In the previous ENSO post I referenced the Rajchenbach article on Faraday waves.
There is a telling assertion within that article:
"For instance, to the best of our knowledge, the dispersion relation (relating angular frequency ω and wavenumber k) of parametrically forced water waves has astonishingly not been explicitly established hitherto. Indeed, this relation is often improperly identified with that of free unforced surface waves, despite experimental evidence showing significant deviations"
What they are suggesting is that too much focus has been placed on natural resonances and the dispersion relationships within a free fluid volume. Whereas the forced response is clearly as important — if not more — and that the forcing will show through in the solution of the equations. I have been pursuing this strategy for a while, having started down the Mathieu equation right away and then eventually realizing the importance of the forced response, yet the Rajchenbach article is the first case that I have found made of what I always thought should be a rather obvious assumption. The fact that the peer-reviewers allowed the "astonishingly" adjective in the paper is what makes it telling. It's astonishing in the equivalent sense that Rajchenbach & Clamond are pointing out that a pendulum's motion will be impacted by a periodic push. In other words, astonishing in the sense that this premise should be obvious!
So, what I take as reinforcement from the article is that there has been a misguided focus on unconstrained or free fluid flow, as opposed to solving the dynamics of a Newtonian fluid within an enclosing container. Read through it and see if you come up with a similar understanding.
Further, what I am also finding is that in the cases of ENSO and QBO, the fundamental seasonality forcing does not jump out at you, but in fact is mixed into the stew and is not revealed by conventional methods. What you need is a good model of the physics and novel means of extracting the parameters from the dynamics, such as using the Mathieu equation for fluid dynamics — as Rajchenbach suggests and I have been advocating for ENSO. That's why it hasn't been obvious over the years, and the continuing claim being made that ENSO (by Tsonis) and QBO (by Lindzen) are driven by natural/chaotic resonances lives on ...
I have off-and-on used the Excel Solver to optimize model fits and decided to try it again to supplement the multiple linear regression approach that I have been using. The regression is very fast for specific parameters but a Solver can explore more possibilities over a longer training time. The Mathieu equation that I have been using reflects that of Rajchenbach's formulation. Rajchenbach considers the Mathieu modulation itself a forcing, which is entirely valid, but I add a RHS term forcing g(t) which is supported by references on sloshing behavior by Frandsen and by Faltinsen.
The Mathieu modulation term shown in the above equation is a biennial period, but I added Hill terms for annual, semiannual, and 2/3 of a year in addition to the 2-year period. The last factor of 2/3 is a result of beating a 2-year period against the annual period as shown in the previous post. Not a problem adding these terms as the Solver should be able to sort them out and place a small magnitude on the coefficient if it's not a strong factor. For the g(t) terms I applied the strong 6.5, 14, 18.6 year angular momentum (Chandler wobble) terms and the seasonally aliased tropical, anomalous, etc tidal terms as seeded input for the Solver.
So that the Excel input looked approximately like the following inset (similar to that described here) , where the A column is time, the B column contains the ENSO values, the C column is the ENSO 2nd-derivative, and the rest E,F,G terms are parameters:
$C4+($G$1+ $E$2*COS(2*PI()/$E$3*$A4-$E$4)+ $E$5*COS(2*PI()/$E$6*$A4-$E$7)
This is actually not as complicated as it looks, and after entering the set and initiating a run, the Solver locks in strongly to the Mathieu terms and barely budges from the periods of 1/2, 2/3, 1, and 2 years. The RHS terms adjust a bit from the seed but not much more than a few percent. The optimizing target is a hybrid metric of correlation coefficient and scaled error.
As I have mentioned in a previous post (pending link), there is a degree of "artificial" correlation that occurs with the Mathieu wave equation fitting algorithm, since the ENSO signal is being modulated by a known waveform on the LHS and so that can be recovered to some extent on the RHS. However, the correlation could just as easily be anti if the modulation happened to not align with the +/- excursion of the data. And yet we see very little of anti-correlated regions in Figure 2 — and considering how noisy the ENSO signal is, the Solver fitting algorithm generates a strong match between data and model.
This is also a very simple model to set up and anybody with proficiency with Excel can duplicate these results. It definitely agrees with the previous regression approach I have been using, but this Solver was also able to identify the strong 2/3 year modulation that is predicted by non-linear period doubling from annual to biennial (note that a 1/3 year period caused by nonlinear interactions between 1 and 1/2 year periods might also be relevant, but I haven't added that yet).
In review, the important observational premises to invoke in this model are:
- A biennial mode is operational
- A phase inversion occurs between 1980 and 1996, justified by metastability of the biennial model with respect to even and odd starting years.
- Mathieu equation formulation for sloshing behavior
- A right-hand-side (RHS) forcing due to known angular momentum and gravity terms, with additional seasonal aliasing
Compare these four premises to the four observational premises made for QBO and you can see how easily one can get stymied in arriving at a complete physical model. None of these are plainly obvious from the research literature on ENSO and QBO. But when the right physics gets complemented with the right math, the results can be quite promising, if not "astonishingly not been explicitly established hitherto".