This is how to apply a slow feature analysis to a sloshing model.

Start with the Mathieu equation and keep it in its differential form

$\frac{d^2x(t)}{dt^2}+[a-2q\cos(2\omega t)]x(t)=F(t)$

The time-modulating parameter multiplying *x(t)* is replaced with a good guess -- which is that it is either an annual or biennial modulation and with a peak near the end of the year. For a biennial modulation, the peak will appear on either an odd or even year. In the modulation below it is on an even year:


The RHS *F(t)* is essentially the same modulation but with a multiplicative forcing corresponding to the known angular momentum and tidal variations. The strongest known is at the Chandler Wobble frequency of ~432 days. There is another wobble predicted at ~14 years and a nutation at 18.6 years due to the lunar nodal variation and a heavily aliased anomalistic period at 27.54 days. Those are the strongest known forcings and are input with unknown amplitude and phase.

The search solver tries to find the best fit by varying these parameters over a "training" window of the ENSO time series. I first take a split window that takes an older time interval and matches with a more recent time interval. The "out-of-band" interval is then used to test the fit.


Then I reverse the fit by using the out-of-band interval as the fitting interval and testing against the former.


The reason that the fit is stationary across the time intervals is because the strength and the phase of the wobble terms remains pretty much constant. The dense chart below is a comparison of the factors used. Note that some of the lesser tidal forcing factors are included as well and they do not fare quite as well.


Here is an animated GIF of how the two fitting intervals compare:


The sloshing model is parsimonious with the data and what remains to be done is to establish the plausibility of the model. In that regard, is there enough of an angular momentum change or torque in the earth's rotation to cause the ocean to slosh? Or is there enough to cause **the thermocline** to slosh, which is actually what's happening? The difference in density of water above and below the thermocline is enough to create a reduced effective gravity that can plausibly be extremely sensitive to momentum changes. The sensitivity is equivalent to an oil/water wave machine

Yet the current thinking in the consensus ENSO research is that prevailing east-west winds are what causes a sloshing buildup. But what causes those winds? Could those be a **result** of ENSO and not a **cause**? They seem to be highly correlated,


but what causes a wind other than a pressure differential? And that pressure differential arises from the pressure dipole measured by the ENSO SOI index. Whenever the pressure is low in the west Pacific, it is high in the east and vice versa.

So the plausibility of this model of torque-assisted sloshing is essentially wrapped around finding out whether this effect is of similar magnitude of any wind-forced mechanism. Perhaps the question to ask -- is it easier to cause a wave machine to slosh by blowing on the surface or to gently rock it?