About that [caveat I mentioned above](discussion/comment/15472/#Comment_15472):

> "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)."

Following up on this gap, I extended the derivation to formally evaluate the velocity.

So from:

$ \zeta(t) = \sin( \sqrt{A} \sum_{i=1}^{i=N} k_i \sin(\omega_i t) + \theta_0 ) $

and [the third simplified Laplace equation](http://contextearth.com/2016/07/04/alternate-simplification-of-qbo-from-laplaces-tidal-equations/)

$ \frac {\partial v}{\partial t} =-\frac {1}{a} \frac {\partial }{\partial \varphi } (g\zeta +U)$

we can derive

$ \frac {\partial v}{\partial t} = \cos( \sqrt{A} \sum_{i=1}^{i=N} k_i \sin(\omega_i t) + \theta_0 ) $

So the *acceleration* of wind, not the velocity, is what obeys the Sturm-Liouville equation. A derivative preserves the periods of the Fourier components, but not the amplitudes, so what we see is a differently shaped envelope for QBO -- i.e. one that is more spiky due to the time derivative applied.

A single lunar Draconic tidal term of 27.212 days multiplied by a yearly modulation peaked at a specific season is enough to capture the QBO acceleration time-series (with a correlation coefficient of 0.35 considering the data waveform has *no filtering applied* and very little optimization went into the fit):

![qboAccel](http://imageshack.com/a/img923/9315/Zgjszt.png)

This makes sense as the acceleration of wind is simply a $ F = ma $ response to a gravitational forcing.

Incidentally, I submitted this research to the AGU meeting for this fall and will find out if the abstract gets accepted in October.

> "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)."

Following up on this gap, I extended the derivation to formally evaluate the velocity.

So from:

$ \zeta(t) = \sin( \sqrt{A} \sum_{i=1}^{i=N} k_i \sin(\omega_i t) + \theta_0 ) $

and [the third simplified Laplace equation](http://contextearth.com/2016/07/04/alternate-simplification-of-qbo-from-laplaces-tidal-equations/)

$ \frac {\partial v}{\partial t} =-\frac {1}{a} \frac {\partial }{\partial \varphi } (g\zeta +U)$

we can derive

$ \frac {\partial v}{\partial t} = \cos( \sqrt{A} \sum_{i=1}^{i=N} k_i \sin(\omega_i t) + \theta_0 ) $

So the *acceleration* of wind, not the velocity, is what obeys the Sturm-Liouville equation. A derivative preserves the periods of the Fourier components, but not the amplitudes, so what we see is a differently shaped envelope for QBO -- i.e. one that is more spiky due to the time derivative applied.

A single lunar Draconic tidal term of 27.212 days multiplied by a yearly modulation peaked at a specific season is enough to capture the QBO acceleration time-series (with a correlation coefficient of 0.35 considering the data waveform has *no filtering applied* and very little optimization went into the fit):

![qboAccel](http://imageshack.com/a/img923/9315/Zgjszt.png)

This makes sense as the acceleration of wind is simply a $ F = ma $ response to a gravitational forcing.

Incidentally, I submitted this research to the AGU meeting for this fall and will find out if the abstract gets accepted in October.