I created a QBO page that is a concise derivation of the theory behind the oscillations:

http://contextEarth.com/compact-qbo-derivation/

Four key observations allow this derivation to work

1. Coriolis effect cancels at the equator and use a small angle (in latitude) approximation to capture any differential effect.

2. Identification of wind acceleration and not wind speed as the measure of QBO.

3. Associating a latitudinal displacement with a tidal elevation via a partial derivative expansion to eliminate an otherwise indeterminate parameter.

4. Applying a seasonal aliasing to the lunar tractive forces which ends up perfectly matching the observed QBO period.

These are obscure premises but all are necessary to derive the equations and match to the observations.

This model should have been derived long ago .... that's what has me stumped. Years ago I spent hours working on transport equations for semiconductor devices so have a good feel for how to handle these kinds of DiffEq's. You literally had to do this otherwise you would never develop the intuition on how a transistor or some other device works. The QBO for some reason reminds me quite a bit of solving the [Hall effect](https://en.wikipedia.org/wiki/Hall_effect). Maybe I am just using a different lens in solving these kinds of problems.