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This paper by Thual et al , shows an incredibly good model fit of an ENSO characteristic:
The math is heavy-duty as it modifies the long-wave approximation used by others.
"The temporal evolution of the Tilt and WWV modes is as in the recharge–discharge model from Jin (1997a,b). One may interpret the Tilt mode as a dynamical mode adjusting very rapidly to wind stress forcing (approximated here as adjusting instantly), and the WWV mode as a dynamical mode adjusting slowly to wind stress forcing with dissipation rate $\sigma$ (Burgers et al. 2005). Consistently with Jin (1997a,b), Fedorov (2010), and Clarke (2010), the WWV mode arises from the propagation and reflection of equatorial waves that are not in phase with wind stress forcing. Here, $\sigma$ is the same as the one solved without wind stress forcing in Eqs. (8)–(11), which is consistent because the WWV mode is obtained from a linear composition of those solutions."
The interesting result is that they claim that the ENSO response is "very rapid" to changes in the wind stress forcing. This is in keeping with the model that I am applying, which uses the QBO as a wind forcing.
What I don't quite comprehend is that it uses the wind directly via reanalysis of wind stress (data from here), and so one can't really isolate the causality. Is the wind stress partly the result of the ocean upwelling/downwelling or the only causal agent?
The correlation between atmospheric pressure and wind is well known, as wind is caused by pressure gradients. But we also know that wind can force surface water to assumulate on the windward side, thus leading to thermocline tilting as gravitational forces take effect.
So let us agree that this fit is great and that we can assume that the wind is causing the ENSO fluctuations; but unless one can actually predict what the wind will do in the future, we are still left with nothing to extrapolate against. That is why the model that I have been using, which applies the QBO as the main forcing, has more potential for prediction. The QBO is highly periodic, with a period of 28 months that can be extrapolated into the future, subject to any jitter that it will experience. On the other hand, the TW model by Thual still needs a model to predict the wind stress.
I think this is important, as the understanding of ENSO form physical principles is advancing rapidly, thanks to research by Clarke and others that have been looking at wind curl.
This is what Thual have to say:
"This ocean adjustment involves considerable energy exchanges through variations of the equatorial thermocline depth (Goddard and Philander 2000; Brown a nd Fedorov 2010b), and explains the relatively high predictability of ENSO (Cane and Zebiak 1985; Meinen and McPhaden 2000;McPhaden 2012)."
Note that Thual state that there is relatively high predictability of ENSO (my emphasis). Yet without predictability of wind we arrive at a Catch-22 situation; in fact, one that all models that depend on erratic EOF data as factors find themselves in. Unless the EOF can be modeled, one is left flapping in the breeze when it comes to predictive power.
S. Thual, B. Dewitte, N. Ayoub, and O. Thual, “An Asymptotic Expansion for the Recharge-Discharge Model of ENSO,” Journal of Physical Oceanography, vol. 43, no. 7, pp. 1407–1416, 2013. https://hal.archives-ouvertes.fr/file/index/docid/951088/filename/jpo-d-12-0161_2E1.pdf