El Niño and the Southern Oscillation remain one of the most studied yet difficult-to-predict climate systems. Traditional models either rely on complex global circulation calculations or treat the process as stochastic, with limited predictive power. This work stands apart: its nearest neighbors in the archive deal with quantum gravity, crystal plasticity, and machine learning, while its closest intellectual relatives are studies linking gravity to wave equations. Here, a deterministic approach is proposed, relying on known astronomical and geophysical periodicities.
The model is built on a second-order equation describing the sloshing of a fluid in a confined volume. The characteristic frequency is set by a period of 4.25 years, corresponding to thermocline dynamics. The modulation q(t) takes the form of a Mathieu term with a two-year period, and the forcing F(t) includes seasonally aliased lunar periods (draconic, anomalistic, tropical), the rate of the Chandler wobble with a period of about 6.46 years, and a 14-year term related to the QBO. For fitting, Southern Oscillation Index data from 1880 onward were used; the solution was found numerically in Mathematica and R's DiffEq.
The model reproduces the historical SOI series with a correlation of 0.8. A key element was introducing a phase inversion of the biennial modulation in the interval 1981–1996, corresponding to a known climate shift. The contribution of solar activity (the Schwabe cycle) did not improve the fit and was therefore excluded. The machine learning tool Eureqa independently confirmed the choice of QBO and Chandler wobble factors. Residual discrepancies are mainly due to measurement noise at the Tahiti and Darwin stations.
This work shows that ENSO can be viewed as a deterministic forced system rather than a random process. This differs from neighboring works in the archive, which either use stochastic models or require full hydrodynamic computation. Here, a simple equation suffices to capture the main features of the phenomenon. This approach opens the way to forecasts years ahead, provided that rare regime shifts can be predicted.
The model could be refined by a more accurate description of the Chandler wobble, including possible splitting of the spectral peak. One could also investigate the connection with the 18.6-year draconic cycle and the triaxial rotation of the Earth. For forecasting, it is critical to learn in advance when the biennial phase inversions, like the 1981 shift, will occur.
The results could be useful for short- and medium-term El Niño forecasting, which is important for agriculture, water resource management, and risk assessment of extreme weather events.
Test the model against paleoclimate reconstructions of ENSO over the past centuries. Attempt to link phase inversions to observed changes in atmospheric angular momentum.
For climatology, the nature of rare regime shifts that break stationarity remains an open question. This model explicitly includes such a shift as a change in the parity of the biennial cycle, but the mechanism triggering the inversion is not yet clear.
The 4.25-year period characteristic of the thermocline is close to half of the 8.5-year cycle of the lunar perigee, which may point to a tidal origin of the main frequency.