This is a nice recent summary (2014) of the major and minor tidal periods
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.
The machine learning found a clear 3.52 year period, which is very close to the listed 1305 day period - 1305/365.25 = 3.57 years.
the following paper considers this cycle as part of the "Lp group" four of the main long periodic tides (9.3 yr and 1305, 205 and 121 days
S. Loyer, J. Hinderer, and J.-P. Boy, “Determination of the gravimetric factor at the Chandler period from Earth orientation data and superconducting gravimetry observations,” Geophysical Journal International, vol. 136, no. 1, pp. 1–7, 1999.
BTW, the paper that [you reference](http://www.nature.com/ngeo/journal/v7/n5/full/ngeo2138.html) is humorous in how they show correlation between ENSO and QBO.
In the upper panel, they really cherry-pick a five year interval where the two measures clearly correlate. Unfortunately over a longer interval, the correlation is *not at all* visibly apparent. Kind of wonder how that escaped peer review. Then again, one can always bring this up to show how we "outsiders" have to work extra hard to get acceptance. All the climate "insiders" have to do is show their affiliation and write a "just-so" story as a narrative. :)