Biennial Stratosphere Mode

In addition to the biennial mode found in ENSO and in GPS readings, a stratospheric biennial mode also exists. This is different than the quasi-biennial oscillation (QBO) modeled previously, as it shows a more strict two year cycle.

From Dunkerton [1]:

Second, concerning their time dependence, subbiennial variations should not be viewed in isolation from other modes. The seasonal dependence of QBO anomalies cannot be described by a single harmonic, whether quasi-biennial or subbiennial; rather, their superposition provides for the seasonality.


Only three parameters were varied to obtain the least squares fit: the QBO period, QBO phase, and subbiennial phase. This calculation was performed on the three tracers independently, but the same QBO period was obtained in each case, namely, 26 months. The corresponding subbiennial period is 22.3 months

What Dunkerton is talking about essentially is akin to a 2.166 year quasi-biennial signal mixed with a 1.858 year sub-biennial signal. These combine in frequency space as Dunkerton states by their superposition, and also couple together via a ~26-year modulation on the biennial signal -- as per the trig identity:

cos(2\pi t/2.166) + cos(2\pi t/1.858) = 2 cos(\pi t) cos(2\pi t/26)


This is substantiated by Remsberg in his methane study [2] where in addition he sees a smaller amplitude factor which is closer to the exact biennial signal.

Three interannual terms are almost always present and are 19 included in the model: an 838-dy (27.5-mo) or QBO1 term; a small amplitude, 718-dy (23.5-mo) biennial or QBO2 term; and a 690-dy (22.6-mo) sub-biennial (SB) term, whose period arises from the difference interaction between QBO and annual terms. The relative amplitudes of these three interannual terms vary with latitude and altitude.

More recently (in 2014) Remsberg reports [5]:

Instead, Fourier analysis of the time series of the residuals after removing the seasonal terms almost always indicates that there are two significant, interannual terms having periods of order of 28 (QBO-like) and 21 months (subbiennial term denoted as IA).

This is akin to a 2.333 year quasi-biennial signal mixed with a 1.75 year sub-biennial signal, leading to a 14-year signal modulating the biennial signal:

cos(2\pi t/2.333) + cos(2\pi t/1.75) = 2 cos(\pi t) cos(2\pi t/14)


Which is strikingly similar to the behavior observed in the ENSO model's 14-year modulation of the biennial cycle. The idea was that this 14 year modulation is close to the additional triaxial wobble suggested by Wang.  Slightly weaker is an 18.6 year modulation which is close to the lunar nodal period.

Fig. 1:  Biennial modulation in the ENSO power spectrum

The similarity between the stratospheric measures and the ENSO observations is based on the common behavior of a symmetrical balance about the biennial cycle. This is starkly observed in the power spectrum of ENSO shown above.

In a closely related finding, the single-minded AGW-denier Murry Salby also discusses this modulation in [6], but ascribes the modulation to the sunspot cycle, as he finds a value closer to an 11-year modulation. That assumption has to be taken with some circumspection, as it is well known that Salby favors a solar-based rationale to global temperature variation, which has gotten himself into some bizarre predicaments.


1. Dunkerton, Timothy J. "Quasi-biennial and subbiennial variations of stratospheric trace constituents derived from HALOE observations." Journal of the atmospheric sciences 58.1 (2001): 7-25.
2. Remsberg, Ellis E. "Methane as a diagnostic tracer of changes in the Brewer–Dobson circulation of the stratosphere." Atmospheric Chemistry and Physics 15.7 (2015): 3739-3754.
3. Remsberg, E. E., and L. E. Deaver. "Interannual, solar cycle, and trend terms in middle atmospheric temperature time series from HALOE." Journal of Geophysical Research: Atmospheres 110.D6 (2005).
4. Remsberg, Ellis E. "On the observed changes in upper stratospheric and mesospheric temperatures from UARS HALOE." NASA Langley Report (2006).
5. Remsberg, E. E. "Decadal-scale responses in middle and upper stratospheric ozone from SAGE II version 7 data." Atmospheric Chemistry and Physics 14.2 (2014): 1039-1053.
6. Salby, Murry, Patrick Callaghan, and Dennis Shea. "Interdependence of the tropical and extratropical QBO: Relationship to the solar cycle versus a biennial oscillation in the stratosphere." Journal of Geophysical Research: Atmospheres 102.D25 (1997): 29789-29798.

\frac{1}{2.333} + \frac{1}{1.75} = 1


Biennial Signals from GPS

This paper is from an open-access journal, yet it's a mind blower.

Pan, Yuanjin, et al. "The Quasi-Biennial Vertical Oscillations at Global GPS Stations: Identification by Ensemble Empirical Mode Decomposition." Sensors 15.10 (2015): 26096-26114.

GPS signals have enough information content that they can pick up sensitive measures related to earth deformation, so the paper describes the results.

That strict biennial signal is readily apparent along even years and it likely has everything to do with the biennial-modulated ENSO behavior. Have to remember that a thermocline separating two-slightly different density liquids acts as a highly sensitive sensor -- any slight forcing will get the sloshing in motion, and the GPS is likely isolating the biennial component.

Also some indication that the measure is picking up the 28 month QBO signal. See Table 1 in the paper, which shows an average period of 2.3 years for the quasi-biennial oscillation.

Biennial Connection to Seasonal Aliasing

I found an interesting mathematical simplification relating seasonal aliasing of short-period cycles with a biennial signal.

If we start with a signal of an arbitrary frequency \omega_L = 2\pi/T_L

L(t) = k \cdot sin(\omega_L t + \phi)

and then modulate it with a delta function array of one spike per year

s(t) = \sum\limits_{i=1}^n a_i sin(2 \pi t i +\theta_i)

This is enough to create a new aliased cycle that is simply the original frequency \omega_L summed with an infinite series of that frequency shifted by multiples of 2\pi .

f(t) = k/2 \sum\limits_{i=1}^n a_i sin((\omega_L - 2 \pi i)t +\psi_i) + ...

This was derived in a previous post. So as a concrete example, the following figure is a summed series of f(t) -- specifically what we would theoretically see for an anomalistic lunar month cycle of 27.5545 days aliased against a yearly delta.

Fig 1: Series expansion of aliasing

If you count the number of cycles in the span of 100 years, it comes out to a little less than 26 cycles, or an approximately 3.9 year aliased period. If a low-pass filter is applied to this time series, which is likely what would happen in the lagged real world, a sinusoid of period 3.9 years would emerge.

The interesting simplification is that the series above can also be expressed exactly as a biennial + odd-harmonic expansion

[cos(\pi t) + cos(3 \pi t) + cos(5 \pi t) + ...] \cdot sin(\frac{2\pi}{T} t)

where T is given by

1/T = 1/2 + int(1/T_L) - 1/T_L

where the function "int" truncates to the integer part of the period reciprocal. Since T_L is shorter than the yearly period of 1, then the reciprocal is guaranteed to be greater than one.

As a check, if we take only the first biennial term of this representation (ignoring the higher frequency harmonics) we essentially recover the same time series.

Fig 2: Biennial modulation term

What is interesting about this factoring is that a biennial modulation may naturally emerge as a result of yearly aliasing, which is possibly related to what we are seeing with the ENSO model in its biennial mode. Independent of how many lunar gravitational terms are involved, the biennial modulation would remain as an invariant multiplicative factor.

For the ENSO signal, an anomalistic term corresponding to a biennial modulation operating on a 4.085 year sinusoid appears significant in the model

1/4.085 = 1/2 + int(365.242/27.5545) - 365.242/27.5545

which is expanded as this pair of biennially split factors (see ingredient #5 in the ENSO model)

1/4.085 ~ 2/3.91 - 2/1.34

ENSO Phase Reversal

As I indicated in this post, lots of good information is available in "unpublished" NASA memorandum.

Returning to the mystery of an inferred phase reversal in the biennial forcing, the best description of the interaction of the Chandler wobble with ENSO is this NASA technical memorandum.

Fig 1: From B.F.Chao, NASA Technical Memorandum 86231

So Chao essentially reports that the Chandler wobble is in-phase with ENSO from 1900-1979, but it flips phase starting around 1980. This is precisely in line with what I am finding.

Here's an example of how you can gain confidence that you are on the right track. I configured the multiple linear regression to train on the ENSO time series applying the wave equation transform from 1880-1940, and then looked at the correlation coefficient over the interval 1940-2013 -- which is completely outside the training interval.

Fig 2: Training from 1880-1940, correlation outside. The ENSO signal is phase reversed from 1980 to 1996.

The in=inside cc is 0.725 and the out=outside cc is 0.758. It is rare that you can get a correlation higher outside of the training interval, but because of the stronger noise in the earliest ENSO measurements this is not out of the question. In terms of variance contributions, the reduced noise promotes the real underlying signal.

Next, we generate a set of several synthesized red-noise random walks that have similar variance to the actual ENSO data. This is essentially the Ornstein-Uhlenbeck algorithm, describing a random walk with a reversion-to-the-mean potential well factor. Ten runs were combined into an animated GIF shown below:

Fig 3: A set of O-U random walk profiles.

Note that although the correlation coefficient is moderate within the training interval (anywhere from 0.4 to 0.6), it is insignificant outside of the training interval, and a few times it actually goes negative. In other words, there is no phase coherence outside of the interval, and any apparent agreement within the training interval is likely the result of over-fitting.

Alas, research is still being published [1] arguing whether or not ENSO is red noise. This is actually a moot point since what the GCMs generate (as far as I can tell) are invariably reported as sets of time series with random outcomes.  If ENSO is actually deterministic -- apart from the rare metastable phase reversal and volcanic activity -- the GCMs should be slight variations of Figure 2 and not Figure 3.

[1] Chen, Xianyao, and John M. Wallace. "Orthogonal PDO and ENSO indexes." Journal of Climate 2016 (2016).

You don't need a weatherman to know which way the wind blows.

Deterministically Locked on the ENSO Model

After several detours and dead-ends, it looks as if I have locked on a plausible ENSO model, parsimonious with recent research.  The sticky widget almost from day 1 was the odd behavior in the ENSO time-series that occurred starting around 1980 and lasting for 16 years. This has turned out to be a good news/bad news/good news opportunity.  Good in the fact that applying a phase inversion during that interval allowed a continuous fit across the entire span.  Yet it's bad in that it gives the impression of applying a patch, without a full justification for that patch. But then again good in that it explains why other researchers never found the deterministic behavior underlying ENSO -- applying conventional tools such as the Fourier transform aren't much help in isolating the phase shift (accepting Astudillo's approach).

Having success with the QBO model, I wasn't completely satisfied with the ENSO model as it stood at the beginning of this year. It didn't quite click in place like the QBO model did. I had been attributing my difficulties to the greater amount of noise in the ENSO data, but I realize now that's just a convenient excuse -- the signal is still in there.  By battling through the inversion interval issue, the  model has improved significantly.  And once the correct forcing and Mathieu modulation is applied, the model locks in place to the data with the potential to work as well as a deterministic tidal prediction algorithm.

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Crucial recent citations for ENSO

I have found the following research articles vital to formulating a basic model for ENSO.

The first citation finds the disturbance after 1980 leading to the identification of a phase reversal in the ENSO behavior. They apply Takens embedding theorem (which works for linear and non-linear systems such as Mathieu and Hill) to the time series, reconstructing current and future behavior from past behavior.

H. Astudillo, R. Abarca-del-Rio, and F. Borotto, “Long-term non-linear predictability of ENSO events over the 20th century,” arXiv preprint arXiv:1506.04066, 2015.

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ENSO Biennial Modulation Transform

Model uses only four factors: including Chandler wobble, an additional triaxial wobble, lunar nodal , and lunar anomalistic period. Yellow highlights regions that show some divergence. The region between 1980 and 1996 is phase inverted -- without that, the fit would degrade significantly

More to come.

Tidal Locking and ENSO

As I have been formulating a model for ENSO, I always try to relate it to a purely physical basis. The premise I have had from the beginning is that some external factor is driving the forcing of the equatorial Pacific thermocline. This forcing stimulus essentially causes a sloshing in the ocean volume due to small changes in the angular momentum of the rotating earth. I keep thinking that the origin is lunar as the success of the QBO model in relating lunisolar forcing to the oscillatory behavior of the QBO winds is enough motivation to keep on a lunar path.

Yet, I am finding that the detailed mechanism for ENSO differs from that of QBO. An interesting correlation I found is in the tidal-locking of the Earth to the moon. I think this is a subset of the more general case of spin-orbit resonance, where the rotation rate of a satellite is an integral ratio of the main body. In the case of the moon and the earth, it explains why the same moon face is always directed at the earth -- as they spin at the same rate during their mutual orbit, thus compensating via a kind of counter-rotation as shown in the left figure below.

Fig 1: Tidal Locking (left) results in the spin of the moon

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