In Chapter 11 of the book, we present the geophysical recipe for the forcing of the QBO of equatorial stratospheric winds. As explained, the fundamental forcing is supplied by the lunar draconic cycle and impulse modulated by a semi-annual (equatorial) nodal crossing of the sun. It’s clear that the QBO cycle has asymptotically approached a value of 2.368 years, which is explained by its near perfect equivalence to the physically aliased draconic period. Moreover, there is also strong evidence that the modulation/fluctuation of the QBO period from cycle to cycle is due to the regular variation in the lunar inclination, thus impacting the precise timing and shape of the draconic sinusoid. That modulation is described in this post.

As the chapter does not go into the detailed nature of the lunisolar orbit, a good review is available at this NASA page. The salient excerpt describes the lunar draconic month variation:

“The mean interval in the periodic variation of both the draconic month and the orbital inclination is 173.3 days. This is the average time it takes for the Sun to travel from one node to the other. It is also equivalent to the interval between the midpoints of two eclipse seasons. The period is slightly less than half a year because of the retrograde motion of the nodes.”

https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html#draconic

In accordance with the NASA chart of lunar orbit inclination variation, we apply precisely that phase of modulation on the draconic sinusoid, only varying the strength of the modulation to fit the QBO signal.

Applying this modulation and optimizing its magnitude by performing an iterative least-squares fit to the QBO will increase the correlation coefficient from ~0.6 for the unmodulated draconic cycle to beyond 0.8. It essentially tracks the locations of the sign reversals much better that the pure draconic cycle while not changing the long-term mean of 2.368 year for the QBO period.

This variation also impacts the model for ENSO, as a similarly modulated draconic signal is applied as a forcing. This is the monthly view of the correlation between the two (the daily view generates a finer detail). The correlation coefficient is ~0.65 which is quite good considering that these time-series are fit independently (and no teleconnection assumed between QBO and ENSO, apart from the common-mode gravitational forcing.

**As a post-script**: It’s perhaps worthwhile to think of this modulation as an amplified forcing due to it’s signalling the points for the alignment of moon and sun in longitude, a la lunar and solar eclipses as described on the NASA site, “*the inclination is always near its maximum value for both solar and lunar eclipses* “. Because of this alignment, there is likely to be an amplified tidal forcing.

This is a comparison of the frequency spectrum of the modulated draconic signal for QBO and ENSO, indicating the common positions for the 173.3 day modulation on the left and the 2.368 year fundamental aliased QBO period to the right of that peak. These are aliased positions due to the monthly sampling.

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