Interesting data from this paper on the equatorial-only Semi-Annual Oscillation (SAO) of the upper stratosphere and lower mesosphere wind pattern.

[1] 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.

The SAO flips by 180 degrees between the stratosphere (the SSAO) and the mesosphere (the MSAO). You can see this in the upper panel 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. The QBO starts at altitudes below the SSAO.

![sao1](https://imagizer.imageshack.com/img923/1894/WJ8gOf.png)

This may be explained by the Laplace's Tidal Equation analytic solution that I have been applying to the ENSO and QBO models.

The equation applied is \\( \sin( A \sin(4 \pi t + \phi) + \theta(z)) ) \\)

If the LTE phase 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. Can kind of see that in the power-spectra above.

So below is the theoretical LTE plot alongside the paper's plot. The contour colors don't quite match up, and I don't have Mathematica any longer to get a matching color density plot

![sao2](https://imagizer.imageshack.com/img921/1654/dLwB8A.png)

[1] 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.

The SAO flips by 180 degrees between the stratosphere (the SSAO) and the mesosphere (the MSAO). You can see this in the upper panel 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. The QBO starts at altitudes below the SSAO.

![sao1](https://imagizer.imageshack.com/img923/1894/WJ8gOf.png)

This may be explained by the Laplace's Tidal Equation analytic solution that I have been applying to the ENSO and QBO models.

The equation applied is \\( \sin( A \sin(4 \pi t + \phi) + \theta(z)) ) \\)

If the LTE phase 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. Can kind of see that in the power-spectra above.

So below is the theoretical LTE plot alongside the paper's plot. The contour colors don't quite match up, and I don't have Mathematica any longer to get a matching color density plot

![sao2](https://imagizer.imageshack.com/img921/1654/dLwB8A.png)