[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
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!
[mathjax]An old game show called Name That Tune asked contestants to identify a song by listening to just a few notes.
This post is the QBO equivalent to that. How short an interval can we train on to reproduce the rest of the time series? The answer is not much is required.
With respect to the ENSO model I have been thinking about ways of evaluating the statistical significance of the fit to the data. If we train on one 70 year interval and then test on the following 70 year interval, we get the interesting effect of finding a higher correlation coefficient on the test interval. The training interval is just below 0.85 while the test is above 0.86.
Although the historical coral proxy measurements are not high resolution (1 year resolution available), they can provide substantiation for models of the modern day instrumental record. This post is a revisit of a previous analysis of the Universal ENSO Proxy (UEP).
The interval from 1881 to 1950 of the ENSO data was used to train the DiffEq ENSO model. This gives a higher correlation coefficient (~0.85) on the test interval (from 1950 to 2014) than the training interval (1881-1950) as shown below:
Fig. 1: ENSO model fit over the modern instrumental record
Since ENSO data shows stationarity and coherence over the interval of 70 years, this fit was re-applied to the UEP data over 4 different ranges 1650-1720, 1720-1790, 1790-1860, and 1860-1930. High correlation coefficients were found for each of these intervals (> 0.70) and compared against fits to a red noise model shown below:
Fig 2: Proxy fits over various ranges as compared to a red noise model (added for clarification: essentially a synthetic data set to evaluate the ENSO model against). The significance is at least at 0.95 for each proxy interval.
Each of the ENSO fits lies within the 0.95 significance level, and only 1 out of 500 red noise simulations obtained a 0.8 correlation coefficient, which is what the 1720-1790 interval achieved.
| Interval | CC |
|---|---|
| 1650-1720 | 0.772 |
| 1720-1790 | 0.807 |
| 1790-1860 | 0.710 |
| 1860-1930 | 0.763 |
The significance of having each of these intervals at least 0.95 is 1-(1-0.95)^4 if these are all independent. That is a small number less than about (1/20)^4 in likelihood.
The caveat is that the ENSO is also not likely to be coherent over intervals much greater than 70 years, as the shift around 1980 in phase for the model demonstrates.
This substatntiates the finding of Hanson, Brier, Maul [1] that ENSO and El Nino frequencies may be relatively constant back to the year 1525.
Astudillo et al [2] also confirmed the ENSO shift after 1980 stating that “the amplitude of the 1982-1983 event is unique”. Further they concisely describe the ENSO behavior as being highly deterministic by stating:
“This is of crucial importance since if a system is deterministic, the vector field at every period of the state space is uniquely defined by a set of ordinary differential equations.”
[1] K. Hanson, G. W. Brier, and G. A. Maul, “Evidence of significant nonrandom behavior in the recurrence of strong El Niño between 1525 and 1988,” Geophysical Research Letters, vol. 16, no. 10, pp. 1181–1184, 1989.
[2] H. Astudillo, R. Abarca-del-Río, and F. Borotto, “Long-term potential nonlinear predictability of El Niño–La Niña events,” Climate Dynamics, pp. 1–11, 2016.
“One of the earth’s most regular climate cycles is disrupted” issued recently by the UK Met office
Plus all these recent papers:
Newman, P. A., L. Coy, S. Pawson, and L. R. Lait (2016), The anomalous change in the QBO in 2015–2016, Geophys. Res. Lett., 43, 8791–8797, doi:10.1002/2016GL070373.
Dunkerton, T. J. (2016), The Quasi-Biennial Oscillation of 2015-16: Hiccup or Death Spiral?, Geophys. Res. Lett., 43, doi:10.1002/2016GL070921.
An unexpected disruption of the atmospheric quasi-biennial oscillation, Scott M. Osprey, Neal Butchart, Jeff R. Knight, Adam A. Scaife, Kevin Hamilton, James A. Anstey, Verena Schenzinger, Chunxi Zhang, Science, 08 Sep 2016, DOI: 10.1126/science.aah4156
Do strong warm ENSO events control the phase of the stratospheric QBO?, Geophysical Research Letters, Sep 2016, Bo Christiansen, Shuting Yang, Marianne S. Madsen, DOI: 10.1002/2016GL070751
At RealClimate.org someone named Nemesis asked:
What might the implications of a disrupted QBO be? Any idea?
Concerning the QBO disruption that Nemesis mentions above. Does anybody really understand the QBO to begin with? The original theory was developed by the contrarian Richard Lindzen and it really is a limited model if you dig into it. For example, he never could derive the rather obvious period (28 months) of the QBO.
Years ago (circa 1998) the observation was about the “continuing difficulties in obtaining a realistic QBO” yet you continue to find references to “obtaining realistic QBOs”
Get a solid theory for QBO in place and only then can you start to reason about anomalies and disruptions that occur. IMO shouldn’t make assertions regarding the source of the latest disruption unless we can agree on the nominal QBO behavior.
I created a QBO page that is a concise derivation of the theory behind the oscillations:
http://GeoEnergyMath.com/compact-qbo-derivation/
Four key observations allow this derivation to work
These are obscure premises but all are necessary to derive the equations and match to the observations.
This model should have been derived long ago …. that’s what has me stumped. Years ago I spent hours working on transport equations for semiconductor devices so have a good feel for how to handle these kinds of DiffEq’s. You literally had to do this otherwise you would never develop the intuition on how a transistor or some other device works. The QBO for some reason reminds me quite a bit of solving the Hall effect. And of course it has geometric resemblance to the physics behind an electric motor or generator. Maybe I am just using a different lens in solving these kinds of problems.
[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.
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.
[mathjax]I have been playing the models of ENSO and QBO off of each other. So for whatever insights I glean from one model, I have tried to apply it to the other. My last ENSO analysis involved simplifying Laplace’s Tidal Equations and finding similarity to the Mathieu-modulation used to describe sloshing of water volumes. However, I had left one equation in a rough state, and stated that I would get back to it. That’s what I will do in this post, and I can flatly assert that the simplification is an eye-opener for understanding the behavior of QBO! (Which shouldn’t be surprising since Richard Lindzen tried vainly to apply Laplace’s equations to QBO, with the outcome of creating a monster of complexity, see [1])
Continue reading