[mathjax]Some of this will go into my presentation at the AGU meeting this December 12.
THE #AGU16 presentation:
Analytical Formulation of Equatorial Standing Wave Phenomena: Application to QBO and ENSOhttps://t.co/qVL2W8NI79 pic.twitter.com/FQQuERoh4g— Paul Pukite (@WHUT) November 10, 2016
[mathjax]Given the fact that very short training intervals will reveal the underlying fundamental frequencies of QBO, could we do the same for ENSO? Not nearly as short, but about 40 years is the interval required to uncover the fundamental frequencies. Again none of this is possible unless we make the assumption of a phase reversal for ENSO between the years 1980 and 1996.
Elaborating on this comment attached to an LOD post, noting this recent paper:
Shen, Wenbin, and Cunchao Peng. 2016. “Detection of Different-Time-Scale Signals in the Length of Day Variation Based on EEMD Analysis Technique.” Special Issue: Geodetic and Geophysical Observations and Applications and Implications 7 (3): 180–86. doi:10.1016/j.geog.2016.05.002.
Because of the law of conservation of momentum sloshing can change the velocity of a container full of liquid, momentarily speeding it up or slowing it down as the liquid sloshes back and forth. By the same token, suddenly slowing or speeding of that container can also cause the sloshing. So there is a chicken and egg quality to the analysis of sloshing, making it difficult to ascertain the origin of the effect.
If ENSO is a manifestation of a liquid sloshing in a container and if the length-of-day (LOD) is a measurement of the angular momentum changes of the Earth’s rotation, then it is perhaps useful to compare the fundamental time-varying signals in each measurement.
[mathjax]I recently posted a bog article called QBO Model Final Stretch. The idea with that post was to give an indication that the physics and analytical math model explaining the behavior of the QBO was in decent shape. I would like to do the same thing with the ENSO model but retain the context of the QBO model. Understanding the QBO was a boon to making progress with ENSO as it provided an excellent training ground for time-series analysis and also provided some insight into the underlying forcing functions. In the literature, there is a clear indication that ENSO and QBO are somehow related, but the causality chain remains unclear.
Glenn Brier is on my list of essential climate science researchers. I am sure he has since retired but his status as both a fellow of the American Meteorological Society and a fellow of the American Statistical Association indicated he knew his stuff. One of his research topics was exploring periodicities in climate data. He was eminently qualified for this kind of analysis as his statistics background provided him with the knowledge and skill in being able to distinguish between stochastic versus deterministic behaviors.
One paper Brier co-authored in 1989 is an obscure gem in terms of understanding the determinism of El Nino and ENSO [1]. He essentially analyzed a 463-year historical chronology of strong El Nino events and was able to find significant periodicities of 6.75 and 14 years in the data.
This compares very well with what I have been able to decipher as the strongest ENSO periodicities (taken from the instrumental record since 1880) of 6.5 and 14 years.
One of the most questioned aspects of the CSALT model of global temperature is the LOD to Temperature factor. This creates a multi-decadal variation in temperature useful for optimizing a multiple-linear regression AGW model dependent on CO2 and other factors.
Lunisolar tides impact variations in Length-of-Day (LOD). So does ENSO and QBO. There is a recursive aspect to these relationships as well, since both LOD and ENSO have the same Chandler wobble match in apparent forcing periodicity. This is what I believe generates a 6-year signal that gets identified routinely in the LOD time-series, such as the latest finding in ref [1] below.
From ref [1], the 6-year signal in LOD series. Counting cycles this is close to an average 6.25 year period, close to the 6.4 year Chandler wobble angular momentum variation.
Now we can add this paper by Marcus [2] to the mix. This is a very detailed look at the correlation between long-range LOD variations (longer than the 6-year variation of Ref [1]) and global surface temperature. His application of a 5-year running mean is essentially similar to a 5-year lag that works as a best fit in the CSALT model. Marcus stops short of assigning a source cause for the LOD-to-Temperature correlation, but the general idea is that angular momentum variations are the forcing terms that slosh the sources of heat to the surface — “via core-induced rotational and/or related global-scale processes”. (I also have to note that Marcus is an independent researcher, who at one time had an affiliation with NASA JPL.)
From Marcus [2], correlation of LOD against various temperature indices.
[1] Duan, Pengshuo, Genyou Liu, Lintao Liu, Xiaogang Hu, Xiaoguang Hao, Yong Huang, Zhimin Zhang, and Binbin Wang. 2015. “Recovery of the 6-Year Signal in Length of Day and Its Long-Term Decreasing Trend.” Earth, Planets and Space 67 (1): 1.
[2] Marcus, Steven L. 2015. “Does an Intrinsic Source Generate a Shared Low-Frequency Signature in Earth’s Climate and Rotation Rate?” Earth Interactions 20 (4): 1–14. doi:10.1175/EI-D-15-0014.1.
From a careful analysis of Length-of-Day (LOD) measurements, we can glean lots of information about phenomenon such as tides, earthquakes, core/mantle motion, and ENSO [1]. In Figure 1 below, one can see how this is manifested at different time scales.
Fig. 1: Differing temporal scales of LOD analysis. Short period tides have scale of diurnal and semidiurnal.
I revisited the LOD data to see if I could detect the angular momentum changes that appear to force the QBO and ENSO oscillations, similar to what Wang et al do in [2]. In each of these natural behaviors, there is a clear indication that the 2.369 year seasonally-aliased long period draconic tide is a primary forcing mechanism.
There are two sets of data to look at. The original set from Gross at JPL called LUNAR97 [3] is not online in a machine-readable format, but it is included in the paper. So it was a simple matter to OCR a GIF of the paper’s table. Another more recent set is available from IERS here.
Fig. 2: Compariscon between the two LOD data sets. The Gross set has less filtering
The issue with the more recent set is that the site claims that they do a “5-point quadratic convolute”, at least up to 1955. I don’t want any filtering, so chose the Gross LUNAR97 set. You can see the difference between the two data sets in Figure 2 above.
To draw out the fine features even more, I performed a straightforward second-derivative on the LUNAR97 time-series (same approach as used on the QBO see result). This was then fed into a symbolic regression machine learning tool to determine the principle components. The results are shown in Figure 3 below.
Fig. 3: Result of machine learning on 2nd-derivative of LOD data.
After about an hour of machine learning, the solution included a strong frequency at 22.481 rads/year. In Figure 3, that solution was one of the simplest along the Pareto front, shown by the red dot. This frequency is aliased to the slower frequency of 2.3694 rads/year, which is correspondingly seasonally aliased to an approximate lunar month of 27.2121 days. Incredibly, but perhaps not surprisingly, this compares to the known draconic or nodal lunar month length of 27.21222 days. The other somewhat weaker term is very close to aliasing the anomalistic lunar month of 27.55 days, in the fortnight mode of 13.777 days.
That was the first run I attempted and subsequent runs are picking out some other potential long-period lunar periods (and also the 6.45 year Chandler wobble), but this is really enough to demonstrate a salient result — that the same draconic period that the QBO model uses and that the ENSO model uses as the primary angular momentum forcing term also appears in the most direct and practical measurement of angular momentum known in Earth geophysics. And I do not think this is a instance of artificial sampling aliasing, since each of the LOD measurements was averaged over at least a month. It is likely a real seasonal aliasing behavior and serves to further substantiate the theorized mechanisms forcing the QBO and ENSO behavior. Read the short paper that I placed into arXiv to see how the other parts fall into place.
And a continuing lesson to be learned is to not filter data unless you can be sure that what you are filtering out is nuisance noise. In this case, it appears the filtering performed by IERS was too aggressive and removed a very important signal in the time series! Contrary to what the statistician Tamino has been preaching, these time-series do not have a lot of statistical content. I donated money to his lessons on time-series analysis, but largely disagree on the applicability of his approaches to earth data such as we see in QBO, ENSO, Chandler wobble, and now LOD. If you followed his instructions, you might not find any of the periodic cycles buried in the data, and you might dismiss it all as red noise. Yet, there is likely nothing statistical or complex about these behaviors, as they are largely driven by known deterministic forces. Stay tuned, as the next post will be titled “Needless Complexity”.
[1] B. Fong Chao, “Excitation of Earth’s polar motion by atmospheric angular momentum variations, 1980–1990,” Geophysical research letters, vol. 20, no. 3, pp. 253–256, 1993.
[2] Shen, Wen-Jun Wang Wen-Bin, and Han-Wei Zhang. “Verifications for Multiple Solutions of Triaxial Earth Rotation.” PDF
Note: the authors in [2] also find a 14-year signal in the LUNAR97 data (see figure below). This is likely a result of a wavelet filter that isolated periods between 8 and 16 years as they described the windowing procedure. This 14 year period is critical in the ENSO model.
From Wang et al, Figure 6 “Spectrum of wavelet for data series LUNAR97 with evident peak at period 14 yr “. The reciprocal of 0.0715 cycles/year is 14 years.
[3] Gross, Richard S. “A combined length-of-day series spanning 1832–1997: LUNAR97.” Physics of the Earth and Planetary Interiors 123.1 (2001): 65-76. PDF
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.
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.
I tend to write a more thorough analysis of research results, but this one is too interesting not to archive in real-time.
First, recall that the behavior of ENSO is a cyclostationary yet metastable standing-wave process, that is forced primarily by angular momentum changes. That describes essentially the physics of liquid sloshing. Setting input forcings to the periods corresponding to the known angular momentum changes from the Chandler wobble and the long-period lunisolar cycles, it appears trivial to capture the seeming quasi-periodic nature of ENSO effectively.
The key to this is identifying the strictly biennial yet metastable modulation that underlies the forcing. The biennial factor arises from the period doubling of the seasonal cycle, and since the biennial alignment (even versus odd years) is arbitrary, the process is by nature metastable (not ergodic in the strictest sense). By identifying where a biennial phase reversal occurs, the truly cyclostationary arguments can be isolated.
The results below demonstrate multiple regression training on 30 year intervals, applying only known factors of the Chandler and lunisolar forcing (no filtering applied to the ENSO data, an average of NINO3.4 and SOI indices). The 30-year interval slides across the 1880-2013 time series in 10-year steps, while the out-of-band fit maintains a significant amount of coherence with the data:
GeoEnergy Math 
