A somewhat hidden cyclic variation in the length-of-day (LOD) in the earth's rotation, of between 6 and 7 years, was first reported in Ref [1] and analyzed in Ref [2]. Later studies further refined this period [3,4,5] closer to 6 years.

It's well known that the moon's gravitational pull contributes to changes in LOD [6]. Here is the set of lunar cycles that are applied as a forcing to the ENSO model using LOD as calibration.

The upper two periods (of 13 and 27 days) are well known to contribute to LOD — as [4] states

As the Moon moves around the Earth, due to the tidal effects, the Earth's inertial moment tensor changes with the lunar orbital periods of

27.7 and 13.7 days. Hence, it is clear that there are variations in the LOD at the periods, charactering [sic] by 27.7 and 13.7 days [6]. Similarly, tidal deformation between the Earth and the Sun system may also causes various periods that are sensed to the Earth's orbit perturbation. It is responsible for about 10% of the annual and about 50% of the semiannual periods, and other causes could be the seasonal exchange of angular momentum among the solid Earth, atmosphere and the ocean [7]. A quasi-biennial oscillation was observed by many researchers [8–11]. The El Niño-Southern Oscillation in the troposphere-ocean system and the Quasi-Biennial Oscillation in the equatorial stratosphere, accounts for most of the observed quasi-biennial oscillation of LOD [8,9].

The third chart shows a mixed period that results from the nonlinear combination of the above Draconic and Anomalistic signals, following from conventional tidal analysis. Below is what the signal appears like over a wider interval, showing a modulation of 6 years, corresponding to the well-known beat frequency between the two lunar month periods of 27.2122 days and 27.5545 days, i.e. 5.9975 years.

If we assume that at the scale of LOD measurements, that this signal may smooth out, we can perform either a moving average or exponential smoothing to the mixture. That is shown below, along with the 6-year periodic signal extracted in ref [5]. The periods are nearly commensurate, but ~90° out-of-phase and diverging only slightly (ref [5] states the period as 6.1 yr, while ref [3] says 5.9 yr, so there is uncertainty here).

The consensus that this 6-year signal may be due to *"a gravitational coupling mode or fast torsional waves"*, and that's why a damping was applied by [5], to take into account a possible damped resonant response. Yet, if this 6-year oscillation is instead a forced response due to the mixed nonlinear lunar forcing, this becomes a steady-state response, which will be stationary over time.

The bottom-line is that if this is the real cause to the correlation, we can use this LOD period as a further calibration to the ENSO model and in particular to the phase of the mixed term, much as we use other lunar data for calibration.

## References

[1] Del Rio, R. Abarca, D. Gambis, and D. A. Salstein. "Interannual signals in length of day and atmospheric angular momentum." *Annales Geophysicae*. Vol. 18. No. 3. Springer-Verlag, 2000.

[2] Mound, Jon E., and Bruce A. Buffett. "Detection of a gravitational oscillation in length-of-day." Earth and Planetary Science Letters 243.3 (2006): 383-389.

[3] Holme, Richard, and O. De Viron. "Characterization and implications of intradecadal variations in length of day." Nature 499.7457 (2013): 202.

[4] Shen, Wenbin, and Cunchao Peng. "Detection of different-time-scale signals in the length of day variation based on EEMD analysis technique." Geodesy and Geodynamics 7.3 (2016): 180-186.

[5] Duan, Pengshuo, et al. "Possible damping model of the 6-year oscillation signal in length of day." Physics of the Earth and Planetary Interiors 265 (2017): 35-42.

[6] B. Fong Chao, “Excitation of Earth’s polar motion by atmospheric angular momentum variations, 1980–1990,” *Geophysical research letters*, vol. 20, no. 3, pp. 253–256, 1993. See fig below: