Following up on the last post on the ENSO forcing, this note elaborates on the math. The tidal gravitational forcing function used follows an inverse power-law dependence, where a(t) is the anomalistic lunar distance and d(t) is the draconic or nodal perturbation to the distance.
Note the prime indicating that the forcing applied is the derivative of the conventional inverse squared Newtonian attraction. This generates an inverse cubic formulation corresponding to the consensus analysis describing a differential tidal force:
For a combination of monthly and fortnightly sinusoidal terms for a(t) and d(t) (suitably modified for nonlinear nodal and perigean corrections due to the synodic/tropical cycle) the search routine rapidly converges to an optimal ENSO fit. It does this more quickly than the harmonic analysis, which requires at least double the unknowns for the additional higher-order factors needed to capture the tidally forced response waveform. One of the keys is to collect the chain rule terms a'(t) and d'(t) in the numerator; without these, the necessary mixed terms which multiply the anomalistic and draconic signals do not emerge strongly.
As before, a strictly biennial modulation needs to be applied to this forcing to capture the measured ENSO dynamics — this is a period-doubling pattern observed in hydrodynamic systems with a strong fundamental (in this case annual) and is climatologically explained by a persistent year-to-year regenerative feedback in the SLP and SST anomalies.
Here is the model fit for training from 1880-1980, with the extrapolated test region post-1980 showing a good correlation.
The geophysics is now canonically formulated, providing (1) a simpler and more concise expression, leading to (2) a more efficient computational solution, (3) less possibility of over-fitting, and (4) ultimately generating a much better correlation. Alternatively, stated in modeling terms, the resultant information metric is improved by reducing the complexity and improving the correlation -- the vaunted cheaper, faster, and better solution. Or, in other words: get the physics right, and all else follows.