Conventional tidal prediction models typically combine constituent lunisolar forcing factors harmonically fit to sets of collected tidal gauge data. A harmonics analysis is favored over a precise orbital ephemeris-based gravitational forcing model, as tides are localized in scope and sensitive to a particular volume geometry. But what happens when the dynamic behavior is much larger in scale? As we have demonstrated previously [1], lunisolar tidal constituents forced by a strong biennial-modulated annual signal will provide a high-quality fit to ENSO – albeit subject to over-fitting of the numerous constituent factors available. Yet as ENSO is a large-scale phenomenon, it should be more amenable to applying a precise set of ephemeris data as the forcing to a Laplace’s tidal equation formulation. This should reflect the underlying physics governing the dynamics more realistically, while severely constraining the degrees of freedom (DOF) in factors which lead to the possibility of over-fitting. We used the NASA JPL Horizons (https://ssd.jpl.nasa.gov/horizons.cgi) ephemeris data for the Sun and Moon as a parametric input to the well-known 1/R^3 gravitational forcing function and verify as good a quality fit as that available from a high-DOF harmonics approach. This extends over the modern-day instrumental record of ENSO but also covers the coral proxy records that span the years from 1650 to 1880. The approach works effectively because the extra DOF (including phases and elliptical nonlinearities in the orbits) needed to precisely define the gravitational forcing are accurately tracked by the Horizons ephemeris algorithm. Importantly, the results are highly sensitive to the relative forcing amplitudes, which is not surprising, since the fast lunisolar cycles are projected over spans of hundreds of years. The challenge is equivalent to attempting to perform a conventional tidal analysis over a similarly lengthy time span, while also dealing with noise and a limited resolution time-series. [1] Pukite, P.R. “Biennial-Aligned Lunisolar-Forcing of ENSO: Implications for Simplified Climate Models.” AGU Fall Meeting, 2017.

By solving Laplace’s tidal equations along the equatorial Pacific thermocline, assuming a delayed-differential effective gravity forcing due to a combined lunar+solar (lunisolar) stimulus, we are able to precisely match ENSO periodic variations over wide intervals. The underlying pattern is difficult to decode by conventional means such as spectral analysis, which is why it has remained hidden for so long, despite the excellent agreement in the time-domain. What occurs is that a non-linear seasonal modulation with monthly and fortnightly lunar impulses along with a biennially-aligned “see-saw” is enough to cause a physical aliasing and thus multiple folding in the frequency spectrum. So, instead of a conventional spectral tidal decomposition, we opted for a time-domain cross-validating approach to calibrate the amplitude and phasing of the lunisolar cycles. As the lunar forcing consists of three fundamental periods (draconic, anomalistic, synodic), we used the measured Earth’s length-of-day (LOD) decomposed and resolved at a monthly time-scale [1] to align the amplitude and phase precisely. Even slight variations from the known values of the long-period tides will degrade the fit, so a high-resolution calibration is possible. Moreover, a narrow training segment from 1880-1920 using NINO34/SOI data is adequate to extrapolate the cycles of the past 100 years (see attached figure). To further understand the biennial impact of a yearly differential-delay, we were able to also decompose using difference equations the historical sea-level-height readings at Sydney harbor to clearly expose the ENSO behavior. Finally, the ENSO lunisolar model was validated by back-extrapolating to Unified ENSO coral proxy (UEP) records dating to 1650. The quasi-biennial oscillation (QBO) behavior of equatorial stratospheric winds derives following a similar pattern to ENSO via the tidal equations, but with an emphasis on draconic forcing. This improvement in ENSO and QBO understanding has implications for vastly simplifying global climate models due to the straightforward application of a well-known and well-calibrated forcing. [1] Na, Sung-Ho, et al. “Characteristics of Perturbations in Recent Length of Day and Polar Motion.” Journal of Astronomy and Space Sciences 30 (2013): 33-41.