In addition to the biennial mode found in ENSO and in GPS readings, a stratospheric biennial mode also exists. This is different than the quasi-biennial oscillation (QBO) modeled previously, as it shows a more strict two year cycle.
From Dunkerton :
Second, concerning their time dependence, subbiennial variations should not be viewed in isolation from other modes. The seasonal dependence of QBO anomalies cannot be described by a single harmonic, whether quasi-biennial or subbiennial; rather, their superposition provides for the seasonality.
Only three parameters were varied to obtain the least squares fit: the QBO period, QBO phase, and subbiennial phase. This calculation was performed on the three tracers independently, but the same QBO period was obtained in each case, namely, 26 months. The corresponding subbiennial period is 22.3 months
What Dunkerton is talking about essentially is akin to a 2.166 year quasi-biennial signal mixed with a 1.858 year sub-biennial signal. These combine in frequency space as Dunkerton states by their superposition, and also couple together via a ~26-year modulation on the biennial signal -- as per the trig identity:
This is substantiated by Remsberg in his methane study  where in addition he sees a smaller amplitude factor which is closer to the exact biennial signal.
Three interannual terms are almost always present and are 19 included in the model: an 838-dy (27.5-mo) or QBO1 term; a small amplitude, 718-dy (23.5-mo) biennial or QBO2 term; and a 690-dy (22.6-mo) sub-biennial (SB) term, whose period arises from the difference interaction between QBO and annual terms. The relative amplitudes of these three interannual terms vary with latitude and altitude.
More recently (in 2014) Remsberg reports :
Instead, Fourier analysis of the time series of the residuals after removing the seasonal terms almost always indicates that there are two significant, interannual terms having periods of order of 28 (QBO-like) and 21 months (subbiennial term denoted as IA).
This is akin to a 2.333 year quasi-biennial signal mixed with a 1.75 year sub-biennial signal, leading to a 14-year signal modulating the biennial signal:
Which is strikingly similar to the behavior observed in the ENSO model's 14-year modulation of the biennial cycle. The idea was that this 14 year modulation is close to the additional triaxial wobble suggested by Wang. Slightly weaker is an 18.6 year modulation which is close to the lunar nodal period.
The similarity between the stratospheric measures and the ENSO observations is based on the common behavior of a symmetrical balance about the biennial cycle. This is starkly observed in the power spectrum of ENSO shown above.
In a closely related finding, the single-minded AGW-denier Murry Salby also discusses this modulation in , but ascribes the modulation to the sunspot cycle, as he finds a value closer to an 11-year modulation. That assumption has to be taken with some circumspection, as it is well known that Salby favors a solar-based rationale to global temperature variation, which has gotten himself into some bizarre predicaments.
1. Dunkerton, Timothy J. "Quasi-biennial and subbiennial variations of stratospheric trace constituents derived from HALOE observations." Journal of the atmospheric sciences 58.1 (2001): 7-25.
2. Remsberg, Ellis E. "Methane as a diagnostic tracer of changes in the Brewer–Dobson circulation of the stratosphere." Atmospheric Chemistry and Physics 15.7 (2015): 3739-3754.
3. Remsberg, E. E., and L. E. Deaver. "Interannual, solar cycle, and trend terms in middle atmospheric temperature time series from HALOE." Journal of Geophysical Research: Atmospheres 110.D6 (2005).
4. Remsberg, Ellis E. "On the observed changes in upper stratospheric and mesospheric temperatures from UARS HALOE." NASA Langley Report (2006).
5. Remsberg, E. E. "Decadal-scale responses in middle and upper stratospheric ozone from SAGE II version 7 data." Atmospheric Chemistry and Physics 14.2 (2014): 1039-1053.
6. Salby, Murry, Patrick Callaghan, and Dennis Shea. "Interdependence of the tropical and extratropical QBO: Relationship to the solar cycle versus a biennial oscillation in the stratosphere." Journal of Geophysical Research: Atmospheres 102.D25 (1997): 29789-29798.