[mathjax]We use seasonal aliasing of the lunar gravitational pull to generate the incredible fits to the QBO.
How does one get the harmonics of the aliasing?
The starting premise is that a known lunar tidal forcing signal is periodic
The seasonal signal is likely a strong periodic delta function, which peaks at a specific time of the year. This can be approximated as a Fourier series of period $$2pi$$.
For now, the exact form of this doesn’t matter, as what we are trying to show is how the aliasing comes about.
The forcing is then a combination of the lunar cycles $$L(t)$$ amplified in some way by the strongly cyclically peaked seasonal signal $$s(t)$$.
Multiplying this out, and pulling the lunar factor into the sum
then with the trig identity
Expanding the lower frequency difference terms and ignoring the higher frequency additive terms
Now you can see how the high frequency $$omega_L$$ term gets reduced in frequency by multiples of $$2pi$$, until it nears the period of the seasonal cycle. And those are the factors that feature in the multiple regression fit. What the regression does is determine the weighting of the $$a_i$$ terms, across the set of lunar $$omega_L$$ terms.
This is a link to the interactive QBO app on the dynamic context server: http://entroplet.com/context_qbo/navigate