I configured the multiple linear regression to train on the ENSO time series from 1880-1940, and then looked at the correlation coefficient over the interval 1940-2013 -- which is completely outside the training interval.

The in=inside cc is 0.725 and the out=outside cc is 0.758. It is rare that you can get a correlation higher outside of the training interval, but because of the stronger noise in the earliest ENSO measurements this is not out of the question. In terms of variance contributions, the reduced noise promotes the real underlying signal.

Next, we generate a set of several synthesized red-noise random walks that have similar variance to the actual ENSO data. This is essentially the Ornstein-Uhlenbeck algorithm, describing a random walk with a reversion-to-the-mean potential well factor. Ten runs were combined into an animated GIF shown below:

Note that although the correlation coefficient is moderate within the training interval (anywhere from 0.4 to 0.6), it is insignificant outside of the training interval, and a few times it actually goes negative. In other words, there is no phase coherence outside of the interval, and any apparent agreement within the training interval is likely the result of over-fitting.

Alas, there is still research being published claiming ENSO is red noise. This is also what the GCM's are generating as far as I can tell, scores of time series with random outcomes.

I have a new posting at http://contextearth.com/2016/05/12/deterministically-locked-on-the-enso-model/

`Here's an example of how you can gain confidence that you are on the right track. I configured the multiple linear regression to train on the ENSO time series from 1880-1940, and then looked at the correlation coefficient over the interval 1940-2013 -- which is completely outside the training interval. ![soi](http://imageshack.com/a/img924/6689/zLbpcb.png) The in=inside cc is 0.725 and the out=outside cc is 0.758. It is rare that you can get a correlation higher outside of the training interval, but because of the stronger noise in the earliest ENSO measurements this is not out of the question. In terms of variance contributions, the reduced noise promotes the real underlying signal. Next, we generate a set of several synthesized red-noise random walks that have similar variance to the actual ENSO data. This is essentially the Ornstein-Uhlenbeck algorithm, describing a random walk with a reversion-to-the-mean potential well factor. Ten runs were combined into an animated GIF shown below: ![ou](http://imageshack.com/a/img922/1023/t6hyd2.gif) Note that although the correlation coefficient is moderate within the training interval (anywhere from 0.4 to 0.6), it is insignificant outside of the training interval, and a few times it actually goes negative. In other words, there is no phase coherence outside of the interval, and any apparent agreement within the training interval is likely the result of over-fitting. Alas, there is still research being published claiming ENSO is red noise. This is also what the GCM's are generating as far as I can tell, scores of time series with random outcomes. I have a new posting at [http://contextearth.com/2016/05/12/deterministically-locked-on-the-enso-model/](http://contextearth.com/2016/05/12/deterministically-locked-on-the-enso-model/)`

How does the ENSO model described in this thread compare to the consensus view?

Searching for the leading lights studying ENSO, the GaTech scientist Peter Webster has the #1 cited paper on ENSO according to Google Scholar. Webster works in the same circles as Judith Curry and Anastasios Tsonis. From what I have read, this circle of scientists claim that ENSO is not very predictable and at most will show a seasonal preference for onset.

Tsonis may be the most identified name for formulating a network ENSO model. He suggests that correlating teleconnections between geospatially-separated climate regions holds the key to the (un)predictability regarding ENSO. Tsonis was able to publish a paper in Physical Review Letters which enhanced his credibility wrt ENSO modeling. He further believes that the climate is very unpredictable to the extent it can undergo spontaneous shifts.

Unfortunately, an issue with Tsonis' credentials regarding ENSO is very similar to the issue Lindzen has regarding his consensus QBO model. They are both global warming deniers according to http://denierlist.wordpress.com/2013/10/04/anastasios-tsonis/

I should act surprised, but find it predictable considering that Tsonis writes papers with Judith Curry, who herself continues to peddle contrarian views before congress whenever she can.

That's perhaps the sorry shape of the discourse; that the leading lights on QBO and ENSO shoot so much from the hip concerning a serious topic such as AGW. The fact that I can come up with a quantitative QBO model to challenge Lindzen's theory and a quantitative ENSO model to contradict Tsonis' model leads me to believe that their non-AGW related science is due for a re-evaluation.

So that's our competition for an ENSO model -- very weak tea.

`How does the ENSO model described in this thread compare to the consensus view? Searching for the leading lights studying ENSO, the GaTech scientist Peter Webster has the #1 cited paper on ENSO according to Google Scholar. Webster works in the same circles as Judith Curry and Anastasios Tsonis. From what I have read, this circle of scientists claim that ENSO is not very predictable and at most will show a seasonal preference for onset. Tsonis may be the most identified name for formulating a network ENSO model. He suggests that correlating teleconnections between geospatially-separated climate regions holds the key to the (un)predictability regarding ENSO. Tsonis was able to publish a [paper in Physical Review Letters](https://pantherfile.uwm.edu/aatsonis/www/publications/2008-06_Tsonis-AA_TopologyandPredictabilityofElNinoandLaNinaNetworks-2.pdf) which enhanced his credibility wrt ENSO modeling. He further believes that the climate is very unpredictable to the extent it can undergo spontaneous shifts. Unfortunately, an issue with Tsonis' credentials regarding ENSO is very similar to the issue Lindzen has regarding his consensus QBO model. They are both global warming deniers according to http://denierlist.wordpress.com/2013/10/04/anastasios-tsonis/ > "Dear Peter > Yes this my quote. > As for your question: at the end of the century we were sitting on the highest global temperature value of the modern record. Since then we have leveled off and we may in fact be cooling. “We have reached the top of the mountain”, therefore it’s not surprising that the last decade is one of the warmest on record. Think about it! The important aspect is that the warming of the 80s and 90s has stopped and the models missed it completely! The important issue is that we have entered a new regime in global temperature tendency. In fact, I find it very misleading that scientists will present “the warmest decade” argument to justify their beliefs (or failures). > Regarding the oceans absorbing heat, it is another argument without solid proof. > Best > Prof. Tsonis" I should act surprised, but find it predictable considering that Tsonis writes papers with Judith Curry, who [herself continues to peddle contrarian views before congress](https://www.youtube.com/watch?v=Oh6zDbWMuP0) whenever she can. That's perhaps the sorry shape of the discourse; that the leading lights on QBO and ENSO shoot so much from the hip concerning a serious topic such as AGW. The fact that I can come up with a quantitative QBO model to challenge Lindzen's theory and a quantitative ENSO model to contradict Tsonis' model leads me to believe that their non-AGW related science is due for a re-evaluation. So that's our competition for an ENSO model -- very weak tea.`

I have commented on Tsonis' work elsewhere and remarked on how he seems to forget the conclusions of the paper he co-authored with Sugihara and others.

`I have <a href="https://hypergeometric.wordpress.com/2014/12/01/tsonis-swanson-chaos-and-s__t-happens/">commented on Tsonis' work elsewhere</a> and remarked on how he seems to forget the conclusions of the paper he co-authored with Sugihara and others.`

Thanks, that is very instructive.

From your blog:

Before I read this I was preparing a table of factors to consider when addressing the quality of a model

I think my third bullet is one that most directly addresses falsifiability. I am contending that ENSO may not be as chaotic as it first looks, and that any one interval contains the information needed to reproduce other intervals. That's the state-space Takens embedding idea. There really is no controlled experiment available and to check against future events will of course take time, so that's why the more that you can turn a model inside and out the higher the confidence one has in its applicability.

The frustrating thing about the way that Tsonis and Lindzen have operated (wrt to ENSO and QBO) is how they stifled other ideas from getting consideration. I think Lindzen cut off new ideas by fire-hosing lots of physics theory in his papers, but I never could find a fit to the data in anything he has written! I think I have gone through much of his work and never did see a graph of data overlaid with a model. Tsonis took a different route by playing the chaos card and suggesting that ENSO is by nature not predictable, yet this seemed more like an assertion than backed up by a thorough analysis. As you point out, Tsonis did say this in a previous paper

So he must have subtracted out the shorter inter-annual contribution, which is consistent with compensating for an ENSO signal factor. I can't really figure Tsonis out, as he seems to write lots of contradictory statements. He shares that proclivity with Judith Curry, who plays the uncertainty card all the time. Confusion to these people is equivalent to scientific uncertainty. It really helps Curry's agenda when she can go before congress and state that climate modeling is impossible.

`> "I have commented on Tsonis' work elsewhere and remarked on how he seems to forget the conclusions of the paper he co-authored with Sugihara and others." Thanks, that is very instructive. From your blog: > "First, there is no instance where their series based explanation makes a prediction that is falsifiable. I challenge thrm to make one. > Second, while they offer a statistical explanation of series, they have not advanced a physical mechanism for its realization, something essential for both taking it seriously as a hypothesis and for supporting additional scientific work based upon it." Before I read this I was preparing a table of factors to consider when addressing the quality of a model * Physical plausibility of model * Fidelity of fit, via correlation coefficient, etc * Stationarity of fit * Choice of plausible known parameters * Sensitivity of parameters * Conciseness of model, simplicity * Information criteria (combines fidelity and conciseness) * Novelty of finding * Robustness of model, is it too brittle? I think my third bullet is one that most directly addresses falsifiability. I am contending that ENSO may not be as chaotic as it first looks, and that any one interval contains the information needed to reproduce other intervals. That's the state-space Takens embedding idea. There really is no controlled experiment available and to check against future events will of course take time, so that's why the more that you can turn a model inside and out the higher the confidence one has in its applicability. The frustrating thing about the way that Tsonis and Lindzen have operated (wrt to ENSO and QBO) is how they stifled other ideas from getting consideration. I think Lindzen cut off new ideas by fire-hosing lots of physics theory in his papers, but I never could find a fit to the data in anything he has written! I think I have gone through much of his work and never did see a graph of data overlaid with a model. Tsonis took a different route by playing the chaos card and suggesting that ENSO is by nature not predictable, yet this seemed more like an assertion than backed up by a thorough analysis. As you point out, Tsonis did say this in a previous paper > "The lack of an oscillatory model signal suggests that the interdecadal global mean surface temperature signal derived from the observations and shown in Figs. 1A and 2B is indeed the signature of natural long-term climate variability." So he must have subtracted out the shorter inter-annual contribution, which is consistent with compensating for an ENSO signal factor. I can't really figure Tsonis out, as he seems to write lots of contradictory statements. He shares that proclivity with Judith Curry, who plays the uncertainty card all the time. Confusion to these people is equivalent to scientific uncertainty. It really helps Curry's agenda when she can go before congress and state that climate modeling is impossible.`

I just found this news about the ENSO theorist Tsonis. Within the last few days he joined the anti-climate science Global Warming Policy Foundation as a council member.

This may be the tipping point in terms of Tsonis' scientific credibility. Unless he has been living under a rock, Tsonis had to have known that this organization is purely political and has little interest in advancing science. Like many of these organizations, they decide on a name that only makes it sound like they are interested in the science.

So both Lindzen and Tsonis belong to GWPF, and from the professions of the other members, these two are likely the most cited and credentialed climate scientists in the council.

** BTW, according to this ENSO paper by Tsonis, the Tsonis criteria is the number of points necessary for the estimation of a correlation dimension D via $10^{2+0.4D}$

`I just found this news about the ENSO theorist Tsonis. Within the last few days [he joined the anti-climate science Global Warming Policy Foundation as a council member](https://notalotofpeopleknowthat.files.wordpress.com/2016/05/image26.png). > "Dr Tsonis is an American atmospheric scientist and distinguished professor at the University of Wisconsin–Milwaukee. His research focuses on the study of climate dynamics and global change. His work has led to the Tsonis criterion**, a research method bearing his name. > The GWPF Academic Advisory Council is composed of scientists, economists and other experts who provide the GWPF with timely scientific, economic and policy advice. It reviews and evaluates new GWPF reports and papers, explores future research projects and makes recommendations on issues related to climate research and policy." This may be the tipping point in terms of Tsonis' scientific credibility. Unless he has been living under a rock, Tsonis had to have known that this organization is purely political and has little interest in advancing science. Like many of these organizations, they decide on a name that only makes it sound like they are interested in the science. So both Lindzen and Tsonis belong to GWPF, and from the professions of the other members, these two are likely the most cited and credentialed climate scientists in the council. ** BTW, according to this [ENSO paper by Tsonis](http://www.geo.cornell.edu/ocean/eas3530/papers/ENSO%20Chaos.pdf), the Tsonis criteria is the number of points necessary for the estimation of a correlation dimension D via $10^{2+0.4D}$`

I mentioned stationarity of fit above. I partioned the ENSO time series into 3 non-overlapping intervals of equal length and did a multiple regression fit over each interval, using the same factors, but allowing the amplitudes and phases to adjust. As I have done elsewhere, I phase inverted the data from 1980 to 1996.

This agreement across all the intervals essentially means that the same behavior, characterized by a forcing and response, is embedded in each interval.

This is the fit over the entire time span.

The question is why Tsonis did not identify this stationarity in his own research. It's entirely possible that the phase inverted interval wreaked havoc on any previous attempts at analysis. Or that they did not apply a wave equation transform.

`I mentioned [stationarity of fit above](#Comment_15340). I partioned the ENSO time series into 3 non-overlapping intervals of equal length and did a multiple regression fit over each interval, using the same factors, but allowing the amplitudes and phases to adjust. As I have done elsewhere, I phase inverted the data from 1980 to 1996. ![fit](http://imageshack.com/a/img924/9451/Br4uM1.png) This agreement across all the intervals essentially means that the same behavior, characterized by a forcing and response, is embedded in each interval. This is the fit over the entire time span. ![all](http://imageshack.com/a/img922/3384/6ppQEI.png) The question is why Tsonis did not identify this stationarity in his own research. It's entirely possible that the phase inverted interval wreaked havoc on any previous attempts at analysis. Or that they did not apply a wave equation transform.`

As I mentioned, Peter Webster from Georgia Tech has the #1 citation on Google Scholar when using the search term "ENSO". I have read his papers on ENSO and he doesn't seem to have anything to add apart from qualitative observations. I have interacted with him and he comes across with a negative attitude at worst and little scientific curiosity at best:

From Curry's blog two years ago:

This is really the heart of the physics argument. Peter Webster, Anastosio Tsonis, and colleagues have asserted that ENSO is not predictable past the current year and "that's that". Yet we really have to look at what deterministic modeling is finding in this Azimuth Project thread and then totally reconsider the Webster/Tsonis view.

Getting some interest should not be difficult considering that Peter Webster has within the past few days been revealed as working an AGW denier angle based on publicly released FOIA emails. From the Rabbett Run climate blog, we can see how Webster tried to protect the agenda of his closest colleagues.

These scientists are really not doing science anymore. They are essentially working to cover for the vested interests of their colleagues. Quite an opening to get some real science done I would think. Same deal as with Lindzen and QBO.

`As I mentioned, Peter Webster from Georgia Tech has the #1 citation on Google Scholar when using the search term "ENSO". I have read his papers on ENSO and he doesn't seem to have anything to add apart from qualitative observations. I have interacted with him and he comes across with a negative attitude at worst and little scientific curiosity at best: From Curry's blog two years ago: >" [Peter Webster | May 26, 2014 at 8:50 pm](https://judithcurry.com/2014/05/26/the-heart-of-the-climate-dynamics-debate/#comment-571761) | > Well, WHT, not so easy. The system is highly nonlinear (hence error growth) which limits forecasts of ENSO across the spring time. Called the “spring predictability barrier” and exists when the noise in the system is greater than the signal. This occur in the boreal spring which is the reason for uncertainty in forecasts at that time of the year. Persistence of ENSO indices between April and July is close to zero. Persistence from June to December is much higher. This once the nonlinear trajectory has occurred, the system is very predictable. Now extend this argument to what the next ENSO cycle will be: zero predictability. I think you fall in the trap of noting that ENSO variability has time scales of 2-4 years and that this seemingly oscillatory nature of the phenomena means predictability. Papers on this if you would like. Bottom line, **ENSO is a nonlinear property of climate, naturally varying but the onset of a phase is unpredictable**. >Papers on this topic if you like. >PW" This is really the heart of the physics argument. Peter Webster, Anastosio Tsonis, and colleagues have asserted that ENSO is not predictable past the current year and "that's that". Yet we really have to look at what deterministic modeling is finding in this Azimuth Project thread and then totally reconsider the Webster/Tsonis view. Getting some interest should not be difficult considering that Peter Webster has within the past few days been revealed as working an AGW denier angle based on publicly released FOIA emails. From the [Rabbett Run climate blog](http://rabett.blogspot.com/2016/05/peter-websters-coming-out-party.html), we can see how Webster tried to protect the agenda of his closest colleagues. ![email](https://1.bp.blogspot.com/-TsM8637KGnA/Vzi-JGvz5pI/AAAAAAAADnE/Tnd-VrL-3eAz9Hby-B7gY6BLQP9PpMJwgCLcB/s1600/Untitled.tiff) These scientists are really not doing science anymore. They are essentially working to cover for the vested interests of their colleagues. Quite an opening to get some real science done I would think. Same deal as with Lindzen and QBO.`

The second derivative of the wave-equation-transformed ENSO data, plotted against model. The lower curve is the correlation coefficient, calculated over 4 year window widths.

The CC only goes negative (anti-correlation) around 1921.

This indicates where some of the ENSO data was interpolated because of missing entries -- marked by -999

http://ftp.cpc.ncep.noaa.gov/wd52dg/data/indices/tahiti.his

`The second derivative of the wave-equation-transformed ENSO data, plotted against model. The lower curve is the correlation coefficient, calculated over 4 year window widths. ![2nd](http://imageshack.com/a/img923/8428/PiHL0F.png) The CC only goes negative (anti-correlation) around 1921. This indicates where some of the ENSO data was interpolated because of missing entries -- marked by -999 http://ftp.cpc.ncep.noaa.gov/wd52dg/data/indices/tahiti.his`

This is a frequency spectrum of the wave-equation transformed ENSO. The strongest peaks that have frequency between 0 and 1/year are all associated with the seasonally aliased Draconic lunar month (27.2 days) and Draconic fortnight (1/2 month = 13.6 days).

The arrows show the relationship between +/- pairs about the 1/(2yr) modulation.

This is the model spectrum comparison. Apart from the 1/f filtering envelope, the symmetry about the f=0.5/yr is evident.

and the actual fit using training from 1881 to 1950. Interesting that the correlation coefficient is higher outside the training interval than within the training interval!

`This is a frequency spectrum of the wave-equation transformed ENSO. The strongest peaks that have frequency between 0 and 1/year are all associated with the seasonally aliased Draconic lunar month (27.2 days) and Draconic fortnight (1/2 month = 13.6 days). ![dracspec](http://imageshack.com/a/img924/9025/ujm0lj.png) The arrows show the relationship between +/- pairs about the 1/(2yr) modulation. This is the model spectrum comparison. Apart from the 1/f filtering envelope, the symmetry about the f=0.5/yr is evident. ![fit](http://imageshack.com/a/img923/6103/e79ure.png) and the actual fit using training from 1881 to 1950. Interesting that the correlation coefficient is higher outside the training interval than within the training interval! ![training](http://imageshack.com/a/img921/1870/SDw7kW.png)`

Would it not be weird if Pierre-Simon Laplace's original tidal equations fit ENSO data better than it does tides?

http://contextEarth.com/2016/06/10/pukites-model-of-enso/

If what I derived is correct, this is the direct connection between the Navier-Stokes formulation that are used in GCM's to the simplified Mathieu wave equation that I have been investigating for a while.

$ \frac{d^2\Theta}{dt^2} + \left[\frac{s}{\sigma} + s^2 + \sigma^2 \gamma \right] \Theta \mu_0 \omega^2 sin(\omega t) = \sigma^2 \mu_0 \omega^2 sin(\omega t) f(t) $

This equation, when the LHS is directly evaluated for $\omega=\pi$ (a biennial cycle) and then fit to the RHS's f(t) with a seasonally aliased Draconic lunar forcing cycle (and a few other lunar periods) results in excellent agreement to the data.

So ... given that when I was taught courses in quantum/solid-state/statistical physics, it was always about taking the approximation when appropriate to draw out the simplified representation. Why am I not seeing this approach anywhere in the climate research literature?

`Would it not be weird if Pierre-Simon Laplace's original tidal equations fit ENSO data better than it does tides? http://contextEarth.com/2016/06/10/pukites-model-of-enso/ If what I derived is correct, this is the direct connection between the Navier-Stokes formulation that are used in GCM's to the simplified Mathieu wave equation that I have been investigating for a while. $ \frac{d^2\Theta}{dt^2} + \left[\frac{s}{\sigma} + s^2 + \sigma^2 \gamma \right] \Theta \mu_0 \omega^2 sin(\omega t) = \sigma^2 \mu_0 \omega^2 sin(\omega t) f(t) $ This equation, when the LHS is directly evaluated for $\omega=\pi$ (a biennial cycle) and then fit to the RHS's f(t) with a seasonally aliased Draconic lunar forcing cycle (and a few other lunar periods) results in excellent agreement to the data. So ... given that when I was taught courses in quantum/solid-state/statistical physics, it was always about taking the approximation when appropriate to draw out the simplified representation. Why am I not seeing this approach anywhere in the climate research literature?`

+Rash Kamel on +Azimuth caught this paper:

Michael L. Griffiths et al. Western Pacific hydroclimate linked to global climate variability over the past two millennia

http://www.nature.com/ncomms/2016/160608/ncomms11719/full/ncomms11719.html

They identify shifts from largely hot Nino to cold Nina conditions in 900 to 1250 and back from 1300 to 1900.

`+Rash Kamel on +Azimuth caught this paper: Michael L. Griffiths et al. Western Pacific hydroclimate linked to global climate variability over the past two millennia http://www.nature.com/ncomms/2016/160608/ncomms11719/full/ncomms11719.html They identify shifts from largely hot Nino to cold Nina conditions in 900 to 1250 and back from 1300 to 1900.`

After studying proxy ENSO data for awhile, the big issue is one of resolution, and the fact that much of the interesting ENSO dynamics occurs on the 1 to 3 year cyclic interval. Unfortunately, because all of the proxy data is sampled at best only on 1 year intervals, then due to Nyquist sampling limitations, the best one can get down to is about 2 year resolution.

In addition, that long period behavior observed via proxy may not even be ENSO. By comparison, the models over the last 100+ years behave like strong oscillations about a mean of zero, and the longest periods are still less than 20 years.

So that paper is interesting but I don't know how to apply that knowledge for fitting ENSO to higher resolution time-frames.

`After studying proxy ENSO data for awhile, the big issue is one of resolution, and the fact that much of the interesting ENSO dynamics occurs on the 1 to 3 year cyclic interval. Unfortunately, because all of the proxy data is sampled at best only on 1 year intervals, then due to Nyquist sampling limitations, the best one can get down to is about 2 year resolution. In addition, that long period behavior observed via proxy may not even be ENSO. By comparison, the models over the last 100+ years behave like strong oscillations about a mean of zero, and the longest periods are still less than 20 years. So that paper is interesting but I don't know how to apply that knowledge for fitting ENSO to higher resolution time-frames.`

On the sampling resolution, is ENSO only sampled at one year? Or are there several samples in bunches and the midpoints of bunches are separated by one year? I have no idea, but on a lark, I'm asking. If there are many samples, but they are not uniform, Lomb-Scargle can still do a reasonable job on resolution. If they are only sampled once every year, then, yeah, Nyquist applies. On the other hand, there may be time domain techniques which are applicable. You'll probably guess what it will be, but I continue to be impressed by the flexibility of various state-space methods, e.g., sketched at https://hypergeometric.wordpress.com/2016/06/06/six-cases-of-models/ and its reference. There are lots of ways this can be cooked. I'm loooking at a General Additive Models one by Simon Wood, recorded in his textbook,

Generalized Additive Models: An Introduction with(2006), Section 6.7.2.R`On the sampling resolution, is ENSO only sampled at one year? Or are there several samples in bunches and the midpoints of bunches are separated by one year? I have no idea, but on a lark, I'm asking. If there are many samples, but they are not uniform, Lomb-Scargle can still do a reasonable job on resolution. If they are only sampled once every year, then, yeah, Nyquist applies. On the other hand, there may be time domain techniques which are applicable. You'll probably guess what it will be, but I continue to be impressed by the flexibility of various state-space methods, e.g., sketched at https://hypergeometric.wordpress.com/2016/06/06/six-cases-of-models/ and its reference. There are lots of ways this can be cooked. I'm loooking at a General Additive Models one by Simon Wood, recorded in his textbook, <em>Generalized Additive Models: An Introduction with <b>R</b></em> (2006), Section 6.7.2.`

For proxies they only sample at one year because measures such as coral rings only allow a single measurement per year -- its essentially the width of that year's coral ring (or the tracer content of that ring, etc). Same as with tree rings, these marker-based measures all have the same low-resolution issue.

I could be wrong, I am not a climatologist ... yet no one wants to discuss this stuff. For example, I was curious about something on Isaac Held's blog from last week:

So I thought I would ask a question to a real climate scientist as a comment to that post. Of course, I always save these comments in case they get lost:

So I waited a couple of days for it to get out of moderation. And it did, but it disappeared. So I placed another question to his blog asking why it disappeared. This time he responded by email:

OK, so perhaps technically, I may have not been precisely on topic. But what's frustrating is that he doesn't even have the consideration to respond to my question

privately. Alas, that's what I find is par for the course when dealing with these guys. Pat you on the head and tell you to run along now.Like I mentioned in the previous comment, these guys are still using the basic equations as formulated by Pierre-Simon Laplace back in 1776. I don't think Laplace was a climatologist either.

I looked at some of the quotations attributed to Laplace, and some are kind of ironic

`For proxies they only sample at one year because measures such as coral rings only allow a single measurement per year -- its essentially the width of that year's coral ring (or the tracer content of that ring, etc). Same as with tree rings, these marker-based measures all have the same low-resolution issue. I could be wrong, I am not a climatologist ... yet no one wants to discuss this stuff. For example, I was curious about something on [Isaac Held's blog from last week](http://www.gfdl.noaa.gov/blog/isaac-held/2016/06/03/70-spherical-rotating-radiative-convective-equilibrium): >**70. Spherical rotating radiative-convective equilibrium** > ".... So the only source of spatial inhomogeneity is the rotating sphere. For our thin rapidly rotating atmospheric spherical shell, the key inhomogeneity is the latitude dependence of the strength of the horizontal component of the Coriolis force, the 'Coriolis parameter', $f \equiv 2 \Omega\sin(\theta)$ where $\theta$ is the latitude. The amplitude of f increases from the equator, where it vanishes, to the poles. This latitudinal gradient in f is critical for TC evolution. " So I thought I would ask a question to a real climate scientist as a comment to that post. Of course, I always save these comments in case they get lost: > "Is it generally agreed that along the the equator the Coriolis forces cancel out and so the wave equation model undergoes a vast simplification? And that this has significant implications for how longitudinal equatorial behaviors such as ENSO and QBO can be more easily modeled? > And also that the clearly observable standing wave mode of ENSO implies that the temporal and spatial modes of the partial differential equation can be separated out. That means that a challenging partial differential equation can be transformed into a more easily solvable (and potentially numerically stable) ordinary differential equation." So I waited a couple of days for it to get out of moderation. And it did, but it disappeared. So I placed another question to his blog asking why it disappeared. This time he responded by email: > "Isaac Held - NOAA Federal <isaac.held@noaa.gov> > I did not accept your comment on post #70 because it was not on topic. The topic of the post is not the Coriolis force, with which most readers of my blog will be familiar." OK, so perhaps technically, I may have not been precisely on topic. But what's frustrating is that he doesn't even have the consideration to respond to my question *privately*. Alas, that's what I find is par for the course when dealing with these guys. Pat you on the head and tell you to run along now. Like I mentioned in the previous comment, these guys are still using the basic equations as formulated by [Pierre-Simon Laplace back in 1776](https://en.wikipedia.org/wiki/Theory_of_tides#Laplace.27s_tidal_equations). I don't think Laplace was a climatologist either. I looked at some of the quotations attributed to Laplace, and some are kind of ironic * "It is therefore obvious that ... " * "We are so far from knowing all the agents of nature and their diverse modes of action that it would not be philosophical to deny phenomena solely because they are inexplicable in the actual state of our knowledge. But we ought to examine them with an attention all the more scrupulous as it appears more difficult to admit them." * "This simplicity of ratios will not appear astonishing if we consider that all the effects of nature are only mathematical results of a small number of immutable laws." * "Infinitely varied in her effects, nature is only simple in her causes." * "What we know is little, and what we are ignorant of is immense. "`

Laplace tidal equation transform applying only tidal periods and biennial modulation compared to ENSO data:

This is much more fundamental than the contrived delayed action oscillator models and the other models described on the Azimuth pages:

http://www.azimuthproject.org/azimuth/show/ENSO#ZCModel

I have never understood this entry, where it says that this is

"somewhat realistic ENSO behavior"when in fact I have never seen a time-series comparison of that model against real data.One of the reasons that I have spent so much time (relatively, as this is still spare time for me) on the ENSO model is that I tend to first consider what other research is finding and then build on that if possible. But if all the other research is going in the wrong direction, it takes a lot of time to unwind all the reported research on ENSO.

So now, by applying the more foundational approach of Laplace's Tidal Equations, I am finding that its the scientists such as Richard Lindzen that may be responsible for leading us down a rabbit hole of complexity.

There are two points that are truly confounding about this:

(1) How did they (Lindzen and others) miss the seasonal aliasing? Everyone knows that the synodic lunar month of 29.5306 days exactly aliased the principal semi-diurnal tide of 12.4206 hours -- at least I think they do. It's actually quite difficult to find the derivation for this anywhere online, even though its this simple:

12.42 hours = 12 hours/(1-1day/29.5306days)

The step from understanding how the monthly/daily relate to how the yearly/monthly relate should not be that difficult.

(2) What is the infatuation with eigenvalues and eigenfunctions of the primitive equations? Everywhere you look in the literature, you find differential equations describing wave interactions, starting with Laplace's or shallow-water wave equation (these are the same, BTW). Yet, for tidal analysis, this is all thrown away anyways because all that matters is the forced response based on the lunisolar periods. In other words, the natural response due to the characteristic frequencies (i.e. eigenvalues) within the wave equation are irrelevant for practical purposes. The current tidal analysis simply applies harmonic analysis of the known diurnal and semi-diurnal periods from (1) above to model the observed tides over various ocean regions.

It seems that only scientists have this infatuation over the eigenvalues, while engineers and other practitioners realize that much of the world runs on forced responses for all practical purposes

The end result of the complexity that Lindzen has foisted on our collective wisdom [1] is that others think that the only way to solve a phenomena such as ENSO is to formulate and computationally solve general circulation models (GCM). And that there are other splinter factions that rely on the aforementioned non-linear models or teleconnections, ala Tsonis, to try to explain ENSO. But what is the moon wrt to the earth but a teleconnection?

So, based on what I am finding, simplified tidal equations interacting with these known forced seasonally aliased lunar periods work remarkably well to capture ENSO behaviors. Typically one will get criticism for making a model more and more complicated to capture behaviors that can't be fit any other way, but that criticism should vanish if all this extraneous complexity evaporates in favor of a simpler more fundamental formulation.

I have a couple of interesting pictures that may help to understanding how sensitive that the ocean's thermocline is to a mechanical forcing. This is a picture of a wavecell sloshing in action illustrating how the interface between slightly different density liquids responds to a forcing.

Of course this is a much larger agitation than what the moon can do to the earth, but scaled back and considering that the buoyancy compensation from the very slight density differences between a warmer upper ocean layer ($\rho_0$) and a colder lower ocean layer ($\rho_1$ below the thermocline) generating a sloshing motion should not be difficult to substantiate.

If you read the research on ENSO, what is currently considered the best candidate for a forcing is the wind! Not only does this feel totally unsubstantiated, but what causes the wind to oscillate? Could it not be that the wind is a side-effect, or just happens to move in concert with the lunar periods? And that's where all the work on QBO comes in. And the QBO is another phenomena that Lindzen completely missed, as this also follows the seasonally aliased lunar periods precisely!

Yet, remember that Lindzen said this as well [2]:

It boggles my mind how Lindzen was able to produce all that detailed math in [1] but fail to make the obvious connection to a simple model.

[1] R. D. Lindzen, “Planetary waves on beta-planes,” Monthly Weather Review, vol. 95, no. 7, pp. 441–451, 1967.

[2] Lindzen, Richard S., and Siu-Shung Hong. "Effects of mean winds and horizontal temperature gradients on solar and lunar semidiurnal tides in the atmosphere." Journal of the Atmospheric Sciences 31.5 (1974): 1421-1446.

`Laplace tidal equation transform applying only tidal periods and biennial modulation compared to ENSO data: ![themod](http://imageshack.com/a/img923/106/j4pvHW.png) This is much more fundamental than the contrived delayed action oscillator models and the other models described on the Azimuth pages: http://www.azimuthproject.org/azimuth/show/ENSO#ZCModel > "Apparently the simplest climate model that exhibits somewhat realistic ENSO behavior is the ‘minimal model’ of the equatorial ocean-atmosphere system called the Zebiak–Cane model or ZC model:" I have never understood this entry, where it says that this is *"somewhat realistic ENSO behavior"* when in fact I have never seen a time-series comparison of that model against real data. One of the reasons that I have spent so much time (relatively, as this is still spare time for me) on the ENSO model is that I tend to first consider what other research is finding and then build on that if possible. But if all the other research is going in the wrong direction, it takes a lot of time to unwind all the reported research on ENSO. So now, by applying the more [foundational approach of Laplace's Tidal Equations](http://contextearth.com/2016/06/10/pukites-model-of-enso/), I am finding that its the scientists such as Richard Lindzen that may be responsible for leading us down a rabbit hole of complexity. There are two points that are truly confounding about this: (1) How did they (Lindzen and others) miss the seasonal aliasing? Everyone knows that the [synodic lunar month of 29.5306 days](https://en.wikipedia.org/wiki/Lunar_month) exactly aliased the [principal semi-diurnal tide of 12.4206 hours](https://en.wikipedia.org/wiki/Lunar_month) -- at least I think they do. It's actually quite difficult to find the derivation for this anywhere online, even though its this simple: 12.42 hours = 12 hours/(1-1day/29.5306days) The step from understanding how the monthly/daily relate to how the yearly/monthly relate should not be that difficult. (2) What is the infatuation with eigenvalues and eigenfunctions of the primitive equations? Everywhere you look in the literature, you find differential equations describing wave interactions, starting with Laplace's or shallow-water wave equation (these are the same, BTW). Yet, for tidal analysis, this is all thrown away anyways because all that matters is the forced response based on the lunisolar periods. In other words, the natural response due to the characteristic frequencies (i.e. eigenvalues) within the wave equation are irrelevant for practical purposes. The current tidal analysis simply applies harmonic analysis of the known diurnal and semi-diurnal periods from (1) above to model the observed tides over various ocean regions. It seems that only scientists have this infatuation over the eigenvalues, while engineers and other practitioners realize that much of the world runs on forced responses for all practical purposes --- The end result of the complexity that Lindzen has foisted on our collective wisdom [1] is that others think that the only way to solve a phenomena such as ENSO is to formulate and computationally solve general circulation models (GCM). And that there are other splinter factions that rely on the aforementioned non-linear models or teleconnections, ala Tsonis, to try to explain ENSO. But what is the moon wrt to the earth but a teleconnection? So, based on what I am finding, simplified tidal equations interacting with these known forced seasonally aliased lunar periods work remarkably well to capture ENSO behaviors. Typically one will get criticism for making a model more and more complicated to capture behaviors that can't be fit any other way, but that criticism should vanish if all this extraneous complexity evaporates in favor of a simpler more fundamental formulation. --- I have a couple of interesting pictures that may help to understanding how sensitive that the ocean's thermocline is to a mechanical forcing. This is a picture of a wavecell sloshing in action illustrating how the interface between slightly different density liquids responds to a forcing. ![wavecell](http://imageshack.com/a/img924/4589/yV6wd9.gif) Of course this is a much larger agitation than what the moon can do to the earth, but scaled back and considering that the buoyancy compensation from the very slight density differences between a warmer upper ocean layer ($\rho_0$) and a colder lower ocean layer ($\rho_1$ below the thermocline) generating a sloshing motion should not be difficult to substantiate. ![rho](http://imageshack.com/a/img924/8817/aUrd7X.png) If you read the research on ENSO, what is currently considered the best candidate for a forcing is the wind! Not only does this feel totally unsubstantiated, but what causes the wind to oscillate? Could it not be that the wind is a side-effect, or just happens to move in concert with the lunar periods? And that's where all the work on QBO comes in. And the QBO is another phenomena that Lindzen completely missed, as this also follows the seasonally aliased lunar periods precisely! ![qbo](http://imageshack.com/a/img923/3016/0xdv5d.png) Yet, remember that Lindzen said this as well [2]: > "it is unlikely that lunar periods could be produced by anything other than the lunar tidal potential." ![linzen](http://imageshack.com/a/img908/7594/NPwSYZ.gif) It boggles my mind how Lindzen was able to produce all that detailed math in [1] but fail to make the obvious connection to a simple model. [1] R. D. Lindzen, “Planetary waves on beta-planes,” Monthly Weather Review, vol. 95, no. 7, pp. 441–451, 1967. [2] Lindzen, Richard S., and Siu-Shung Hong. "Effects of mean winds and horizontal temperature gradients on solar and lunar semidiurnal tides in the atmosphere." Journal of the Atmospheric Sciences 31.5 (1974): 1421-1446.`

I marvel at this wave equation transform:

This is what it looks like with no filtering on the ENSO data:

The transform takes the second derivative and without filtering, the noise swamps the signal, yet the multiple regression can still pull out the ENSO peaks.

The more I work on signal processing, the more I can understand how engineers can pull a signal out of any data, especially weak transmissions from space missions millions of miles away.

`I marvel at this wave equation transform: ![m](http://imageshack.com/a/img923/5283/IS5qfE.png) This is what it looks like with no filtering on the ENSO data: ![nof](http://imageshack.com/a/img923/1061/nw2ecr.png) The transform takes the second derivative and without filtering, the noise swamps the signal, yet the multiple regression can still pull out the ENSO peaks. The more I work on signal processing, the more I can understand how engineers can pull a signal out of any data, especially weak transmissions from space missions millions of miles away.`

There's an aero-engineer from Boeing named David Young that does battle with climate modelers on various forums. He has the credentials for doing computational fluid dynamics (CFD) judging from his papers. His thesis is that the GCM simulations do not apply physics in a computationally correct manner.

One of the reasons for coming up with a simplified yet accurate model for ENSO (and QBO) is that this subverts the criticisms concerning the more detailed computational models. That will also marginalize the arguments of skeptics that use complexity to challenge the correctness of climate models.

I doubt that people like David Young will challenge the accepted tidal analysis models, and if my latest ENSO model works as well as tidal analysis does, then that will negate his argument, at least for this scale of model.

`There's an aero-engineer from Boeing named David Young that does battle with climate modelers on various forums. He has the credentials for doing computational fluid dynamics (CFD) judging from his papers. His thesis is that the GCM simulations do not apply physics in a computationally correct manner. > ![dy](http://imageshack.com/a/img922/3357/hIE8bw.png) One of the reasons for coming up with a simplified yet accurate model for ENSO (and QBO) is that this subverts the criticisms concerning the more detailed computational models. That will also marginalize the arguments of skeptics that use complexity to challenge the correctness of climate models. I doubt that people like David Young will challenge the accepted tidal analysis models, and if my latest ENSO model works as well as tidal analysis does, then that will negate his argument, at least for this scale of model.`

The ENSO data transformed with a biennial Mathieu modulation show strong stationarity over 30 year intervals.

The following sequence is training over 4 different ~32yr intervals, followed by one over the entire interval.

1880-1912 -- This is considered the poorest quality ENSO data

1912-1945

1945-1980

1980-2013 -- From 1980 to 1996 the ENSO data shows a phase inversion

Entire -- the only problematic years are 1936 and inconsistently 1908 and 1904.

The strongest forcing periods show exceptional phase coherence across the intervals. In the top panel in the chart below, the positive periods correspond to forcings that were modulated by a biennial cycle, and the negative periods are the complementary none-modulated terms. The training fit is via multiple regression so that these regions have high correlation coefficients with the data -- almost over-fitted, in fact. Yet the over-fitting does not degrade the correlation in the out-of-band validation interval significantly. The CCs are still at least 0.7 in each out-of-band interval.

In the lower panel, the amplitudes for each forcing differ more significantly across the intervals. This is likely due to overfitting of amplitude in the training period, as noise impacts the amplitude more than the phase. In other words, the overfitting is essentially applying the extra degrees-of-freedom to compensate for possible noise excursions.

Of course there is correlation due to the biennial modulation applied to the

f(t)on the LHS and a similar modulation applied to the forcing terms on the RHS, but the factor could just as easily generate the opposite phase, asf(t)is erratic on its own. For example, attempting this with a red-noise model forf(t), the CC would be anywhere from -0.2 to 0.20 on the out-of-band interval, even though it may be quite high on a training 32 year interval. Indeed, the ENSO behavior is definitely not red noise but deterministic/stationary with the only caveat of possible metastability with respect to a biennial phase reversal.The number of factors that go into this model are comparable to the number that go into a tidal analysis model. For tidal analysis, the shortest period for fitting is 18.6 years, corresponding to the longest long period tide. Here, we go with at least 30 years since biennial sidebands that extend past 18.6 years occur according to the power spectrum.

`The ENSO data transformed with a biennial Mathieu modulation show strong stationarity over 30 year intervals. The following sequence is training over 4 different ~32yr intervals, followed by one over the entire interval. 1880-1912 -- This is considered the poorest quality ENSO data ![1](http://imageshack.com/a/img924/3815/mv3tkg.png) 1912-1945 ![2](http://imageshack.com/a/img921/9696/yTesQb.png) 1945-1980 ![3](http://imageshack.com/a/img923/5442/RNRkX0.png) 1980-2013 -- From 1980 to 1996 the ENSO data shows a phase inversion ![4](http://imageshack.com/a/img923/8030/zjhWAd.png) Entire -- the only problematic years are 1936 and inconsistently 1908 and 1904. ![all](http://imageshack.com/a/img923/5029/ic9am3.png) The strongest forcing periods show exceptional phase coherence across the intervals. In the top panel in the chart below, the positive periods correspond to forcings that were modulated by a biennial cycle, and the negative periods are the complementary none-modulated terms. The training fit is via multiple regression so that these regions have high correlation coefficients with the data -- almost over-fitted, in fact. Yet the over-fitting does not degrade the correlation in the out-of-band validation interval significantly. The CCs are still at least 0.7 in each out-of-band interval. In the lower panel, the amplitudes for each forcing differ more significantly across the intervals. This is likely due to overfitting of amplitude in the training period, as noise impacts the amplitude more than the phase. In other words, the overfitting is essentially applying the extra degrees-of-freedom to compensate for possible noise excursions. ![phases](http://imageshack.com/a/img924/876/MkirHa.png) Of course there is correlation due to the biennial modulation applied to the *f(t)* on the LHS and a similar modulation applied to the forcing terms on the RHS, but the factor could just as easily generate the opposite phase, as *f(t)* is erratic on its own. For example, attempting this with a red-noise model for *f(t)*, the CC would be anywhere from -0.2 to 0.20 on the out-of-band interval, even though it may be quite high on a training 32 year interval. Indeed, the ENSO behavior is definitely not red noise but deterministic/stationary with the only caveat of possible metastability with respect to a biennial phase reversal. The number of factors that go into this model are comparable to the number that go into a tidal analysis model. For tidal analysis, the shortest period for fitting is 18.6 years, corresponding to the longest long period tide. Here, we go with at least 30 years since biennial sidebands that extend past 18.6 years occur according to the power spectrum. ![bien](http://imagizer.imageshack.us/a/img910/4637/tI5DlD.gif) > "(insert meta-discussion here)"`

The quality of the ENSO model above reveals itself in the hidden symmetry in the time series data. I am testing out various methods to reveal the symmetry.

Consider the wave equation transform I am applying to the data:

$ f''(t) + (A + B cos(\pi t)) f(t) = cos(\pi t) g(t) $

This applies a biennial modulation to the signal, which will reinforce any natural modes. If f(t) doesn't have a phase relationship with $cos(\pi t)$ then the time-series will get further scrambled. The model then is essentially adjusting A and B so that a minimal forcing signal g(t) will reproduce the waveform.

To pull out the symmetry, first take the power spectrum and then plot the equivalent periods as sidebands of the biennial signal.

$ T = 1 / (1/2 - \omega / {2\pi}) $

This will produce one exclusion zone where the modulation frequency is greater than 0.5 per year, which is shown to the immediate right of the center axis in the chart below :

The mirror symmetry is very apparent with each of the sidebands pairing up sharp peaks on each side of the origin. Since the power spectrum is a discrete transform over an arbitrary sampling period (one month), the values for period will never match one-for-one on each side, yet the +/- values do match very closely. A few values do not match, but first consider the matches.

An example of a strong match is shown by the labels colored red near the Chandler wobble period of 6.4 years. In bright green are the sidebands that I believe are associated with seasonally aliased anomalistic tidal forcing -- these are split into a further pair because of the proximity to the doubled 4 year period. The violet pair at +/- 2.9 is due to the Chandler wobble splitting off against the 2-year period. The values at 18 and 9 align with the nodal period of 18.6 years.

The weakest pairings, such as those at +/- 7.6 and at -5.2 (the complement +5.2 is broad) may be spurious, and are kept in the model to get a sense of how much overfitting can be tolerated during multiple regression on various training intervals.

To give an idea of how the symmetry translates to a simulated red noise time series

Some pairs do happen to match up but this is the result of some part of the time series being in phase with a biennial cycle. Overall, the sharpness and symmetry isn't there. That is just one sample and others O-U random walk trials will show even worse asymmetry or marginally better. To get symmetry at the level of the real ENSO data would take an extremely ordered set of cycles.

This is really the most "mechanical" analysis I have done yet on ENSO. The transform w/power spectrum I am applying is really meat-and-potatoes signal processing, which is typically applied to reveal the behaviors not apparent in the raw data.

Performing a Hilbert transform is another potential analysis one can apply. Not sure if that would help but I have noted its application in some recent ENSO papers. They are really searching and scrambling to find something that works. I believe that working with a mix of real physics and signal processing math is what will eventually prove most useful.

`The quality of the ENSO model [above](#Comment_15430) reveals itself in the hidden symmetry in the time series data. I am testing out various methods to reveal the symmetry. Consider the wave equation transform I am applying to the data: $ f''(t) + (A + B cos(\pi t)) f(t) = cos(\pi t) g(t) $ This applies a biennial modulation to the signal, which will reinforce any natural modes. If f(t) doesn't have a phase relationship with $cos(\pi t)$ then the time-series will get further scrambled. The model then is essentially adjusting A and B so that a minimal forcing signal g(t) will reproduce the waveform. To pull out the symmetry, first take the power spectrum and then plot the equivalent periods as sidebands of the biennial signal. $ T = 1 / (1/2 - \omega / {2\pi}) $ This will produce one exclusion zone where the modulation frequency is greater than 0.5 per year, which is shown to the immediate right of the center axis in the chart below : ![power](http://imageshack.com/a/img923/919/0RjPtp.png) The mirror symmetry is very apparent with each of the sidebands pairing up sharp peaks on each side of the origin. Since the power spectrum is a discrete transform over an arbitrary sampling period (one month), the values for period will never match one-for-one on each side, yet the +/- values do match very closely. A few values do not match, but first consider the matches. An example of a strong match is shown by the labels colored red near the Chandler wobble period of 6.4 years. In bright green are the sidebands that I believe are associated with seasonally aliased anomalistic tidal forcing -- these are split into a further pair because of the proximity to the doubled 4 year period. The violet pair at +/- 2.9 is due to the Chandler wobble splitting off against the 2-year period. The values at 18 and 9 align with the nodal period of 18.6 years. The weakest pairings, such as those at +/- 7.6 and at -5.2 (the complement +5.2 is broad) may be spurious, and are kept in the model to get a sense of how much overfitting can be tolerated during multiple regression on various training intervals. To give an idea of how the symmetry translates to a simulated red noise time series ![ou](http://imageshack.com/a/img921/7010/LO7AhR.png) Some pairs do happen to match up but this is the result of some part of the time series being in phase with a biennial cycle. Overall, the sharpness and symmetry isn't there. That is just one sample and others O-U random walk trials will show even worse asymmetry or marginally better. To get symmetry at the level of the real ENSO data would take an extremely ordered set of cycles. --- This is really the most "mechanical" analysis I have done yet on ENSO. The transform w/power spectrum I am applying is really meat-and-potatoes signal processing, which is typically applied to reveal the behaviors not apparent in the raw data. Performing a Hilbert transform is another potential analysis one can apply. Not sure if that would help but I have noted its application in some recent ENSO papers. They are really searching and scrambling to find something that works. I believe that working with a mix of real physics and signal processing math is what will eventually prove most useful. > "(insert meta-discussion here)"`

This just turned up in the Independent:

"Lillo countered that the tweet referred to some unusual behavior of a phenomenon known as the QBO or Quasi-Biennial Oscillation, which is “a separate story” from the cross-equator flow. The QBO, he said, is an oscillation in equatorial stratospheric winds, which has been “out of phase.” He chalked up the weird QBO behavior to natural variability “even though I’m an advocate for identifying connections to human-caused climate change".

http://www.independent.co.uk/news/science/jet-stream-crossing-equator-global-climate-emergency-dismissed-debunked-a7113241.html

PS rant: people write tl;dr which I assume is supposed to mean, too long, did not read. If it doesn't I'd like to know. If it does then it offends my career efforts at unambiguous, precise and easily meaningful variable names - surely it should be tl;dnr? grrrrrr...

`This just turned up in the Independent: "Lillo countered that the tweet referred to some unusual behavior of a phenomenon known as the QBO or Quasi-Biennial Oscillation, which is “a separate story” from the cross-equator flow. The QBO, he said, is an oscillation in equatorial stratospheric winds, which has been “out of phase.” He chalked up the weird QBO behavior to natural variability “even though I’m an advocate for identifying connections to human-caused climate change". http://www.independent.co.uk/news/science/jet-stream-crossing-equator-global-climate-emergency-dismissed-debunked-a7113241.html PS rant: people write tl;dr which I assume is supposed to mean, too long, did not read. If it doesn't I'd like to know. If it does then it offends my career efforts at unambiguous, precise and easily meaningful variable names - surely it should be tl;dnr? grrrrrr...`

Jim, I read that and I have been tweeting with a few of the meteorology grad students that were pointing out the unusual behavior.

The behavior that they are referring to is happening starting in late 2015 to early 2016. In the figure below, the width if the peak is essentially too narrow and is not staying at a peak as long as it has in the past.

`Jim, I read that and I have been tweeting with a few of the meteorology grad students that were pointing out the unusual behavior. The behavior that they are referring to is happening starting in late 2015 to early 2016. In the figure below, the width if the peak is essentially too narrow and is not staying at a peak as long as it has in the past. ![tweet](https://pbs.twimg.com/media/ClLeq6cWMAAWbEf.jpg)`

More on what Jim Studdard mentioned last month

There is a refereed letter out concerning this behavior:

This is the view that they say is anomalous, starting after 2015. Look at the lower right

Since I have a model of QBO trained on data up to a few years ago, I can use it to judge whether the behavior is anomalous according to a projection. Recall that this model only applies periods derived from lunar tidal cycles, so that the outcomes are deterministic. This is a fit over each of the stratospheric levels (10 hPa to 70 hPa) reported monthly since 1953. The training is up until 2010, so the last few years are pure projections.

About the only anomaly I see is in the 70 hPa QBO fit. The upper stratosphere readings below 10 hPa start to be dominated by semiannual cycles while the lower stratosphere readings with atmospheric pressures higher than 70 hPa start to be impacted by strong seasonal cycles and by surface drag. So it's not surprising that the strong El Nino of late 2015/early 2016 could influence the 70 hPa QBO.

Reasoning about why they think that the QBO behavior is anomalous and "unprecedented", we need to remember that no predictive model for QBO exists. All they have is GCMs that apply Lindzen's original formulation. So what everything believes to be anomalous is predicated on qualitative reasoning based on past observations.

As Jim mentioned, the meteorologists on twitter and elsewhere get excited about this stuff:

I do think the 70 hPa behavior is new but that again can't easily be separated by the strong El Nino we experienced and the huge temperature spike in equatorial SST that occurred. That will certainly impact the QBO forcing terms to some extent.

`More on what [Jim Studdard mentioned last month](https://forum.azimuthproject.org/discussion/comment/15435/#Comment_15435) > This just turned up in the Independent: >"Lillo countered that the tweet referred to some unusual behavior of a phenomenon known as the QBO or Quasi-Biennial Oscillation, which is “a separate story” from the cross-equator flow. The QBO, he said, is an oscillation in equatorial stratospheric winds, which has been “out of phase.” He chalked up the weird QBO behavior to natural variability “even though I’m an advocate for identifying connections to human-caused climate change". >http://www.independent.co.uk/news/science/jet-stream-crossing-equator-global-climate-emergency-dismissed-debunked-a7113241.html There is a refereed letter out concerning this behavior: > [1] P. A. Newman, L. Coy, S. Pawson, and L. R. Lait, “The anomalous change in the QBO in 2015-16,” Geophysical Research Letters, 2016. [PDF](http://contextearth.com/wp-content/uploads/2016/08/10.1002@2016GL070373.pdf) > "The quasi-biennial oscillation (QBO) is a tropical lower stratospheric, downward propagating zonal wind variation, with an average period of ~28 months. The QBO has been constantly documented since 1953. Here we describe the evolution of the QBO during the Northern Hemisphere winter of 2015-16 using radiosonde observations and meteorological reanalyses. Normally, the QBO would show a steady downward propagation of the westerly phase. In 2015-16, there was an anomalous upward displacement of this westerly phase from ~30 hPa to 15 hPa. These westerlies impinge on, or “cut-off” the normal downward propagation of the easterly phase. In addition, easterly winds develop at 40 hPa. Comparisons to tropical wind statistics for the 1953-present record demonstrate that this 2015-16 QBO disruption is unprecedented." This is the view that they say is anomalous, starting after 2015. Look at the lower right --- ![newman](http://imageshack.com/a/img924/5552/du4suv.png) --- Since I have a [model of QBO](http://contextearth.com/2016/07/04/alternate-simplification-of-qbo-from-laplaces-tidal-equations/) trained on data up to a few years ago, I can use it to judge whether the behavior is anomalous according to a projection. Recall that this model only applies periods derived from lunar tidal cycles, so that the outcomes are deterministic. This is a fit over each of the stratospheric levels (10 hPa to 70 hPa) reported monthly since 1953. The training is up until 2010, so the last few years are pure projections. ![projection](http://imageshack.com/a/img921/8197/ai8hA7.png) About the only anomaly I see is in the 70 hPa QBO fit. The upper stratosphere readings below 10 hPa start to be dominated by semiannual cycles while the lower stratosphere readings with atmospheric pressures higher than 70 hPa start to be impacted by strong seasonal cycles and by surface drag. So it's not surprising that the strong El Nino of late 2015/early 2016 could influence the 70 hPa QBO. Reasoning about why they think that the QBO behavior is anomalous and "unprecedented", we need to remember that no predictive model for QBO exists. All they have is GCMs that apply Lindzen's original formulation. So what everything believes to be anomalous is predicated on qualitative reasoning based on past observations. As Jim mentioned, the meteorologists on twitter and elsewhere get excited about this stuff: ![ant](http://imageshack.com/a/img924/7370/IFcpGr.png) I do think the 70 hPa behavior is new but that again can't easily be separated by the strong El Nino we experienced and the huge temperature spike in equatorial SST that occurred. That will certainly impact the QBO forcing terms to some extent.`

I agree that especially by looking at the whole time series:

this looks anomalous. But I wouldn't rely on Singapur/Berlin data only.

There must be more measurements.

I haven't found though other recent visualizations. There are some visualizations from around 2004 in an article which seems -amongst others- to be based on saber emission radiometry measurements": http://www.atmos-chem-phys.net/9/3957/2009/acp-9-3957-2009.pdf

`>This is the view that they say is anomalous, starting after 2015. Look at the lower right I agree that especially by looking at the whole time series: ![fu](http://www.geo.fu-berlin.de/met/ag/strat/produkte/qbo/qbo_wind.jpg) this looks anomalous. But I wouldn't rely on Singapur/Berlin data only. There must be more measurements. I haven't found though other recent visualizations. There are some visualizations from around 2004 in an article which seems -amongst others- to be based on <a href="http://saber.gats-inc.com/index.php">saber emission radiometry measurements"</a>: <a href="http://www.atmos-chem-phys.net/9/3957/2009/acp-9-3957-2009.pdf">http://www.atmos-chem-phys.net/9/3957/2009/acp-9-3957-2009.pdf</a>`

nad said:

The paper claims that the data is not spurious because they say:

There is a nice analogy to this behavior in the reporting of ocean tide data. Consider a case in which the tidal behavior synchronizes with a model apart from occasional storm surges. So the anomalous behavior looks like this:

Everything agrees with the model except for the storm surge spike. The spike doesn't impact the rest of the time-series because that is set by the boundary-conditions of the lunar-solar gravitational tide periodicities. Same thing in this QBO case. Some of the QBO series appear to not even be impacted very strongly by whatever the perturbation (likely El Nino) is.

Yet my model and the consensus QBO model are so far apart in terms of fundamental agreement there is not much more to suggest what they are thinking. Their entire understanding is apparently based on the foundation set forward by Richard Lindzen. And I don't think anyone truly understands that. So their view of what is or isn't an anomaly is all based on a qualitative view or on their intuition. OTOH, mine is based on a quantitative tidal model, which is the view that Lindzen had considered at one time, but then dismissed on two different occasions that I can find.

`nad said: >"this looks anomalous. But I wouldn't rely on Singapur/Berlin data only. >There must be more measurements." The paper claims that the data is not spurious because they say: > "The consistency of the westerlies-to-easterlies development at all of the radiosonde locations means that the QBO anomaly cannot be attributed to data errors or radiosonde problems from one station." --- There is a nice analogy to this behavior in the reporting of ocean tide data. Consider a case in which the tidal behavior synchronizes with a model apart from occasional storm surges. So the anomalous behavior looks like this: ![oceantidesurge](http://contextearth.com/wp-content/uploads/2016/05/screenshot1.jpg) Everything agrees with the model except for the storm surge spike. The spike doesn't impact the rest of the time-series because that is set by the boundary-conditions of the lunar-solar gravitational tide periodicities. Same thing in this QBO case. Some of the QBO series appear to not even be impacted very strongly by whatever the perturbation (likely El Nino) is. Yet my model and the consensus QBO model are so far apart in terms of fundamental agreement there is not much more to suggest what they are thinking. Their entire understanding is apparently based on the foundation set forward by Richard Lindzen. And I don't think anyone truly understands that. So their view of what is or isn't an anomaly is all based on a qualitative view or on their intuition. OTOH, mine is based on a quantitative tidal model, which is the view that Lindzen had considered at one time, but then dismissed on two different occasions that I can find.`

As far as what I understood from the FU website the Singapur/Berlin data is from one location, namely from a radio sonde in Singapur:

And well there could also be errors in the handling of the data. Like if there is a change in scientific personel.

I don't know Richard Linzens theory but it seems there are also many other theories and models on QBO out there:

Gravity waves seem to be induced in part by tidal forces, which is also what you are at so I don't understand your argumentation.

But I have to say I dont quite understand the Gravity wave mechanisms theories and I dont have the time to look into them profoundly.

Moreover I am in particular more concerned about whats going on in the stratosphere. Unfortunately the Fig 3 diagram goes up only to 50kms, although they have apparently measurements which go up to 100kms ("The SABER instrument measures temperatures and several trace gases from the tropopause region to above 100 km"). But anyways in those Altitude-time cross sections of Kelvin wave temperature variances you can see that the easterly QBO phases are correlated with big temperature variations in the SAO. The QBO phases seem to be initiated by those phases of the SAO.

I dont know yet wether this plays a role but apparently there were quite some geomagnetic storms earlier this year.

`>The paper claims that the data is not spurious because they say: >> "The consistency of the westerlies-to-easterlies development at all of the radiosonde locations means that the QBO anomaly cannot be attributed to data errors or radiosonde problems from one station." As far as what I understood from the <a href="http://www.geo.fu-berlin.de/en/met/ag/strat/produkte/qbo/index.html">FU website</a> the Singapur/Berlin data is from one location, namely from a radio sonde in Singapur: >From these daily values the monthly mean zonal wind components were calculated for the levels 70, 50, 40, 30, 20, 15, and 10 hPa and a data set from 1953 to the present was produced by combining the observations of the three radiosonde stations Canton Island (closed 1967), Gan/Maledive Islands (closed 1975), and Singapore (data file: qbo.dat). And well there could also be errors in the handling of the data. Like if there is a change in scientific personel. I don't know Richard Linzens theory but it seems there are also many other theories and models on QBO out there: >The QBO was discovered in the 1950s by researchers at the UK Meteorological Office (Graystone 1959), but its cause remained unclear for some time. Radiosonde soundings showed that its phase was not related to the annual cycle, as is the case for many other stratospheric circulation patterns. In the 1970s it was recognized by Richard Lindzen and James Holton that the periodic wind reversal was driven by atmospheric waves emanating from the tropical troposphere that travel upwards and are dissipated in the stratosphere by radiative cooling. The precise nature of the waves responsible for this effect was heavily debated; in recent years, however, gravity waves have come to be seen as a major contributor and the QBO is now simulated in a growing number of climate models (Takahashi 1996, Scaife et al. 2000, Giorgetta et al. 2002) Gravity waves seem to be induced in part by tidal forces, which is also what you are at so I don't understand your argumentation. But I have to say I dont quite understand the Gravity wave mechanisms theories and I dont have the time to look into them profoundly. Moreover I am in particular more concerned about whats going on in the stratosphere. Unfortunately the <a href="http://www.atmos-chem-phys.net/9/3957/2009/acp-9-3957-2009.pdf">Fig 3 diagram</a> goes up only to 50kms, although they have apparently measurements which go up to 100kms ("The SABER instrument measures temperatures and several trace gases from the tropopause region to above 100 km"). But anyways in those Altitude-time cross sections of Kelvin wave temperature variances you can see that the easterly QBO phases are correlated with big temperature variations in the SAO. The QBO phases seem to be initiated by those phases of the SAO. I dont know yet wether this plays a role but apparently there were quite some geomagnetic storms earlier this year.`

nad,

Gravity waves are a general category. For example, if one perturbs a water surface, the effects of the

earth'sgravitational force are what causes the characteristic response. The elevated part of the liquid has more gravitational potential energy that the lower part of the liquid and so it will start to oscillate as it returns to equilibrium. Same general Newtonian principle as a pendulum, but applied to a fluid.The current consensus on QBO is that the period is an emergent resonance after applying a numerical computation to Lindzen's original formulation.

The same equations are used to calculate oceanic tides. The issue is that ocean tides can also be directly computed from the lunisolar periods. Yet somehow this was missed when people applied Lindzen's QBO formulation. Lindzen himself seemed to understand this possibility and I have quotes where he explains this, but never appeared to follow through.

There are no lunar parameterizations in any of the GCM models last I looked.

`nad, Gravity waves are a general category. For example, if one perturbs a water surface, the effects of the *earth's* gravitational force are what causes the characteristic response. The elevated part of the liquid has more gravitational potential energy that the lower part of the liquid and so it will start to oscillate as it returns to equilibrium. Same general Newtonian principle as a pendulum, but applied to a fluid. The [current consensus on QBO is that the period is an emergent resonance](http://contextearth.com/2016/06/19/recent-research-say-qbo-frequency-is-emergent-property/) after applying a numerical computation to Lindzen's original formulation. The same equations are used to calculate oceanic tides. The issue is that ocean tides can also be directly computed from the lunisolar periods. Yet somehow this was missed when people applied Lindzen's QBO formulation. Lindzen himself seemed to understand this possibility and I have quotes where he explains this, but never appeared to follow through. ![l2](http://imageshack.com/a/img921/5338/wmEyEU.png) ![l1](http://imageshack.com/a/img923/2756/rq6xwM.png) There are no lunar parameterizations in any of the GCM models last I looked.`

nad said:

At the upper atmosphere, the oscillations turn from being QBO to semi-annual oscillations (SAO) -- two full periods per year. So the forcing is obviously due to strong solar effects as the sun crosses the equator twice each year.

Below this altitude, the atmosphere gets denser and so the lunar tidal potential starts interacting with the sharpened (due to nonlinear amplification) solar pulses that occur twice a year. The solar pulses will synchronize with the moon as it crosses the equator every 13.606 days to create a constructive amplification every 2.36 years and progressively weaker amplifications every 0.7 years and 0.41 years. And that's exactly what you find in the QBO time-series waveform decomposition.

Two ways to look at this agreement.

That the odds of this happening by coincidence is quite small (see the Lindzen quotes in the previous post above).

Someone will have to come up with a theory that rules this out as a mechanism. They will have to demonstrate that the lunar forcing magnitude is physically too small to have any effect.

Historically, that is essentially the same argument used by mathematicians that first looked at ocean tides. They first had an exact agreement with a composite of lunar cycles, and then when Newton's theories were formalized, they were satisfied that the lunar gravitational forcing was great enough to have an observable effect.

Laplace's tidal equations were used to formalize the effect for ocean tides, and I apply them to the QBO along the equator on this blog post. This is one of the most concise and elegant physical models that I have developed in a long while, a great melding of applied physics and applied mathematics. I wouldn't be surprised if someone else has done this somewhere in the research literature, but I have yet to uncover it.

`nad said: > "Moreover I am in particular more concerned about whats going on in the stratosphere. Unfortunately the Fig 3 diagram goes up only to 50kms, although they have apparently measurements which go up to 100kms ("The SABER instrument measures temperatures and several trace gases from the tropopause region to above 100 km"). But anyways in those Altitude-time cross sections of Kelvin wave temperature variances you can see that the easterly QBO phases are correlated with big temperature variations in the SAO. The QBO phases seem to be initiated by those phases of the SAO." At the upper atmosphere, the oscillations turn from being QBO to semi-annual oscillations (SAO) -- two full periods per year. So the forcing is obviously due to strong solar effects as the sun crosses the equator twice each year. Below this altitude, the atmosphere gets denser and so the lunar tidal potential starts interacting with the sharpened (due to nonlinear amplification) solar pulses that occur twice a year. The solar pulses will synchronize with the moon as it crosses the equator every 13.606 days to create a constructive amplification every 2.36 years and progressively weaker amplifications every 0.7 years and 0.41 years. And that's exactly what you find in the QBO time-series waveform decomposition. ![dr](http://imagizer.imageshack.us/a/img921/6223/oi6jJE.png) Two ways to look at this agreement. - That the odds of this happening by coincidence is quite small (see the Lindzen quotes in the previous post above). - Someone will have to come up with a theory that rules this out as a mechanism. They will have to demonstrate that the lunar forcing magnitude is physically too small to have any effect. Historically, that is essentially the same argument used by mathematicians that first looked at ocean tides. They first had an exact agreement with a composite of lunar cycles, and then when Newton's theories were formalized, they were satisfied that the lunar gravitational forcing was great enough to have an observable effect. [Laplace's tidal equations](https://en.wikipedia.org/wiki/Theory_of_tides) were used to formalize the effect for ocean tides, and I apply them to the [QBO along the equator on this blog post](http://contextearth.com/2016/07/04/alternate-simplification-of-qbo-from-laplaces-tidal-equations/). This is one of the most concise and elegant physical models that I have developed in a long while, a great melding of applied physics and applied mathematics. I wouldn't be surprised if someone else has done this somewhere in the research literature, but I have yet to uncover it.`

What solar pulses occur twice a year? How should that synchronizing take place? Do you mean solar wind?

I have no idea how strong the gravitation pull by the moon manifests itself on the solar wind, I would say small but in principle there could be an effect of that sort - if that is what you mean. (The moon is of course also sometimes in the way between sun and earth.) And if your machine learning model has been done rightly (of which I am not convinced yet) and you wonderously really only need the lunar periods as input (thats what I understood from your model) then the 2.36 years together with the semiannual SAO could eventually result in something that looks strictly every 2 years with intermediate phase flips (thats what I found the QBO looks like). But then if the moon crosses the equator and this "solarwindpulling"- effect is of significant size than the QBO would rather probably show longitudinal differences. But on the FU website it is written:

`>Below this altitude, the atmosphere gets denser and so the lunar tidal potential starts interacting with the sharpened (due to nonlinear amplification) solar pulses that occur twice a year. The solar pulses will synchronize with the moon as it crosses the equator every 13.606 days to create a constructive amplification every 2.36 years and progressively weaker amplifications every 0.7 years and 0.41 years. And that's exactly what you find in the QBO time-series waveform decomposition. What solar pulses occur twice a year? How should that synchronizing take place? Do you mean solar wind? I have no idea how strong the gravitation pull by the moon manifests itself on the solar wind, I would say small but in principle there could be an effect of that sort - if that is what you mean. (The moon is of course also sometimes in the way between sun and earth.) And if your machine learning model has been done rightly (of which I am not convinced yet) and you wonderously really only need the lunar periods as input (thats what I understood from your model) then the 2.36 years together with the semiannual SAO could eventually result in something that looks strictly every 2 years with intermediate phase flips (thats what I found the QBO looks like). But then if the moon crosses the equator and this "solarwindpulling"- effect is of significant size than the QBO would rather probably show longitudinal differences. But on the <a href="http://www.geo.fu-berlin.de/en/met/ag/strat/produkte/qbo/index.html">FU website</a> it is written: >This data set is supposed to be representative of the equatorial belt since all studies have shown that longitudinal differences in the phase of the QBO are small.`

nad asked:

The QBO and SAO only occurs along the equator and the solar equinox occurs twice a year, so that explains the synchronization of the SAO. That's when the maximum solar energy is directed to the equatorial latitudes. And at the altitude of the SAO, more of the UV is trapped by ozone, which means that this layer is more sensitive to solar than lower stratospheric layers.

You also questioned:

I don't rely on the machine-learning model. I actually derived the QBO equations from first principles and completely independently, fully documented here, and then incorporated the seasonally aliased nodal tidal cycles as a latitudinal forcing that couples into Laplace's equations as a cross term. All the machine-learning does is confirm that this is the best-fitting symbolic representation available. It's similar to predicting that a given trajectory should be parabolic and then using a symbolic regression tool to determine that a parabolic arc is indeed the most concise or least complex representation. In other words, the symbolic regression machine learning doesn't prove anything, it's just evidence for a plausible fit.

I don't know about that. The earth rotating once per day is probably enough to spread the solar energy out evenly across the longitudes. The embedded video below is how the QBO appears as a collective standing wave that encircles the earth. That's essentially a spatial wave-number of zero. No real longitudinal differences.

`nad asked: > "What solar pulses occur twice a year? How should that synchronizing take place? Do you mean solar wind?" The QBO and SAO only occurs along the equator and the [solar equinox](https://en.wikipedia.org/wiki/Equinox) occurs twice a year, so that explains the synchronization of the SAO. That's when the maximum solar energy is directed to the equatorial latitudes. And at the altitude of the SAO, more of the UV is trapped by ozone, which means that this layer is more sensitive to solar than lower stratospheric layers. You also questioned: > "... and if your machine learning model has been done rightly" I don't rely on the machine-learning model. I actually derived the QBO equations from first principles and completely independently, [fully documented here](http://contextearth.com/2016/07/04/alternate-simplification-of-qbo-from-laplaces-tidal-equations/), and then incorporated the seasonally aliased nodal tidal cycles as a latitudinal forcing that couples into Laplace's equations as a cross term. All the machine-learning does is confirm that this is the best-fitting symbolic representation available. It's similar to predicting that a given trajectory should be parabolic and then using a symbolic regression tool to determine that a parabolic arc is indeed the most concise or least complex representation. In other words, the symbolic regression machine learning doesn't prove anything, it's just evidence for a plausible fit. > " than the QBO would rather probably show longitudinal differences" I don't know about that. The earth rotating once per day is probably enough to spread the solar energy out evenly across the longitudes. The embedded video below is how the QBO appears as a collective standing wave that encircles the earth. That's essentially a spatial wave-number of zero. No real longitudinal differences. <iframe src="https://www.youtube.com/embed/I_-QlDmicIw" allowfullscreen="" height="480" width="854" frameborder="0"></iframe>`

thanks for the video.

It seems the differences may be on the height level of SAO. Even in the video one can observe fringes.

OK but if this 2.3 ys periodicity exists (I havent checked) the lunar influence could be a major source for QBO, as an ansatz.

`thanks for the video. >No real longitudinal differences. It seems the differences may be on the height level of SAO. Even in the video one can observe fringes. OK but if this 2.3 ys periodicity exists (I havent checked) the lunar influence could be a major source for QBO, as an ansatz.`

I don't really know what an "ansatz" means in this context. Is that what we could call a "premise"? All I know is how to apply the foundational physics of tides to the problem and simplify the equations according to a small angle approximation at the equator. It's essentially one simplifying step removed from what Laplace proposed in the 1700's.

It's not well known but Laplace's tidal equations are closely related to the numerically solved general circulation models used to simulate the climate. The primitive equations

"are a set of nonlinear differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models". The complete set of primitive equations are fairly complex so that"The primitive equations may be linearized to yield Laplace's tidal equations, an eigenvalue problem from which the analytical solution to the latitudinal structure of the flow may be determined."So what I did was take Laplace's equations and simplify them further here. Then we put in the forcing terms according to the lunar and solar cycles and come up with the QBO behavior straightforwardly.

Is that what you call an ansatz?

`I don't really know what an "ansatz" means in this context. Is that what we could call a "premise"? All I know is how to apply the foundational physics of tides to the problem and simplify the equations according to a small angle approximation at the equator. It's essentially one simplifying step removed from what Laplace proposed in the 1700's. It's not well known but [Laplace's tidal equations](https://en.wikipedia.org/wiki/Theory_of_tides) are closely related to the numerically solved [general circulation models](https://en.wikipedia.org/wiki/General_circulation_model) used to simulate the climate. The [primitive equations](https://en.wikipedia.org/wiki/Primitive_equations) *"are a set of nonlinear differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models"*. The complete set of primitive equations are fairly complex so that *"The primitive equations may be linearized to yield Laplace's tidal equations, an eigenvalue problem from which the analytical solution to the latitudinal structure of the flow may be determined."* So what I did was take Laplace's equations and simplify them further [here](http://contextearth.com/2016/07/04/alternate-simplification-of-qbo-from-laplaces-tidal-equations/). Then we put in the forcing terms according to the lunar and solar cycles and come up with the QBO behavior straightforwardly. Is that what you call an ansatz?`

The discussion of QBO probably should be in this thread: https://forum.azimuthproject.org/discussion/1640/predictability-of-the-quasi-biennial-oscillation#latest

Yet that's OK as ENSO and QBO are related in terms of being the major standing wave modes along the equator for (respectively) oceanic & atmospheric behaviors.

My confidence level on getting the model for QBO right is about 95% and for ENSO its about 75%. I don't see any holes in the QBO, being based on a straightforward physical derivation. The ENSO model is more tricky as it requires the invocation of a strict biennial modulation, which is known to be metastable on odd vs even years.

The QBO is best fit as a solution to a Sturm-Liouville equation, while ENSO fits best as a cyclically-forced Mathieu equation. I have yet to try what a combined Sturm-Liouville and Mathieu equation would look like.

The essential difference between the two is described by the following parametric DiffEq

$ f(t) (1+cos(w t) ) + d/dt(f'(t)/(1+Kcos(w t))) = 0 $

For a Mathieu equation, K=0 while K=1 gives a solvable Sturm-Liouville formulation.

The character of the solutions differ. Mathieu gives an erratic amplitude and less predictable period (much like ENSO) while the Sturm-Liouville clamps the amplitude and has a more regular period (much like QBO).

The Red curve below is a Sturm-Liouville superimposed on a family of Mathieu solutions. The Mathieu modulation is very slight so the erratic nature is suppressed.

`The discussion of QBO probably should be in this thread: https://forum.azimuthproject.org/discussion/1640/predictability-of-the-quasi-biennial-oscillation#latest Yet that's OK as ENSO and QBO are related in terms of being the major standing wave modes along the equator for (respectively) oceanic & atmospheric behaviors. My confidence level on getting the model for QBO right is about 95% and for ENSO its about 75%. I don't see any holes in the QBO, being based on a straightforward physical derivation. The ENSO model is more tricky as it requires the invocation of a strict biennial modulation, which is known to be metastable on odd vs even years. The QBO is best fit as a solution to a Sturm-Liouville equation, while ENSO fits best as a cyclically-forced Mathieu equation. I have yet to try what a combined Sturm-Liouville and Mathieu equation would look like. The essential difference between the two is described by the following parametric DiffEq $ f(t) (1+cos(w t) ) + d/dt(f'(t)/(1+Kcos(w t))) = 0 $ For a Mathieu equation, K=0 while K=1 gives a solvable Sturm-Liouville formulation. The character of the solutions differ. Mathieu gives an erratic amplitude and less predictable period (much like ENSO) while the Sturm-Liouville clamps the amplitude and has a more regular period (much like QBO). The Red curve below is a Sturm-Liouville superimposed on a family of Mathieu solutions. The Mathieu modulation is very slight so the erratic nature is suppressed. ![vs](http://imageshack.com/a/img924/4739/5tRnVh.png)`

Sorry about findig the right Thread I am currently One Finger Typing with partially Crazy autocorrection and very bad connection. I wrote Ansatz because the 2.4 Period Appears to me to be a lot for having those rather long strictly 2 year periods before a "flip" of phase. I have briefly glanced at your derivation of the sturm-liouville equation from your hough tidal operator comment linked as comment 15433 in Jim's slab model thread (sorry can't link right now) . It's not clear to me how you motivate your constants and I am not sure how much machine learning is entering this process. By the way I think there was also a sign mistake when you were setting \mu to very small against sigma I n front of one of the non derivative terms (edgy brackets).

`Sorry about findig the right Thread I am currently One Finger Typing with partially Crazy autocorrection and very bad connection. I wrote Ansatz because the 2.4 Period Appears to me to be a lot for having those rather long strictly 2 year periods before a "flip" of phase. I have briefly glanced at your derivation of the sturm-liouville equation from your hough tidal operator comment linked as comment 15433 in Jim's slab model thread (sorry can't link right now) . It's not clear to me how you motivate your constants and I am not sure how much machine learning is entering this process. By the way I think there was also a sign mistake when you were setting \mu to very small against sigma I n front of one of the non derivative terms (edgy brackets).`

nad, This is what

ansatzmeans according to WikipediaI also find that ansatz is also understood as what we call an "educated guess". In my case, the educated guess I made was in predicting that solving Laplace's tidal equation at the equator for a small latitude approximation (i.e. Coriolis forces cancel to zero) would lead to a possible answer. That's educated because it applies what I know about the physics of the situation. It's also educated because all GCM's use this same set of equations to simulate the QBO.

The way you are using

ansatzsounds more like acaveatorprovisowhich means a clause, qualification, or condition. You are expressing concerns about the derivation based on inconsistencies that you observe.The caveat that I expressed is that my derivation was for delta variations in atmospheric pressure, whereas QBO is a measure of velocity. I brushed this aside for the moment because the two - velocity and pressure - are closely related at the differential level (for example see Bernoulli's equation).

I can't find the possible mistake that you are referring to because I solved the equations here without a $\mu$ in the derivation

http://contextearth.com/2016/07/04/alternate-simplification-of-qbo-from-laplaces-tidal-equations/

If you can be more specific, I would appreciate it.

I also don't understand how you would think machine learning is entering the process. I am simply reducing and solving a set of partial differential equations, just as one does in a college calculus course. I don't ever recall needing to invoke a machine learning tool to do that. Even if I did, there are plenty of symbolic tools available for reducing and checking equation derivations. I am sure many mathematicians use these to check their derivations, whether they are willing to admit to it or not!

`> " I wrote Ansatz because the 2.4 Period Appears to me to be a lot for having those rather long strictly 2 year periods before a "flip" of phase." nad, This is what *ansatz* means according to Wikipedia > "An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing a mathematical or physical problem or solution. It can take into consideration boundary conditions." I also find that ansatz is also understood as what we call an "educated guess". In my case, the educated guess I made was in predicting that solving Laplace's tidal equation at the equator for a small latitude approximation (i.e. Coriolis forces cancel to zero) would lead to a possible answer. That's educated because it applies what I know about the physics of the situation. It's also educated because all GCM's use this same set of equations to simulate the QBO. The way you are using *ansatz* sounds more like a *caveat* or *proviso* which means a clause, qualification, or condition. You are expressing concerns about the derivation based on inconsistencies that you observe. The caveat that I expressed is that my derivation was for delta variations in atmospheric pressure, whereas QBO is a measure of velocity. I brushed this aside for the moment because the two - velocity and pressure - are closely related at the differential level (for example see Bernoulli's equation). I can't find the possible mistake that you are referring to because I solved the equations here without a $\mu$ in the derivation http://contextearth.com/2016/07/04/alternate-simplification-of-qbo-from-laplaces-tidal-equations/ If you can be more specific, I would appreciate it. I also don't understand how you would think machine learning is entering the process. I am simply reducing and solving a set of partial differential equations, just as one does in a college calculus course. I don't ever recall needing to invoke a machine learning tool to do that. Even if I did, there are plenty of symbolic tools available for reducing and checking equation derivations. I am sure many mathematicians use these to check their derivations, whether they are willing to admit to it or not!`

ansatz means here one sets up some mathematical structure in order to get closer to finding a more or less meaningful model. I can't look through all your calculations. I just saw that there should be a minus sign in front of the [...] term in your Jun30 comment 15433 formula after "the equation reduces to".

`ansatz means here one sets up some mathematical structure in order to get closer to finding a more or less meaningful model. I can't look through all your calculations. I just saw that there should be a minus sign in front of the [...] term in your Jun30 comment 15433 formula after "the equation reduces to".`

Concerning the machine learning eg Jul 16 a little further down from comment 15433 You wrote:

This sounded to me as if you had adjusted some constants with the help of experimental data.

`Concerning the machine learning eg Jul 16 a little further down from comment 15433 You wrote: >this is a caveat to the Laplace's tidal equation fit to the QBO data. Although the seasonally aliased Draconic period works very well, a This sounded to me as if you had adjusted some constants with the help of experimental data.`

OK, if its what I think you are referring to, that gets merged into a spatial wavenumber constant. In any case, it won't change the structure of the final equation. The next comment redoes the derivation from an earlier starting point, and that same part gets swept in to the SW(s) constant spatial term before the sign is even considered.

In fact, that's how tidal equations were originally devised and that's how they have worked for the last 100 years. The basic equations are parameterized and then the known lunar and solar periods are filled in and the amplitudes of each are adjusted to give the best fit. I was simply showing the sensitivity of the fit to the exact value of the known tidal periods. See that here on a variable scale. The 4 main main tidal periods are very close, and the fifth "evection" is a second order solar forcing on the moon term that may not be detectable at this scale.

Whether an optimal 27.209 day period is more correct than the nominal 27.212 day period for that lunar period is something that I am indeed interested in. I don't see any real difference in eyeballing the visual quality of the fits, but the correlation coefficient calculation does capture a slight improvement. Whether that is meaningful or within the margins of the uncertainty (measured or natural) I can't tell. I did that exercise because if I didn't, someone was going to ask for it.

Here is a set of nutation periods experienced by earth due to the lunisolar forces. This is reported by Bizouard, and you can see the value is 27.20986 for the nodal force period. Why that isn't 27.212, I don't understand either. If the period is not exact, then the phases will eventually interfere, like after several hundred years.

`> I just saw that there should be a minus sign in front of the [...] term in your Jun30 comment 15433 formula after "the equation reduces to". OK, if its what I think you are referring to, that gets merged into a spatial wavenumber constant. In any case, it won't change the structure of the final equation. The [next comment](https://forum.azimuthproject.org/discussion/comment/15434/#Comment_15434) redoes the derivation from an earlier starting point, and that same part gets swept in to the SW(s) constant spatial term before the sign is even considered. > "This sounded to me as if you had adjusted some constants with the help of experimental data." In fact, that's how tidal equations were originally devised and that's how they have worked for the last 100 years. The basic equations are parameterized and then the known lunar and solar periods are filled in and the amplitudes of each are adjusted to give the best fit. I was simply showing the sensitivity of the fit to the exact value of the known tidal periods. See that here on a variable scale. The 4 main main tidal periods are very close, and the fifth "evection" is a second order solar forcing on the moon term that may not be detectable at this scale. ![sens](http://imagizer.imageshack.us/a/img911/5238/fBNR2Q.png) Whether an optimal 27.209 day period is more correct than the nominal 27.212 day period for that lunar period is something that I am indeed interested in. I don't see any real difference in eyeballing the visual quality of the fits, but the correlation coefficient calculation does capture a slight improvement. Whether that is meaningful or within the margins of the uncertainty (measured or natural) I can't tell. I did that exercise because if I didn't, someone was going to ask for it. Here is a set of nutation periods experienced by earth due to the lunisolar forces. This is reported by Bizouard, and you can see the value is 27.20986 for the nodal force period. Why that isn't 27.212, I don't understand either. If the period is not exact, then the phases will eventually interfere, like after several hundred years. ![biz](http://imageshack.com/a/img923/5129/9UOz0v.png)`

Yes that won't change the structure of the equation you set up but it might eventually be important when fixing parameters.

I don't know the meaning of all those parameters and periods and so I don't understand what you wanted to show with this diagram and what with this table.

`Yes that won't change the structure of the equation you set up but it might eventually be important when fixing parameters. I don't know the meaning of all those parameters and periods and so I don't understand what you wanted to show with this diagram and what with this table.`

nad, The primary period is simply the time it takes the moon to complete one nodal cycle, 27.212 days. If the maximum latitude swing occurs at the most sensitive time of the year at the equator, that will cause the maximum gravitational forcing inputs for the tidal equations. So the repeat period for QBO is defined by this value according to the model. That turns out be about 28 months, matching the observed fundamental period.

But it is not only that value of 28 months, but there are other minor but sharp Fourier components at 5 months and 8.4 months in the QBO spectrum which theoretically must be there if the 28 month period is observed. See this derivation

And then there are the other lunar periods of 27.32 days and 27.55 days which will reinforce the fundamental period, just like it does for ocean tides. Those show up distinctly as well but they are not as strong as the nodal period. And the 27.266 day Mf' component shows up sharply as well, which is essentially the average of 27.32 Draconic and 27.212 Tropical cycles.

It would be the oddest coincidence if these periods all appeared by chance in the QBO. Of course there is always that possibility, but that can't be used to invalidate the model. OTOH, if the cycles were not aligned, then the model could be easily invalidated.

`nad, The primary period is simply the time it takes the moon to complete one nodal cycle, 27.212 days. If the maximum latitude swing occurs at the most sensitive time of the year at the equator, that will cause the maximum gravitational forcing inputs for the tidal equations. So the repeat period for QBO is defined by this value according to the model. That turns out be about 28 months, matching the observed fundamental period. But it is not only that value of 28 months, but there are other minor but sharp Fourier components at 5 months and 8.4 months in the QBO spectrum which theoretically must be there if the 28 month period is observed. [See this derivation](http://contextearth.com/2015/11/17/the-math-of-seasonal-aliasing/) And then there are the other lunar periods of 27.32 days and 27.55 days which will reinforce the fundamental period, just like it does for ocean tides. Those show up distinctly as well but they are not as strong as the nodal period. And the 27.266 day Mf' component shows up sharply as well, which is essentially the average of 27.32 Draconic and 27.212 Tropical cycles. It would be the oddest coincidence if these periods all appeared by chance in the QBO. Of course there is always that possibility, but that can't be used to invalidate the model. OTOH, if the cycles were not aligned, then the model could be easily invalidated.`