In the tidal data there are strong frequency components at 1 year and 1/2 year. These are understandable from seasonal variations caused by the earth's rotation.
So I placed a simple 1-year box-filter on the Sydney tidal data and I also filtered out the low-freq components and overall trend.
Next I added a forcing function of exactly a 2 year period on the RHS of a Mathieu-type DiffEq and optimized over an interval that I used for the validation fit earlier:
This is a promising path because the formulation is very simple yet seems to be able to capture the erratic nature of the waveform.
The need for the 2-year delay differencing may now make more sense. It is possible that ENSO is *not* sensitive to the 2-year tidal cycle so that to extract the ENSO factor from the tidal data, it is important to filter out that periodicity. As I said before the delay difference of 2 years acts as a notch filter to selectively remove that period. It may be that what remains after this filtering is an erratic gradient that tracks the ENSO behavior.
This is like untying a knot ! As one is never sure if one is making things more knotty or less knotty.
BTW, I have no good idea as to what physical mechanism drives the 2-year period, unless some nonlinear behavior is causing a period-doubling of the seasonal behavior -- perhaps a bifurcation?