[mathjax]In the last post on characterizing ENSO, I discussed the trend I observe in the modeling process — the more I have worked it, the simpler the model has become. This has culminated in what I call a “baby” differential equation model, a model that has been reduced to the bare minimum.

This has partly been inspired by Professor John Carlos Baez’s advice over at the Azimuth Project

“I think it’s important to take the simplest model that does a pretty good fit, and offer extensive evidence that the fit is too good to be due to chance. This means keeping the number of adjustable parameters to a bare minimum, not putting in all the bells and whistles, and doing extensive statistical tests”

Well, I am now working with the bare minimal model. All I incorporate is a QBO forcing that matches the available data from 1953 to current. Then I apply it to a 2nd-order differential equation modeled as a characteristic resonant frequency $$omega_0$$.

$$ f”(t) + omega_0^2 f(t) = Forcing(t) = qbo(t) $$

I adjust the initial conditions and then slide the numerically computed Mathematica output to align with the data.

The figure below is the empirical fit to the QBO, configured as a Fourier series so I can extrapolate backwards to 1880.

Figure 2 is a fit using the first 400 months of ENSO SOI data as a training interval, and validated to the last 800 months (100 years total)

The next fit uses the second 400 months of data as a training interval, which is then validated forward and backward by 400 months

The final fit using the last 400 months of data, and validates to the first 800 months

In each case, the overall correlation coefficient is above 0.7, which is as much as can be expected given the correlation coefficient of the empirical model fit to the raw QBO data.

At this point we can show a useful transformation.

$$ f”(t) + omega_0^2 f(t) = qbo(t) $$

If we take the Fourier transform of both sides, then:

$$ (-omega^2 + omega_0^2) F(omega) = QBO(omega) $$

This implies that the power spectra of SOI and QBO should contain many of the same spectral components, but scaled at values proportional to the frequency squared. That is why the SOI, i.e. f(t) or F($$omega$$), contains stronger long time-period components; while the QBO shows the higher frequency, e.g. the principal 2.33 year period, more strongly.

One may ask, what happened to the Chandler wobble component that has been discussed in previous posts?

It is still there but now the wobble forcing is absorbed into the QBO. Using the recently introduced and experimental “FindFormula” machine learning algorithm in Mathematica, the 6+ year Chandler wobble beat period shows up in the top 3 cyclic components of QBO! See the following figure with the yellow highlight. It is not nearly as strong as the 2.33 year period, but because of the spectral scaling discussed in the previous paragraph, it impacts the ENSO time profile as importantly as the higher frequency components.

## Discussion

The caveat in this analysis is that I didn’t include data after 1980, which is explained by the non-stationary disturbance of the climate shift that started being felt after 1980. This will require an additional 400 month analysis interval that I am still working on.

“Can you tell me in words roughly what these 20 lines do, or point me to something you’ve written that explains this latest version?”

It is actually now about half that, if the fitting to the QBO is removed. And the part related to just the DiffEq integration computation is just a couple of lines of Mathematica code. The rest is for handling the data and graphing.

The fit to the QBO can be made as accurate as desired, but I stopped when it got to the correlation coefficient as shown in the first figure above. The idea is that the final modeling result will have a correlation coefficient about that value as well (as the residual is propagating an error signal), which it appears to have.

The geophysics story is very simple to explain, as it is still sloshing but the forcing is now reduced to one observable component. If I want to go this Baby D Model route, I can shorten my current ENSO sloshing paper from its current 3 pages to perhaps a page and a half.

The statistical test proposed is to compare against an alternate model and then see which one wins with an information criteria such as Akaike. I only have a few adjustable parameters (2 initial conditions, a characteristic frequency, and a time shift from QBO to ENSO), so that this will probably beat any other model proposed. It certainly will beat a GCM, which contains hundreds or thousands of parameters

Power spectra of ENSO and QBO, showing similarities in components, especially in primary 2.33 year period, indicated by the red tic mark

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Now this is looking really interesting, plus it’s an easy-to-understand plausible physical model. Most likely the causal route runs QBO-tradewinds-ENSO but that’s still very much in line with what we already know about ENSO. Plus the predictability stays in the picture, at least to some extent.

I’d go ahead and publish your one-pager, but be sure to include your graphs showing the strong correlations, and the power spectra too. Nice work!

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Keith, Thanks

Another plausible explanation that must be considered is that the same forcing is driving both ENSO and QBO. The twist is that the forcing drives QBO without a lagged inertial response so that we detect the forcing directly. The ocean however shows the resonant sloshing response of several years.

Or that the inertial changes in the QBO are being transferred to the ocean as a consequence of conservation of angular momentum.

And there are still papers that claim that ENSO drives QBO, which I can’t understand at all given the baby model results.

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What’s the lag time from QBO to ENSO?

Also, just FYI, the QBO waves start in the stratosphere and move downward, so the QBO-leads-tradewinds again has a plausible physicality.

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I don’t know if I can pin down a lag time because I am not even sure where the maximum exists spatially. On top of that, the response of ENSO to a forcing is complicated by the lag from the resonant response time period.

Saying that, it is probably pretty immediate because I don’t have to shift the QBO time series much to provide a decent fit.

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Another thing to consider is that the tradewinds, via pressure on continents (especially the Andes) may also be a causal forcing for the Chandler Wobble.

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Some of that is also related to sudden stratospheric warming events. Which is another source of angular momentum variations/

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