In Chapter 12 of the book, we concentrated on the mechanism behind the QBO of stratospheric equatorial winds. In a related topic (but only briefly touched on in the book), there is interesting data from a presentation on the equatorial-only Semi-Annual Oscillation (SAO) of the upper stratosphere and lower mesosphere wind pattern . The distinction between QBO and the SAO is that the QBO has a longer periodic cycle and exists at altitudes lower in the stratosphere than the SAO.
 T. Hirooka, T. Ohata, and N. Eguchi, “Modulation of the Semiannual Oscillation Induced by Sudden Stratospheric Warming Events,” in ISWA2016, Tokyo, Japan, 2016, p. 16.— presentation slides from International Symposium on the Whole Atmosphere
What’s interesting at the core fundamental level is that the SAO is understood by consensus to be forced by a semi-annual cycle (a resonant condition happening to match 1/2 year is just too coincidental) whereas there is no consensus behind the mechanism behind the QBO period (the tidal connection is only available from Chapter 12). To make the mathematical connection, the following shows how the SAO draws from the QBO tidal model.
According to the Hirooka paper, the SAO flips by 180 degrees between the stratosphere (the SSAO) and the mesosphere (the MSAO). You can see this in the upper panel in Figure 1 below where the intense westerlies (in RED) occur during the beginning and middle of each year for the MSAO, and they occur between these times (Spring and Fall) for the SSAO. The direction times are complementary for the easterlies in BLUE. At altitudes between the MSAO and SSSAO, the strength of the SAO is significantly reduced as you can see in the lower panel showing the spectral lines.
This may be explained by the Laplace’s Tidal Equation analytic solution that we have been applying to the ENSO and QBO models. The analytic solution expressed in a parametric form is :
If the LTE model phase (θ) varies in altitude (z) due to differing characteristics of the atmospheric density, the sense of the sinusoidal modulation will flip. This is for a value of A that is large enough to cause a strong modulation. For phases halfway between where the sign flips, the modulation bifurcates the semi-annual oscillation such that the 1/2-year period disappears and is replaced by (in-theory) a 1/4-year or 90-day cycle. This can be seen in the Figure 1 power-spectra .
Below in Figure 2 is the theoretical LTE model plot alongside the Hirooka et al plot. The contour colors don’t quite match up, because of the limitations of the plotting tool — the data is represented as a true density plot whereas the model includes contour line artefacts.
As an additional observation from Figure 1: If one considers that an annual oscillation occurs both above and below the altitudes of the QBO, it would be odd if the stratospheric layer for QBO wasn’t similarly forced by a tidal oscillation.