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## Comments

nad, Thanks for following through on this idea of yours.

I have been looking at the possibility of an

exactbiannual period as you have suggested for the QBO.The model I am using for QBO is that there is a characteristic resonant frequency associated with the wave equation, but that an exact biannual modulation is applied.

$ f''(t) + (a + q \sin(4 \pi t + \theta)) f(t) = 0 $

This differs from applying a RHS forcing of a biannual modulation, because that would form strong modulated beat frequencies in the output. But that kind of modulation is not seen in the QBO -- as the amplitudes are roughly constant over the decades.

This is one fit that I created with the biannual Mathieu modulation

In addition to the biannual modulation, I added two longer term modulations -- a ~12 year period and a slow 47 year time dilation. The latter two corrections improves correlation coefficient until the residual starts to appear closer to white noise.

The above data has a median filter and a long-range mean filter applied. If I don't filter, the comparison looks like:

I do think the biannual modulation is a real effect. The details of the waveform, in particular the flattening of the peaks with the characteristic shoulders are very difficult to reproduce any other way. It is possible that the flattening is a saturation effect but that is definitely something that can be added to the model.

This also ties in to the tide gauge model for ENSO and the seasonal alignment thread. The difference is that for the ocean, the exact biennial oscillation shows up as a RHS

forcing. Perhaps the way to understand this is that the atmosphere has less inertia and thus is able to respond to a faster modulation perturbing the characteristics of the media, i.e. the natural resonance.Of course this could use additional machine learning -- once a correlation coefficient gets above 0.8 it often provides enough of a guide that a search algorithm can really start to zone in on a solution. I see that many times with the Eureqa tool ; once it locks in to a correlation approaching unity, it often moves quickly.

`nad, Thanks for following through on this idea of yours. I have been looking at the possibility of an *exact* biannual period as you have suggested for the QBO. The model I am using for QBO is that there is a characteristic resonant frequency associated with the wave equation, but that an exact biannual modulation is applied. $ f''(t) + (a + q \sin(4 \pi t + \theta)) f(t) = 0 $ This differs from applying a RHS forcing of a biannual modulation, because that would form strong modulated beat frequencies in the output. But that kind of modulation is not seen in the QBO -- as the amplitudes are roughly constant over the decades. This is one fit that I created with the biannual Mathieu modulation ![QBO](http://imageshack.com/a/img911/2246/KqvGek.gif) In addition to the biannual modulation, I added two longer term modulations -- a ~12 year period and a slow 47 year time dilation. The latter two corrections improves correlation coefficient until the residual starts to appear closer to white noise. The above data has a median filter and a long-range mean filter applied. If I don't filter, the comparison looks like: ![QBO noFilter](http://imageshack.com/a/img745/1064/BpZIrB.gif) I do think the biannual modulation is a real effect. The details of the waveform, in particular the flattening of the peaks with the characteristic shoulders are very difficult to reproduce any other way. It is possible that the flattening is a saturation effect but that is definitely something that can be added to the model. This also ties in to the [tide gauge model for ENSO](http://forum.azimuthproject.org/discussion/1480/tidal-records-and-enso/#Item_50) and the [seasonal alignment thread](http://forum.azimuthproject.org/discussion/1497/nino-3-and-seasonal-alignment/#Item_3). The difference is that for the ocean, the exact biennial oscillation shows up as a RHS *forcing*. Perhaps the way to understand this is that the atmosphere has less inertia and thus is able to respond to a faster modulation perturbing the characteristics of the media, i.e. the natural resonance. Of course this could use additional machine learning -- once a correlation coefficient gets above 0.8 it often provides enough of a guide that a search algorithm can really start to zone in on a solution. I see that many times with the Eureqa tool ; once it locks in to a correlation approaching unity, it often moves quickly.`

Paul

I am going to join you with these equations, but I will go the route of machine learning and fitting and visualization

DAra

`Paul I am going to join you with these equations, but I will go the route of machine learning and fitting and visualization DAra`

That would be a good idea to either validate or find alternate solutions. We need some more horsepower.

`That would be a good idea to either validate or find alternate solutions. We need some more horsepower.`

This is another view of how the biannual modulation creates the extra shoulder features on the QBO oscillations

Note the red arrows which indicate the biannual accentuation on the quasi-biennial oscillations. Those appear at periods of half a year.

There may also be some confusion here. The three "exact" frequencies that seem operable are as follows

I have seen the influence of each of these frequencies but with different emphasis depending on the physical behavior we are looking at.

It may be useful to make a chart to keep track of these factors.

`This is another view of how the biannual modulation creates the extra shoulder features on the QBO oscillations ![QBO features](http://imageshack.com/a/img905/3964/73w9is.gif) Note the red arrows which indicate the biannual accentuation on the quasi-biennial oscillations. Those appear at periods of half a year. There may also be some confusion here. The three "exact" frequencies that seem operable are as follows * biannual = twice a year (also known as semi-annual) * annual = once a year * biennial = once every two years I have seen the influence of each of these frequencies but with different emphasis depending on the physical behavior we are looking at. It may be useful to make a chart to keep track of these factors.`

In the code it seems you are not using the above equation but the following Hill equation

$$y''(x) + (const1+ const2

cos(4 \pi x + const3) + const4cos(0.5195 \pi x + const5))*y(x)=0$$ I am not sure (anymore) how the solutions of Hills equation look like, but I would think that if the solution is periodic then it would have periodicities as in the given fourier expansion, so this points to a periodicity of 1/0.5195 and not 1/0.5 (which would be the exact biannuity).I assume that "First" plots out y(x) of NDSolve in particular I don't have Mathematica to check. Moreover in your plot you have that strange $y(x+const*sin(const x +....) is that what you call a filter?

Where do you have the QBO data from? Did you compare that with the one in the blogpost you had linked to?

`>The model I am using for QBO is that there is a characteristic resonant frequency associated with the wave equation, but that an exact biannual modulation is applied. >f″(t)+(a+qsin(4πt+θ))f(t)=0 In the code it seems you are not using the above equation but the following <a href="http://en.wikipedia.org/wiki/Hill_differential_equation">Hill equation</a> $$y''(x) + (const1+ const2*cos(4 \pi x + const3) + const4*cos(0.5195 \pi x + const5))*y(x)=0$$ I am not sure (anymore) how the solutions of Hills equation look like, but I would think that if the solution is periodic then it would have periodicities as in the given fourier expansion, so this points to a periodicity of 1/0.5195 and not 1/0.5 (which would be the exact biannuity). I assume that "First" plots out y(x) of NDSolve in particular I don't have Mathematica to check. Moreover in your plot you have that strange $y(x+const*sin(const x +....) is that what you call a filter? Where do you have the QBO data from? Did you compare that with the one in the blogpost you had linked to?`

Indeed there seems to be an ambiguity that is at least the german english dictionary LEO gives here:

This is crazy. I mean with biannual : EVERY TWO YEARS

`>There may also be some confusion here. The three “exact” frequencies that seem operable are as follows > biannual = twice a year (also known as semi-annual) > annual = once a year > biennial = once every two years Indeed there seems to be an ambiguity that is at least the german english dictionary LEO gives [here](http://dict.leo.org/#/search=biannual&searchLoc=0&resultOrder=basic&multiwordShowSingle=on): biannual also: bi-annual adj. - semiannual halbjährlich biannual also: bi-annual adj. - every two years jedes zweite Jahr biannual also: bi-annual adj. - semiannual zweimal jährlich This is crazy. I mean with biannual : EVERY TWO YEARS`

Neither the Mathieu nor the Hill equation is easily understood in terms of a Fourier series. The expansion is a study in recursion of the coefficients, to put it mildly.

That's what makes the analysis challenging yet potentially practical to this domain. None of these timeseries is periodic, but nor are they chaotic. They are closer to quasiperiodic and thus amenable to nonconventional analysis approaches.

If people have ideas, I am all ears.

`Neither the Mathieu nor the Hill equation is easily understood in terms of a Fourier series. The expansion is a study in recursion of the coefficients, to put it mildly. That's what makes the analysis challenging yet potentially practical to this domain. None of these timeseries is periodic, but nor are they chaotic. They are closer to quasiperiodic and thus amenable to nonconventional analysis approaches. If people have ideas, I am all ears.`

I am not sure what you want to say with that. The Mathieu equation is a special case of the Hill equation in terms of its Fourier expansion, as written in the : Hill equation (or almost the same here)

Periodicity is (as I know it) a special case of Quasiperiodicity, you probably meant aperiodic. ?? By the way it seems it is actually rather the constant term, which dictates the periodicity, like according to Wikipedia the "a" here in the Wikipedia article about the Mathieu function enters the Mathieu sine/cosine. Weisstein says the same. Apart from the special cos/sin case in order to find periodic solutions one would need to find characteristic values.

`>Neither the Mathieu nor the Hill equation is easily understood in terms of a Fourier series. I am not sure what you want to say with that. The Mathieu equation is a special case of the Hill equation in terms of its Fourier expansion, as written in the : <a href="http://en.wikipedia.org/wiki/Hill_differential_equation">Hill equation</a> (or almost the same [here](http://mathworld.wolfram.com/HillsDifferentialEquation.html)) >They are closer to quasiperiodic. Periodicity is (as I know it) a special case of Quasiperiodicity, you probably meant aperiodic. ?? By the way it seems it is actually rather the constant term, which dictates the periodicity, like according to Wikipedia the "a" here in the Wikipedia article about the [Mathieu function](http://en.wikipedia.org/wiki/Mathieu_function) enters the Mathieu sine/cosine. [Weisstein](http://mathworld.wolfram.com/MathieuFunction.html) says the same. Apart from the special cos/sin case in order to find periodic solutions one would need to find characteristic values.`

Yes, the MathieuC and MathieuS functions are derived as Fourier coefficients, but they are recursively established by the +/- perturbation of the $\omega$ frequency on the wave equation. It does take lots of computations to get the Mathieu functions to converge properly ( see here ). It definitely is not something as simple as the Taylor series expansion of a Sine function, which is what I meant by not "easily understood".

The line between quasi-periodicity versus aperiodicity is a fine one. In the limit of small q (which is the amplitude of the Mathieu modulation), the MathieuC and MathieuS functions converge to Sine and Cosine functions, which are known to be periodic. The way to think about this is that as the modulation gets stronger, the period of repeat starts to extend, and unless one has a long-enough time series to deal with, one might not pick up the repeat sequence.

Here is an example of a MathieuC function where one can pick out some of the repeat period

That repeat sequence appears to be about 57 time units, but even that evolves somewhat. It is all a matter of practical application. Look up the Floquet theorem and I think there are more references to the solutions of these equations being quasiperiodic than aperiodic, but I might be wrong.

Actually that might not be a bad idea for machine learning from paleo records -- look for patterns of a repeat sequence in the historical data.

`Yes, the MathieuC and MathieuS functions are derived as Fourier coefficients, but they are recursively established by the +/- perturbation of the $\omega$ frequency on the wave equation. It does take lots of computations to get the Mathieu functions to converge properly ( [see here](http://dlmf.nist.gov/28.4) ). It definitely is not something as simple as the Taylor series expansion of a Sine function, which is what I meant by not "easily understood". The line between quasi-periodicity versus aperiodicity is a fine one. In the limit of small q (which is the amplitude of the Mathieu modulation), the MathieuC and MathieuS functions converge to Sine and Cosine functions, which are known to be periodic. The way to think about this is that as the modulation gets stronger, the period of repeat starts to extend, and unless one has a long-enough time series to deal with, one might not pick up the repeat sequence. Here is an example of a MathieuC function where one can pick out some of the repeat period ![MathieuC](http://imageshack.com/a/img742/1842/wYewOa.gif) That repeat sequence appears to be about 57 time units, but even that evolves somewhat. It is all a matter of practical application. Look up the Floquet theorem and I think there are more references to the solutions of these equations being quasiperiodic than aperiodic, but I might be wrong. Actually that might not be a bad idea for machine learning from paleo records -- look for patterns of a repeat sequence in the historical data.`

I don't understand what you mean by "+/- perturbation of the $\omega$ frequency on the wave equation" and I have troubles to understand this recursion in the link you gave, finally m is supposed to go to infinity in the fourier expansion, but then in the recursion explanation it says m goes until $a_{2n}$, where $a_{2n}$ seems to be a characteristic value.

Yes. Taylor expensions are funny, aren't they?

I don't understand what you mean with repeat sequence.

But apart from that Mathieu function discussion, I am not so sure wether there exists at all a mathematical function, which would describe such a "forced to periodicity" behaviour as it seems to be the case for the QBO (at least if one believes that blog post diagram)

Did you check the QBO diagram?

`>Yes, the MathieuC and MathieuS functions are derived as Fourier coefficients, but they are recursively established by the +/- perturbation of the $\omega$ frequency on the wave equation. It does take lots of computations to get the Mathieu functions to converge properly ( see here ). I don't understand what you mean by "+/- perturbation of the $\omega$ frequency on the wave equation" and I have troubles to understand this recursion in the link you gave, finally m is supposed to go to infinity in the fourier expansion, but then in the recursion explanation it says m goes until $a_{2n}$, where $a_{2n}$ [seems](http://mathworld.wolfram.com/MathieuFunction.html) to be a characteristic value. >It definitely is not something as simple as the Taylor series expansion of a Sine function, which is what I meant by not “easily understood”. Yes. Taylor expensions are funny, aren't they? >and unless one has a long-enough time series to deal with, one might not pick up the repeat sequence. I don't understand what you mean with repeat sequence. But apart from that Mathieu function discussion, I am not so sure wether there exists at all a mathematical function, which would describe such a "forced to periodicity" behaviour as it seems to be the case for the QBO (at least if one believes that blog post diagram) Did you check the QBO diagram?`

The repeat sequence is the time span at which the function appears to repeat. Humans are pretty good at picking up patterns. Stare at it for a moment.

`The repeat sequence is the time span at which the function appears to repeat. Humans are pretty good at picking up patterns. Stare at it for a moment.`

nad earlier asked:

That term is borne out of experience by how additional small-perturbation Mathieu (or Hill) terms impact the solution. I indicated that this is a kind of "time dilation", which is what an additional periodic modulation term will do to the characteristic frequency. So a slight change in characteristic frequency will add a long-term modulation in time --i.e. the definition of frequency modulation.

If you don't believe this partly hand-wavy explanation, I moved the time dilation from the y solution term to a 3rd frequency modulation term in the LHS of the DiffEq and recalculated below. The correlation coefficient is even better -- compare to the chart in #2

Thus we have a single strong Mathieu modulation term along with a pair of weaker Hill terms operating on different scales (1) a strong biannual modulation (2) a weak 12 year modulation and (3) and even weaker 50 year modulation.

`nad earlier asked: >I assume that “First” plots out y(x) of NDSolve in particular I don’t have Mathematica to check. Moreover in your plot you have that strange $y(x+const*sin(const x +….) is that what you call a filter? That term is borne out of experience by how additional small-perturbation Mathieu (or Hill) terms impact the solution. I indicated that this is a kind of "time dilation", which is what an additional periodic modulation term will do to the characteristic frequency. So a slight change in characteristic frequency will add a long-term modulation in time --i.e. the definition of frequency modulation. If you don't believe this partly hand-wavy explanation, I moved the time dilation from the y solution term to a 3rd frequency modulation term in the LHS of the DiffEq and recalculated below. The correlation coefficient is even better -- compare to the chart in #2 ![qbo3terms](http://imageshack.com/a/img540/2719/uYyO6E.gif) Thus we have a single strong Mathieu modulation term along with a pair of weaker Hill terms operating on different scales (1) a strong biannual modulation (2) a weak 12 year modulation and (3) and even weaker 50 year modulation.`

Paul your QBO modulation looks impressive, even if I don't understand how you arrived at it. That in particular I think you have now a different model than before, because I don't think that introducing an extra term into Hills equation is the same as letting the solution depend on an oscillatory extra term that would probably be big news if this were the case.

Anyways I give up trying to understand your calculations. It would though be interesting for me to hear your opinion about that blog posts QBO data you linked to. That is your QBO data, although it is hard to see, seems to look quite differently from that.

`>Thus we have a single strong Mathieu modulation term along with a pair of weaker Hill terms operating on different scales (1) a strong biannual modulation (2) a weak 12 year modulation and (3) and even weaker 50 year modulation. Paul your QBO modulation looks impressive, even if I don't understand how you arrived at it. That in particular I think you have now a different model than before, because I don't think that introducing an extra term into Hills equation is the same as letting the solution depend on an oscillatory extra term that would probably be big news if this were the case. Anyways I give up trying to understand your calculations. It would though be interesting for me to hear your opinion about that blog posts QBO data you linked to. That is your QBO data, although it is hard to see, seems to look quite differently from that.`

nad said

As it turns out, the other component in QBO is what they call a Semi-Annual Oscillation (SAO)

The theory of the QBO

See the clear semiannual oscillation above the biennial oscillation? This Mathieu model fit may be a direct clue that the semiannual=biannual variation at altitudes higher than (in the mesopshere) the QBO modulation influences the formation of the biennial cycles below (in the stratosphere).

It may also refute what Lindzen found. If it wasn't for the strong biannual/semiannual modulation ala the Mathieu DiffEq, then the generated oscillation would likely have a completely different characteristic.

This is a more recent description [1]V. S. Babu and G. Ramkumar, “Planetary waves–major forcing agent in generating stratospheric and mesospheric quasi biennial oscillation,” CURRENT SCIENCE, vol. 106, no. 9, p. 1260, 2014. http://www.currentscience.ac.in/Volumes/106/09/1260.pdf

Amazing what machine learning and search optimization can find!

`nad said >"Anyways I give up trying to understand your calculations." As it turns out, the other component in QBO is what they call a Semi-Annual Oscillation (SAO) [The theory of the QBO](http://www.ugamp.nerc.ac.uk/hot/ajh/qbo.htm) "Holton and Lindzen (1972) were the first to propose a model of the QBO based on vertically propagating waves. Originally it was thought that the Semi Annual Oscillation (SAO) in the upper stratosphere played an important role in the QBO. More recently they showed that while the SAO was important, it was not necessary for the formation of the QBO.The mechanism was further explained by Plumb (1977) who showed that the maximum acceleration occurs just below the maximum phase speed, leading to descent of the maximum with time. " ![SAO](http://www.ugamp.nerc.ac.uk/hot/ajh/qboland.gif) "Fig 1. The monthly zonal mean wind in m/s against pressure in mb as seen the the UKMO assimilated dataset at 1.25 ° north of the equator. Easterlies are coloured yellow to blue and westerlies orange to red. The zero line is in thick black and every 5m/s is delineated in thin black. The QBO is roughly between 10mb and 100mb in height extent. Above 3mb the Semi-Annual Oscillation (SAO), a harmonic of the seasonal cycle can be seen, being westerly near the equinox and easterly near the solstice."* See the clear semiannual oscillation above the biennial oscillation? This Mathieu model fit may be a direct clue that the semiannual=biannual variation at altitudes higher than (in the mesopshere) the QBO modulation influences the formation of the biennial cycles below (in the stratosphere). It may also refute what Lindzen found. If it wasn't for the strong biannual/semiannual modulation ala the Mathieu DiffEq, then the generated oscillation would likely have a completely different characteristic. This is a more recent description [1]V. S. Babu and G. Ramkumar, “Planetary waves–major forcing agent in generating stratospheric and mesospheric quasi biennial oscillation,” CURRENT SCIENCE, vol. 106, no. 9, p. 1260, 2014. <http://www.currentscience.ac.in/Volumes/106/09/1260.pdf> Amazing what machine learning and search optimization can find!`

Biannual at ~0.7 hPa and biennial at ~40 hPa. That's a great graph!

`Biannual at ~0.7 hPa and biennial at ~40 hPa. That's a great graph!`

There are other interesting relationships in the QBO data. This site http://www.geo.fu-berlin.de/met/ag/strat/produkte/qbo/qbo.dat contains QBO data for various altitudes, 70,50,40,30,20,15,10 hPa (or mbar).

I did a machine learning exercise on the 40 hPa time series in terms of the other series. That involves essentially looking at cuts across the contour map.

What it found was that the 40 hPa was strongly linked as a multiplication of the 30 hPa with the 50 hPa time series -- with a very good correlation coefficient, about 0.83 without filtering.

It also shows positive spikes that split the biennial cycle roughly in half. This is where both the 30 and 50 both have strong positive or negative values. The downward notches are where one or the other crosses zero.

This is what the QBO page says, and it points out a real delay between 30 and 50 hPa

the transition to easterlies is often delayed between 30 and 50 hPaNow, of course this looks kind of suspicious, so I created my own contour/relief map using the data, see below. What is odd about the view are these strange box-like artifacts that occur along the 40 hPa altitude. I drew one with an enclosing red dashed box below. These appear regularly across the decades.

It is entirely possible that whoever created the data may have tried to average the 40 hPa as (30 hPa + 50 hPa)/2 but inserted a multiplication by mistake?

Otherwise, I can't see this happening, unless the physics says that the 50 hPa is causally related by 40 hPa / 30 hPa, so that the higher altitude can boost the speed as the stratospheric wind direction reverses?

This is either very interesting, or a bust due to data problems. That's what happens with machine learning, as it can find possible artifactual errors -- and these routinely have strong correlations because they are artificially created!

`There are other interesting relationships in the QBO data. [This site](http://www.geo.fu-berlin.de/met/ag/strat/produkte/qbo) <http://www.geo.fu-berlin.de/met/ag/strat/produkte/qbo/qbo.dat> contains QBO data for various altitudes, 70,50,40,30,20,15,10 hPa (or mbar). I did a machine learning exercise on the 40 hPa time series in terms of the other series. That involves essentially looking at cuts across the contour map. ![cuts](http://imageshack.com/a/img540/8854/HiuBd9.gif) What it found was that the 40 hPa was strongly linked as a multiplication of the 30 hPa with the 50 hPa time series -- with a very good correlation coefficient, about 0.83 without filtering. ![](http://imageshack.com/a/img540/1031/Q9nT8E.gif) It also shows positive spikes that split the biennial cycle roughly in half. This is where both the 30 and 50 both have strong positive or negative values. The downward notches are where one or the other crosses zero. This is what the QBO page says, and it points out a real delay between 30 and 50 hPa * The time-height section derived from these data in Fig.30 shows the observed structure of the QBO at equatorial latitudes: * alternating easterly and westerly wind regimes propagate downward with time; * westerlies move down faster and more regularly than easterlies; * **the transition to easterlies is often delayed between 30 and 50 hPa** * easterlies are generally stronger (30-35 m/s) than westerlies (15-20 m/s); * maximum amplitudes of both phases typically occur near 20-hPa; * the average period is about 27 months; * both period and amplitude considerably vary from cycle to cycle. Now, of course this looks kind of suspicious, so I created my own contour/relief map using the data, see below. What is odd about the view are these strange box-like artifacts that occur along the 40 hPa altitude. I drew one with an enclosing red dashed box below. These appear regularly across the decades. ![](http://imageshack.com/a/img537/5456/RIKFZJ.gif) It is entirely possible that whoever created the data may have tried to average the 40 hPa as (30 hPa + 50 hPa)/2 but inserted a multiplication by mistake? Otherwise, I can't see this happening, unless the physics says that the 50 hPa is causally related by 40 hPa / 30 hPa, so that the higher altitude can boost the speed as the stratospheric wind direction reverses? This is either very interesting, or a bust due to data problems. That's what happens with machine learning, as it can find possible artifactual errors -- and these routinely have strong correlations because they are artificially created!`

A simulation that captures both the QBO and the transition from the Semi-Annual Oscillations [1]

One can see how the semi-annual oscillations bleed into the QBO, with the ripple barely visible at 20 hPa, but still there. My bet is that a useful first-order model is either a modulation of the wave equation, ala Mathieu, or of a RHS forcing from a semi-annual cycle. Or it could be some of both. A similar contour plot would not be hard to duplicate as a continuous transition, coded in Mathematica.

Part of the reason for doing this is to come up with first-order parameterizations that don't require the complexity of a GCM. If the QBO is strongly associated with ENSO, this might be a useful approximation.

The strange box-like artifacts observed at 40 hPa in #17 are not seen in this simulation. That one is still a mystery. It is very tempting to pursue that behavior but if it turns out to be a data transformation artifact, that would be wasted time.

[1]S. Schirber, E. Manzini, and M. J. Alexander, “A convection‐based gravity wave parameterization in a general circulation model: Implementation and improvements on the QBO,” Journal of Advances in Modeling Earth Systems, vol. 6, no. 1, pp. 264–279, 2014.

`A simulation that captures both the QBO and the transition from the Semi-Annual Oscillations [1] ![qbo-smo](http://imageshack.com/a/img673/6172/a5JRFg.gif) One can see how the semi-annual oscillations bleed into the QBO, with the ripple barely visible at 20 hPa, but still there. My bet is that a useful first-order model is either a modulation of the wave equation, ala Mathieu, or of a RHS forcing from a semi-annual cycle. Or it could be some of both. A similar contour plot would not be hard to duplicate as a continuous transition, coded in Mathematica. Part of the reason for doing this is to come up with first-order parameterizations that don't require the complexity of a GCM. If the QBO is strongly associated with ENSO, this might be a useful approximation. The strange box-like artifacts observed at 40 hPa in #17 are not seen in this simulation. That one is still a mystery. It is very tempting to pursue that behavior but if it turns out to be a data transformation artifact, that would be wasted time. [1]S. Schirber, E. Manzini, and M. J. Alexander, “A convection‐based gravity wave parameterization in a general circulation model: Implementation and improvements on the QBO,” Journal of Advances in Modeling Earth Systems, vol. 6, no. 1, pp. 264–279, 2014.`

Frankly I have problems with this article. In particular the QBO visualizations on page 3 are decribed with two time intervals one extending over two years (which looks more likely) and one extending over ten years, but this seems not to be just a typo that is the visualization capture says explicitly again that the visualization extends over ten years.

The Gif animation is a very nice visualization! It seems you have quite a talent for detecting special webpages. In particular as a matter of fact one can't find this webpage via a search on the main page. I find it only irritatating that they seem to use red and blue for wind speeds into the east or west direction (minus is east). Especially with the bar below and the corresponding numbers one can very easily mistake that as temperatures, since this color choice is meanwhile quite hardwired for temperatures in visualizations. But then this is a 20 year old visualization from 1994! And yes it looks in those visualization as if the semiannual pressure oscillation is exactly semiannual (and the QBO exactly biannual, especially in the 200mB region the exact annuity is rather good visible).

I don't understand though what they mean with: "who showed that the maximum acceleration occurs just below the maximum phase speed, leading to descent of the maximum with time. " and their other explanations like:

From the visualization it looks to me on a first glance as if there is semiannually (In winter from south and in summer from north) easterlies (blue) "pouring down" from the stratosphere in the course of a quarter year (yes thats strange but thats how it looks) alternating with westerlies in autumn from north and in spring from south. And every second year the easterlies are not "strong enough" (in even years) to leave a "blotch" around the equator at around 20-30mB (which gives the QBO). Would be interesting to see that wind profiles together with temperature measurements....but given the state of temperature measurements I have some doubts that those exist, but may be I am wrong, satellite-drone galore or balloons???

`>This is a more recent description [1]V. S. Babu and G. Ramkumar, “Planetary waves–major forcing agent in generating stratospheric and mesospheric quasi biennial oscillation,” CURRENT SCIENCE, vol. 106, no. 9, p. 1260, 2014. http://www.currentscience.ac.in/Volumes/106/09/1260.pdf Frankly I have problems with this article. In particular the QBO visualizations on page 3 are decribed with two time intervals one extending over two years (which looks more likely) and one extending over ten years, but this seems not to be just a typo that is the visualization capture says explicitly again that the visualization extends over ten years. >As it turns out, the other component in QBO is what they call a Semi-Annual Oscillation (SAO) >The theory of the QBO The Gif animation is a very nice visualization! It seems you have quite a talent for detecting special webpages. In particular as a matter of fact one can't find this webpage via a [search on the main page](http://www.nerc.ac.uk/site/search/index.asp?q=QBO). I find it only irritatating that they seem to use red and blue for wind speeds into the east or west direction (minus is east). Especially with the bar below and the corresponding numbers one can very easily mistake that as temperatures, since this color choice is meanwhile quite hardwired for temperatures in visualizations. But then this is a 20 year old visualization from 1994! And yes it looks in those visualization as if the semiannual pressure oscillation is exactly semiannual (and the QBO exactly biannual, especially in the 200mB region the exact annuity is rather good visible). >"Holton and Lindzen (1972) were the first to propose a model of the QBO based on vertically propagating waves. Originally it was thought that the Semi Annual Oscillation (SAO) in the upper stratosphere played an important role in the QBO. More recently they showed that while the SAO was important, it was not necessary for the formation of the QBO.The mechanism was further explained by Plumb (1977) who showed that the maximum acceleration occurs just below the maximum phase speed, leading to descent of the maximum with time. " I don't understand though what they mean with: "who showed that the maximum acceleration occurs just below the maximum phase speed, leading to descent of the maximum with time. " and their other explanations like: >The more freely propagating westerlies are dissipated at higher altitudes and produce a westerly acceleration leading to a new westerly regime From the visualization it looks to me on a first glance as if there is semiannually (In winter from south and in summer from north) easterlies (blue) "pouring down" from the stratosphere in the course of a quarter year (yes thats strange but thats how it looks) alternating with westerlies in autumn from north and in spring from south. And every second year the easterlies are not "strong enough" (in even years) to leave a "blotch" around the equator at around 20-30mB (which gives the QBO). Would be interesting to see that wind profiles together with temperature measurements....but given the state of temperature measurements I have some doubts that those exist, but may be I am wrong, satellite-drone galore or balloons???`

Yes, the animation clearly shows that the semi-annual is due to alternate heating of the 30N and 30S equinoxes as a year progresses.

Staring at the equator vertical, one can see the higher velocity winds move downward in altitude and then flip direction. That is what is thought to be one of the forcing drivers for ENSO. Also alternating "tongues" coming from the north and south latitudes are clearly seen at sea level , and these winds have a strong annual frequency component.

I can't pick up a strict biennial period from the animation, but only the quasi-biennial. If the QBO is 28 months, that makes the period 2.33=7/3 years, and so the beat frequency when an annual cycle would constructively interfere with a QBO cycle would be once every 7 years. In other words, there would be 3 full QBO cycles in every 7 annual cycles.

`Yes, the animation clearly shows that the semi-annual is due to alternate heating of the 30N and 30S equinoxes as a year progresses. Staring at the equator vertical, one can see the higher velocity winds move downward in altitude and then flip direction. That is what is thought to be one of the forcing drivers for ENSO. Also alternating "tongues" coming from the north and south latitudes are clearly seen at sea level , and these winds have a strong annual frequency component. I can't pick up a strict biennial period from the animation, but only the quasi-biennial. If the QBO is 28 months, that makes the period 2.33=7/3 years, and so the beat frequency when an annual cycle would constructively interfere with a QBO cycle would be once every 7 years. In other words, there would be 3 full QBO cycles in every 7 annual cycles.`

You had probably the visualization of FU Berlin in front of your eyes because it is ordered in seven year rows:

In the description it is written:

translation without guarantee:

And indeed if I count the dark lashes (you called them tongues I prefer to call them lashes) then one countes between 1953 and 2014 more or less 27 lashes, which is 60/27=2.222, which is 26.666 months. Nevertheless as described above and in the previous comments the QBO appears to me to be a consequence of a more regular pattern and thus irregularities might happen like it could happen that some blotches merge etc. One would need thus to look at the respective years, where the exact 2-year pattern breaks down, like wether there were vulcano's etc. and in particular one should look at this together with the SOI. In the Indian article (if one takes the 2 year period for the images) there seem to be some extra SOI lashes in some years, but if one would have the plot for all time intervals then it could be the case, that those appear e.g. as retardations, prolongations etc. And yes the University of Reading animation is only for roughly 6 years and it would definitely be better to have that for 60 years, but however in that animation the phenomenom really looks exact biannual. I was about to collect the different QBO readings on the Wiki, but I currently have no access to it.

`>I can’t pick up a strict biennial period from the animation, but only the quasi-biennial. If the QBO is 28 months, that makes the period 2.33=7/3 years, and so the beat frequency when an annual cycle would constructively interfere with a QBO cycle would be once every 7 years. In other words, there would be 3 full QBO cycles in every 7 annual cycles. You had probably the visualization of [FU Berlin](http://www.geo.fu-berlin.de/met/ag/strat/produkte/qbo) in front of your eyes because it is ordered in seven year rows: ![qbo-FU](http://www.geo.fu-berlin.de/met/ag/strat/produkte/qbo/qbo_wind.jpg) In the description it is written: >Die QBO ist eine annähernd zweijährige Schwingung des zonalen Windes in der unteren und mittleren tropischen Stratosphäre. Das heißt, dass sich zwischen 100 hPa und 10 hPa Westwinde mit Ostwinden abwechseln. Die Periode dieser Schwingung schwankt zwischen 22 und 34 Monaten. Im Mittel beträgt sie 27 Monate. translation without guarantee: >The QBO is an approximately biannual oscillation of the zonal winds in the lower and mid tropical stratosphere. This means that between 100hPa and 10hPa westwinds are interchanging with east winds. The period of this oscillation oscillates between 22 and 24 months. On average it is 27 months. And indeed if I count the dark lashes (you called them tongues I prefer to call them lashes) then one countes between 1953 and 2014 more or less 27 lashes, which is 60/27=2.222, which is 26.666 months. Nevertheless as described [above](http://forum.azimuthproject.org/discussion/1498/is-there-an-exact-biannual-global-temperature-oscillation/?Focus=12864#Comment_12864) and in the previous comments the QBO appears to me to be a consequence of a more regular pattern and thus irregularities might happen like it could happen that some blotches merge etc. One would need thus to look at the respective years, where the exact 2-year pattern breaks down, like wether there were vulcano's etc. and in particular one should look at this together with the SOI. In the Indian article (if one takes the 2 year period for the images) there seem to be some extra SOI lashes in some years, but if one would have the plot for all time intervals then it could be the case, that those appear e.g. as retardations, prolongations etc. And yes the University of Reading animation is only for roughly 6 years and it would definitely be better to have that for 60 years, but however in that animation the phenomenom really looks exact biannual. I was about to collect the different QBO readings on the Wiki, but I currently have no access to it.`

I know that earlier you said that biannual can mean either twice per year or every two years, so you will have to be less ambiguous, otherwise we can not be certain what period you are referring to. Sorry.

There is really no doubt that the twice yearly frequency is exact as the sun crosses the equator exactly twice per year. However the exact once every two year period is less certain. I think I have found evidence here, but that is inferred from tidal gauge records.

`> And yes the University of Reading animation is only for roughly 6 years and it would definitely be better to have that for 60 years, but however in that animation the phenomenom really looks exact biannual. I know that earlier you said that biannual can mean either twice per year or every two years, so you will have to be less ambiguous, otherwise we can not be certain what period you are referring to. Sorry. There is really no doubt that the twice yearly frequency is exact as the sun crosses the equator exactly twice per year. However the exact once every two year period is less certain. I think I have found evidence [here](http://forum.azimuthproject.org/discussion/1480/tidal-records-and-enso/?Focus=12629#Comment_12629), but that is inferred from tidal gauge records.`

but I also said that if I say biannual then I always mean every second year. If I mean twice a year I would use semiannual, because bi means two and semi means half in latin (strictly speaking it seems it should probably be better "duannual" than biannual). This should also be clear from the diagrams I showed and talked about. But I am not a native speaker if you have an unambigous way for saying every second year, let me know should I use duannual?

I haven't looked much at your sloshing calculation, partly because I got no real answer to the question about the 3 month delay between sydney and darwin and somehow I didn't understand why the atmosspheric pressure at Darwin (the SOI) shouldn't anyways be related to the water level in sydney, like by the barometric pressure you mentioned so I gave up early in that discussion and thus I don't understand your evidence.

I have also not so much time to look at all postings. But I do not want sound to discouraging with respect to that discussion that is in principle I think it could be worthwhile to look at sloshings of the pacific and may be the tides have also some influence - I don't know, but I actually think -if I would look at sloshings- then rather in connection with unsmoothnesses in the earths spin around itself, which seems to have an annual component.

`>I know that earlier you said that biannual can mean either twice per year or every two years, so you will have to be less ambiguous, otherwise we can not be certain what period you are referring to. Sorry. but I also [said](http://forum.azimuthproject.org/discussion/1498/is-there-an-exact-biannual-global-temperature-oscillation/?Focus=12834#Comment_12834) that if I say biannual then I always mean every second year. If I mean twice a year I would use semiannual, because bi means two and semi means half in [latin](http://en.wikipedia.org/wiki/Numeral_prefix) (strictly speaking it seems it should probably be better "duannual" than biannual). This should also be clear from the diagrams I showed and talked about. But I am not a native speaker if you have an unambigous way for saying every second year, let me know should I use duannual? >There is really no doubt that the twice yearly frequency is exact as the sun crosses the equator exactly twice per year. However the exact once every two year period is less certain. I think I have found evidence here, but that is inferred from tidal gauge records. I haven't looked much at your sloshing calculation, partly because I got no real answer to the question about the 3 month delay between sydney and darwin and somehow I didn't understand why the atmosspheric pressure at Darwin (the SOI) shouldn't anyways be related to the water level in sydney, like by the barometric pressure [you mentioned](http://forum.azimuthproject.org/discussion/1480/tidal-records-and-enso/?Focus=12566#Comment_12566) so I gave up early in that discussion and thus I don't understand your evidence. I have also not so much time to look at all postings. But I do not want sound to discouraging with respect to that discussion that is in principle I think it could be worthwhile to look at sloshings of the pacific and may be the tides have also some influence - I don't know, but I actually think -if I would look at sloshings- then rather in connection with unsmoothnesses in the earths spin around itself, which seems to have an annual component.`

It is not necessarily tides that we are discussing, but the instrument that measures sea-level height is called a tidal gauge. The impact of the actual tidal variation is filtered out so what is left is the variation due to buildup of water in one location. One factor in the sea-level buildup is the sloshing accumulation and the other factor is the inverted barometer effect.

In other units, this is 1 cm drop per 1 hPa (=1mbar) change in atmospheric pressure

ATMOSPHERIC LOADING AND THE OCEANIC "INVERTED BAROMETER" EFFECT, Wunsch and Stamper

This is what the sea-level looks like at Sydney

When calculating SOI, the average pressure can change by +/- 5 mbar, which equates to +/- 5 cm or +/- 50 mm, which is roughly the peak-to-peak fluctuations in he above plot. In a bit, I will show a graph where these do in fact align.

But this is also the basis for sloshing, whereby one part of the ocean has an elevated sea-level and the other part has a depressed sea-level. So in fact the sloshing and inverted barometer effects are conflating factors. By conflating, it means that they are hard to disentangle. Consider that some of the sloshing is due to wind forcing accumulation of water in a region, but the wind itself is due in part to pressure gradients! I think that is why Wunsch refers to this as a complex dynamical response.

This site has more information on sea-level and relationship to El Nino http://www.aviso.altimetry.fr/en/applications/climate/el-nino/

Figure caption: "Sea level tilt increases, rising more in the west and deepening in the east. The thermocline tilt also increases, most noticeably at the western and eastern ends of the basin."

Note the thermocline change -- which is a sloshing variation that is subsurface.

This is a graph of mean sea-level (MSL) averaged over the NINO 3.4 area

I may have to pick a location where the sea-level changes are most pronounced and see if I can isolate the conflating factors -- if in fact they can be isolated.

`It is not necessarily tides that we are discussing, but the instrument that measures sea-level height is called a tidal gauge. The impact of the actual tidal variation is filtered out so what is left is the variation due to buildup of water in one location. One factor in the sea-level buildup is the sloshing accumulation and the other factor is the inverted barometer effect. Inverted Barometer Effect: "Adjustment of sea level to changes in barometric pressure; in the case of full adjustment, an increase in barometric pressure of 1 mb corresponds to a fall in sea level of 0.01 m. If there is this full adjustment, the observed pressures at the sea bed are unchanged." In other units, this is 1 cm drop per 1 hPa (=1mbar) change in atmospheric pressure [ATMOSPHERIC LOADING AND THE OCEANIC "INVERTED BAROMETER" EFFECT, Wunsch and Stamper ](http://onlinelibrary.wiley.com/doi/10.1029/96RG03037/pdf) Recently,Fu and Pihos [1994] showed that the TOPEX/POSEIDON altimetric missionmeasurements were qualitatively consistent with a primarily static ("equilibrium") oceanic response to the atmospheric load; this is described by an inverted barometer (IB) response. However, an equilibrium response is somewhat difficult to understand because the ocean is capable of a complex dynamical response; consider that the motions induced through the analogous forcing by tides are distinctly nonequilibrium This is what the sea-level looks like at Sydney ![tide](http://www.psmsl.org/data/obtaining/rlr.annual.plots/65.png) When calculating SOI, the average pressure can change by +/- 5 mbar, which equates to +/- 5 cm or +/- 50 mm, which is roughly the peak-to-peak fluctuations in he above plot. In a bit, I will show a graph where these do in fact align. But this is also the basis for sloshing, whereby one part of the ocean has an elevated sea-level and the other part has a depressed sea-level. So in fact the sloshing and inverted barometer effects are conflating factors. By conflating, it means that they are hard to disentangle. Consider that some of the sloshing is due to wind forcing accumulation of water in a region, but the wind itself is due in part to pressure gradients! I think that is why Wunsch refers to this as a complex dynamical response. This site has more information on sea-level and relationship to El Nino <http://www.aviso.altimetry.fr/en/applications/climate/el-nino/> ![elnino](http://www.aviso.altimetry.fr/typo3temp/pics/5736f52cf0.jpg) Figure caption: "Sea level tilt increases, rising more in the west and deepening in the east. The thermocline tilt also increases, most noticeably at the western and eastern ends of the basin." Note the thermocline change -- which is a sloshing variation that is subsurface. This is a graph of mean sea-level (MSL) averaged over the NINO 3.4 area ![msla](http://bulletin.aviso.oceanobs.com/images/produits/indic/enso/time_curves/Msla_nino34_StzdF6.png) I may have to pick a location where the sea-level changes are most pronounced and see if I can isolate the conflating factors -- if in fact they can be isolated.`

I mentioned that I would get a comparison of SOI against sea-level height data in the ENSO region. This is the comparison with of SOI with MSLA averaged over the NINO 3 region over the time period where satellite altimetry measurements are available -- the circles are SOI data and there are two linear combinations of MSLA overlaid :

Note that the agreement is good but the mean sea level anomaly (MSLA) does not show the overall trend that the SOI exhibits

`I mentioned that I would get a comparison of SOI against sea-level height data in the ENSO region. This is the comparison with of SOI with MSLA averaged over the NINO 3 region over the time period where satellite altimetry measurements are available -- the circles are SOI data and there are two linear combinations of MSLA overlaid : ![msla_vs_nino3](http://imageshack.com/a/img674/4703/DFHblN.gif) Note that the agreement is good but the mean sea level anomaly (MSLA) does not show the overall trend that the SOI exhibits`

I just wanted to remark for completeness:

comment

Likewise I wrote here:

In that image the regular odd year pattern is disturbed for the years (63) 65, 67, (77), 79,81,(83),(89),91,(03),05,07,09,(11)

the years in brackets are years where the QBO is only "somewhat" out of sync, this is usually a year before or after a series of "out of sync" signals. So the pattern of the "extrasystoles" is also directly visible in the images.

`I just wanted to remark for completeness: <a href="http://forum.azimuthproject.org/discussion/1471/qbo-and-enso/?Focus=12446#Comment_12446">comment</a> >Moreover it looks to me as if there is also an annual oscillation contained in the signal, like there seems to be always a little “bump” on the downslide between peaks. This bump is sometimes rather high (“a small peak”, an “extrasystole”) and seems to lead in that case to a lagging behind of the biannual signal peak. This happens slightly before the year 63 (almost nonvisible) , before 65 the “extrasystole” is comparibly big leading to the lag between 65 and 67 the small peak behind 67 would have been the original 67 peak if there wouldn’t have been the lagging, in 69 the signal is again in sync, before 77 an extrasystole is again leading to a lagging, again before 79, leading to a lagging, the small peak at 81 would again be the peak at 81 if not for the lagging, here the signal has now a 90 degrees phaseshift, i.e. the beat is quite out of sync, it catches though up again in 85, in 87 small extrasystole with small lagging, in 89 big extrasystole, signal again out quite out of sync, catches up in 97, extrasystole before 01, 03 and 05, out of sync until it “catches up again” in 2013. If there is no extrasystole in 2014 I conjecture that the QBO index will peak again in summer 2015. Likewise I wrote <a href="http://forum.azimuthproject.org/discussion/1498/is-there-an-exact-biannual-global-temperature-oscillation/?Focus=12871#Comment_12871">here:</a> >Nevertheless as described above and in the previous comments the QBO appears to me to be a consequence of a more regular pattern and thus irregularities might happen like it could happen that some blotches merge etc. One would need thus to look at the respective years, where the exact 2-year pattern breaks down, like wether there were vulcano’s etc. and in particular one should look at this together with the SOI. In that image the regular odd year pattern is disturbed for the years (63) 65, 67, (77), 79,81,(83),(89),91,(03),05,07,09,(11) the years in brackets are years where the QBO is only "somewhat" out of sync, this is usually a year before or after a series of "out of sync" signals. So the pattern of the "extrasystoles" is also directly visible in the images.`