# Biennial Connection from QBO to ENSO

I see these as loose-end issues in trying to tie the forcing of the Quasi-Biennial Oscillation (QBO) to the forcing behind the El Nino Southern Oscillation (ENSO).

1. The nature of the biennial oscillations in ENSO [1] — and specifically, what drives the differences in forcing between QBO and ENSO .
2. Why do the tides in the Southern Pacific have a more strictly biennial (i.e. =2 year) periodicity than the quasi-biennial (i.e. ~2.33 year) oscillations in atmospheric wind?
3. The tie-in to the Chandler wobble on the triaxial earth [2], which appears more significant for ENSO than for QBO.
4. Phase reversals in the ENSO standing wave, particularly in 1981.

While collectively trying to resolve these issues, I discovered an intriguing pattern in the wave-equation transformation of the ENSO signal.  This new pattern is based on defining precise sidebands +/- on each side of the exact biennial period. A pair of sinusoidal sidebands are formed when a primary frequency is modulated by another sinusoid.

$sin(pi t) cdot sin(omega_m t) = frac{1}{2} ( cos(pi t - omega_m t) - cos(pi t + omega_m t) )$

The sidebands appear to match the period of three identified wobbles in the angular momentum of the rotating triaxial earth [2]. These sidebands are sufficient to extrapolate most of the wave-equation transformed curve when fitting to either a large interval or to a short interval within the time series. The latter is simply a consequence of a shorter interval containing enough information to reconstruct the rest of the stationary time series.  See Figure 1 below for examples of the effectiveness of the fit across various cross-sectional intervals and how well the short interval sampling extrapolates over the rest of the time series. Even as short a training interval as 15 years results in a fairly effective extrapolation, since 15 years is comparable to the longest constituent modulation period.

Fig. 1: Examples of stationary model fits to the wave equation transformed  ENSO data. The top panel is a fit to the entire interval and the three below are extrapolated from training intervals of varying lengths. Click to enlarge.

This new pattern is essentially a refined extension of the sloshing formulation I started with — but now the symmetry and canonical form is becoming much more readily apparent. The identified side-bands have periods of 6.5, 14.3, and 18.6 years, which you can understand from reading the fractured English in reference [2]. These three periods are known modulations of the earth’s rotation (ala the Chandler wobble) and all fit in to the F(t) term of the biennial-modulated wave equation.

$f''(t) + omega_0^2 f(t) = sin(pi t) cdot F(t)$

So the strict biennial period figures into the forcing via a modulation with the longer period modulations.  From the first equation, multiplying out the individual sinusoidal terms produces pairs of + and sidebands about the 2-year period.  So that the 2.33 year QBO-like period is a consequence of the biennial cycle mixing with the 14.3 year cycle.

## Discussion

Here are some observations related to the bullet point issues at the top

(1) I have long considered that the 2 year biennial factor is likely a strong forcing factor for ENSO starting with the observation that it shows up in tidal gauge measurements in the south Pacific. In this post on the analysis of Sydney, Australia tidal gauge data, the two-year cycle is quite evident.  In fact when the two-year factor is removed, the characteristics of ENSO are revealed in the tide-gauge readings. The 2-year sinusoid is likely a real factor arising likely from non-linear interactions of the seasonal (i.e. yearly forcing). Interestingly, the 2-year sinusoid modulated against itself yields a yearly sinusoid — which we know is relevant.

(2) The connection to the QBO is likely that the biennial modulated with the 14.3 year triaxial wobble component gives a strong sideband very close to the 28-month = 2.33 year QBO frundamental period. This is by far the strongest factor in the ENSO model fit. Certainly these factors reinforce each other in some way as the closeness of these two values is likely not just a coincidence.

$frac{1}{2} - frac{1}{14.3} = frac{1}{2.325}$

(3) The composite sloshing behavior of two slightly different densities of stratified liquid (i.e the thermocline, see Figure 2) is extremely sensitive to changes in angular momentum, especially if the rate of change matches the resonant frequency of the medium. It should not be surprising that forcing factors ranging from 2 years to 20 years will have the greatest impact, considering that the characteristic (resonant) frequency of ENSO may be greater than 4 years (according to Clarke). This resonance period is longer than that required for QBO, simply on account of the inertia of the ocean being much greater than that of the upper atmosphere.

Fig. 2:  The sensitivity of a stratified liquid is very sensitive to linear or angular momentum changes. This wave motion machine provides an analog to the thermocline within the equatorial Pacific ocean, where the more dense cooler water below the thermocline (shown as blue) interacts with the less dense warmer waters (shown as clear) above the thermocline. The underlying thermocline will see much greater fluctuations than the surface level for a given periodic forcing.  (note that this is much more exaggerated than would occur in the ocean)

(4) In terms of understanding a potential phase reversal in the ENSO data, a precise biennial forcing provides a highly plausible mechanism. Since a lengthy interval featuring a coherent biennial  factor is either strictly odd year or even year , all that is required is for the biennial factor to switch from an odd to an even parity (or vice versa) to initiate a phase reversal in the wave equation solution.  As it appears metastable and in a sense is definable as metastable, the duration of a phase reversal may not last long, and so will likely switch back to the lower energy state. That may be an adequate explanation for the 1981-1996 phase reversal. This metastability also appears in the biennial periodicity in tidal gauge measurements.

Potentially resolving these four issues is a huge step forward in creating a comprehensive model of oceanic and atmospheric oscillations.  The basic foundation is becoming more solid in terms of identifying the forcing factors derived from angular momentum effects (due to Chandler wobble and gravitational tidal forcing) while putting to bed the red herring of cyclic solar radiation phenomena such as TSI having a significant impact.  In fact, the identification of the 14 year triaxial factor [2] provides a forcing that fits somewhere in between the 11 and 22 year solar radiation/sunspot cycles.

As a bottom-line, the refined model is still simple in scope, but the individual elements are subtly odd enough to explain why it has been so difficult to hone in on a solution for ENSO. As some have said, this is not the end, but just the beginning of the end. We have tied up some loose-ends but now it is on to making it fit more snugly.

## References

[1] S.-R. Yeo and K.-Y. Kim, “Global warming, low-frequency variability, and biennial oscillation: an attempt to understand the physical mechanisms driving major ENSO events,” Climate Dynamics, vol. 43, no. 3–4, pp. 771–786, 2014. http://link.springer.com/article/10.1007/s00382-013-1862-1/fulltext.html
[2] Wang, Wen-Jun, W.-B. Shen, and H.-W. Zhang, “Verifications for Multiple Solutions of Triaxial Earth Rotation,” IERS Workshop on Conventions Bureau International des Poids et Mesures (BIPM), Sep. 2007. http://www1.bipm.org/utils/en/events/iers/Wang.pdf

## 22 thoughts on “Biennial Connection from QBO to ENSO”

1. The biennial factor is not perfect so some fraction of the unmodulated F(t) bleeds through, which is the k factor below.

$f''(t) + \omega_0^2 f(t) = (sin(\pi t) + k) \cdot F(t)$

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• what happens when we estimate the ENSO signal and not just the wave-equation transformed ENSO signal

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• In retrospect, these guys were tantalizingly close to figuring out ENSO. They had the biennial factor in their Eq(1) but gave up.

[1]K.-Y. Kim, J. J. O’Brien, and A. I. Barcilon, “The principal physical modes of variability over the tropical Pacific,” Earth Interactions, vol. 7, no. 3, pp. 1–32, 2003.

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2. Wang, Wenjun. “Free acceleration of Earth rotation and energy source of ENSO.” 37th COSPAR Scientific Assembly. Vol. 37. 2008.

From the latter cite:
“Earth is a triaxial body. Triaxial body rotation has Euler dynamic model of nonlinearity with A¡B¡C. We develop a decomposition theorem for obtaining three true solutions for the rotation with two stable wobbles of Chandler and another decadal of period 14.6 a. The other may be an unstable inverted pendulum of one way sway. The nonlinear coupling of the two free wobbles provide free accelerations for the rotation or LOD with a period of 7 month and another 14.6 a. The fluctuation of period 14.6 a may cause bifurcation cascade in nonlinear coupling and a series of periods 7.3, 3.65 and 1.825 a may be observed with free accelerations and decelerations so that cause ocean water near the equator flow eastward and westward. This makes the energy source of ENSO in eigen periods of 7, 4, 2 a. Therefore in this study we verify that the energy source of ENSO is the free acceleration of Earth rotation. ”

I modeled a 7 month periodic forcing as Wang suggested, and sure enough, it showed up as a sharp factor centered at 0.5791 years = 6.95 months (albeit within the noise of ENSO).

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3. Dennis Coyne

Hi Paul,

Can you post the link to the QBO and SOI models, they are hard to find on Entroplet, at least for me. Nice work as always.

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• Dennis, They aren’t quite in final form of course (which is why they are hidden) but these are the prototyped extensions to entroplet:
/context_soi/navigate
/context_qbo/navigate

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• That’s an inventive answer to an inseretting question

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• It’s much easier to understand when you put it that way!

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4. Bill H

Hi, WHUT,

Season’s Greetings to you. Just wondering if you’re still planning to publish your work in a peer reviewed journal. What spurred me was Tallbloke’s latest self-congratulatory post on his blog about some SPIE conference proceedings “referencing” a paper of his in the now defunct and truly execrable Pattern Recognition in Physics.

I assume the SPIE paper is not peer reviewed, so the citation is pretty worthless. Nevertheless it annoys me no end that there’s all this peer-reviewed (PRP) garbage out there, giving its authors a glow of smugness while your work isn’t being published.

Incidentally, TB’s home city of Leeds has been just been flooded in the rains that are now pounding not just Cumbria, but much of N. England and N. Wales. Who knows: a sense of reality may be kindled at the Talkshop?

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• Bill,
Tryin to.

I’ve been to the lake district and can imagine the probs. Lots of eroded valleys where water can sluice through.

In good time, everything will come back and bite Rog. Right now, he is siding with determinism and not chaos, which is rare for a denialist. He will start to squirm I bet.

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• Bill H

There’s a significant anthropogenic contribution to the problem as well: mountains bare of trees, indeed significant vegetation of any kind, as a result of landowners grazing heavily sheep, supported by generous subsidies. By contrast, the mountains of continental Europe have remained well-forested.

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5. This is the periodogram of the sidebands, where I transformed the x-axis to reflect the period instead of the frequency.

Note the symmetry about a 2.0077 biennial year period, which was chosen to align the discrete 2048 point FFT with the 1 month per sample data series.

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6. Pingback: QBO Model Validation | context/Earth

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