(following the last post) For posterity, the idea is to simplify the Laplace Tidal Equation along the equator, where the QBO resides.

![laplace](http://contextearth.com/wp-content/uploads/2016/06/laplaceTidalOperator.gif)

Many of the terms evaluated along the equator are second-order .

With the substitution

$ \frac{d\Theta}{d\mu} = \frac{d\Theta}{dt} \frac{dt}{d\mu} $

and for $\sigma \gg \mu$ and $\mu = sin \phi \cong\phi $ the equation reduces to

$ \frac{1}{\sigma^2} \left( \frac{d}{dt}(\frac{d\Theta}{dt} \frac{dt}{d\mu}) \frac{dt}{d\mu} + \left[\frac{s}{\sigma} + s^2\right] \Theta \right) + \gamma \Theta = 0 $

With an educated guess that $\frac{d\mu}{dt}$ is a sinusoid plus a constant, the equation solves as a Sturm-Liouville variant, where $\Theta(t)$ is replaced by $f(t)$ (to keep it consistent with the generic DiffEq I have been using).

![sl](http://imageshack.com/a/img923/302/q857zj.png)

The solution is a sinusoid of a sinusoid, which has interesting properties that map to the QBO time-series.

One of these properties is a gradually clipping for large sinusoidal excursions, which matches to what is observed with QBO (choosing B=0). Note the squaring of the waveform.

![fit](http://imageshack.com/a/img924/4915/S7OmbX.png)

The generic solution's inner modulation :

$ f(t) = k \cdot sin( \sqrt{A} \cdot sin(\omega t + \psi_0) + \phi_0) $

can also be replaced by a set of sinusoids of different frequency. That's where the decomposition of the lunar forcing into a seasonally-aliased set of factors is applied. The sinusoidal clipping also suppresses the beat modulation caused by adding cycles of differing frequency.

It really does collapse almost perfectly into a concise formulation. That's the quasi-elevator-pitch, [more here](http://contextearth.com/2016/06/10/pukites-model-of-enso/).

![laplace](http://contextearth.com/wp-content/uploads/2016/06/laplaceTidalOperator.gif)

Many of the terms evaluated along the equator are second-order .

With the substitution

$ \frac{d\Theta}{d\mu} = \frac{d\Theta}{dt} \frac{dt}{d\mu} $

and for $\sigma \gg \mu$ and $\mu = sin \phi \cong\phi $ the equation reduces to

$ \frac{1}{\sigma^2} \left( \frac{d}{dt}(\frac{d\Theta}{dt} \frac{dt}{d\mu}) \frac{dt}{d\mu} + \left[\frac{s}{\sigma} + s^2\right] \Theta \right) + \gamma \Theta = 0 $

With an educated guess that $\frac{d\mu}{dt}$ is a sinusoid plus a constant, the equation solves as a Sturm-Liouville variant, where $\Theta(t)$ is replaced by $f(t)$ (to keep it consistent with the generic DiffEq I have been using).

![sl](http://imageshack.com/a/img923/302/q857zj.png)

The solution is a sinusoid of a sinusoid, which has interesting properties that map to the QBO time-series.

One of these properties is a gradually clipping for large sinusoidal excursions, which matches to what is observed with QBO (choosing B=0). Note the squaring of the waveform.

![fit](http://imageshack.com/a/img924/4915/S7OmbX.png)

The generic solution's inner modulation :

$ f(t) = k \cdot sin( \sqrt{A} \cdot sin(\omega t + \psi_0) + \phi_0) $

can also be replaced by a set of sinusoids of different frequency. That's where the decomposition of the lunar forcing into a seasonally-aliased set of factors is applied. The sinusoidal clipping also suppresses the beat modulation caused by adding cycles of differing frequency.

It really does collapse almost perfectly into a concise formulation. That's the quasi-elevator-pitch, [more here](http://contextearth.com/2016/06/10/pukites-model-of-enso/).