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


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).


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.


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/).