... because they are so jaw-dropping in their intricacy, and so it's fun to see how far we can take it.

If this model wasn't simply a plausible theoretical explanation of the quasi-biennial oscillations, it would also make for a practical heuristic.

**Fig. 1**: Factors contributing to QBO

Most of the fit is the result of the aliasing of the Draconic or nodal month of 27.21 days (aliased to 2.36 years), but the other known lunar factors contribute as well. Important that these are the *same* periodic factors that go into well-established length-of-day [1] and ocean tidal prediction programs, but are aliased to show resonance or amplification with the seasonal cycle.

**Fig. 2:** Factors contributing to the second derivative of the QBO, these are weighted to shorter cycles.

Taking the second derivative adds much more structure to the profile, yet the same set of factors continue to provide a more than adequate fit.

The QBO is an example of a forced-response system. The natural response could be buried in there as well, and may even be the hazily-attributed 2.96 year cyclic factor indicated above, but the weight clearly favors the known lunar gravitational forcing factors.

[1] R. D. Ray and S. Y. Erofeeva, “Long‐period tidal variations in the length of day,” *Journal of Geophysical Research: Solid Earth*, vol. 119, no. 2, pp. 1498–1509, 2014.

*Related*

And so here are some more fits

This is fitting to the noisy 2nd derivative of the 30 hPa QBO data:

The training is up to year 2000 using multiple linear regression. There is also much more structure to the 2nd derivative, yet you can see how the projection beyond 2000 finds that structure.

Here is the backcast version which uses data from 1975 to 2015, working backward

So today I find that Gavin Schmidt and company at NASA Goddard have written this lengthy paper on how to generate a model of the QBO [1]. I can't seem to find anything remotely trivial in what they are doing. That's 26 pages of stuff that I am not going to try to figure out.

Based on the zero-level of complexity in my own model fit to QBO, I am willing to go on the line and say that the physics behind the forcing of QBO is trivial, and that the math behind it is also trivial.

It's like finding that the hum in your stereo system is caused by the 60 Hz power line -- kind of trivial

Or it's like finding that the ocean tides are caused by the moon and the sun.

[1]D. Rind, J. Jonas, N. Balachandran, G. Schmidt, and J. Lean, “The QBO in two GISS global climate models: 1. Generation of the QBO,” Journal of Geophysical Research: Atmospheres, vol. 119, no. 14, pp. 8798–8824, 2014.

This is the 2nd-derivative model fit for QBO at 70 hPa which is the lowest atmospheric reading. Just about every peak and valley aligns, and the only question is the relative strength and whether it appears as a clear peak or as a shoulder.

The model may still have one missing factor that isn't showing up, or it may be a good idea to solve the full wave equation, which is what works well for the ENSO model.

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