# ENSO model maps to LOD cycles

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.

# ENSO Proxy Revisited

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."

## References

[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.

# QBO Disruption?

"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.

# Compact QBO Derivation

I created a QBO page that is a concise derivation of the theory behind the oscillations:

http://contextEarth.com/compact-qbo-derivation/

Four key observations allow this derivation to work

1. Coriolis effect cancels at the equator and use a small angle (in latitude) approximation to capture any differential effect.
2. Identification of wind acceleration and not wind speed as the measure of QBO.
3. Associating a latitudinal displacement with a tidal elevation via a partial derivative expansion to eliminate an otherwise indeterminate parameter.
4. Applying a seasonal aliasing to the lunar tractive forces which ends up perfectly matching the observed QBO period.

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.

# ENSO Model Final Stretch (maybe)

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.

# Obscure paper on ENSO determinism

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.

# Geophysical Fluid Dynamics first, and then CFD

A recent perfunctorily-peer-reviewed paper in the Proceedings of the Royal Society

Vallis GK. 2016 "Geophysical fluid dynamics: whence, whither and why?" Proc. R. Soc. A 472: 20160140. http://dx.doi.org/10.1098/rspa.2016.0140 and PDF

explains the distinction between analytical physics models of the climate and purely numerical models — i.e. the field known as computational fluid dynamics (CFD). The lack of an intense review cycle makes for a very readable paper, with a refreshing conversational writing style that the editors apparently allowed. The gist of the piece is that the formulation of analytically-based Geophysical Fluid Dynamics (GFD) models of the atmosphere & ocean are essential to making sure that the CFD's are on the right track.

# QBO Model Final Stretch

One caveat in this derivation is that the QBO may actually be the $v(t)$ term - the horizontal longitudinal velocity of the fluid, the wind in other words - which can be derived from the above by applying the solution to Laplace's third tidal equation in simplified form above.

Following up on this gap, I extended the derivation to formally evaluate the velocity.

So from:

$\zeta(t) = \sin( \sqrt{A} \sum_{i=1}^{i=N} k_i \sin(\omega_i t) + \theta_0 )$

and the third simplified Laplace equation

$\frac {\partial v}{\partial t} =-\frac {1}{a} \frac {\partial }{\partial \varphi } (g\zeta +U)$

we can derive

$\frac {\partial v}{\partial t} = \cos( \sqrt{A} \sum_{i=1}^{i=N} k_i \sin(\omega_i t) + \theta_0 )$

So the acceleration of wind, not the velocity, is what obeys the Sturm-Liouville equation. A derivative preserves the periods of the Fourier components, but not the amplitudes, so what we see is a differently shaped envelope for QBO —  i.e. one that is more spiky due to the time derivative applied.

A single lunar Draconic tidal term of $\lambda_d=27.212$ days multiplied by a yearly modulation sharply peaked at a specific season is enough to capture the QBO acceleration time-series:

$\cos^N(2 \pi t) cos(\frac{2\pi}{\lambda_d} t)$

The seasonal sharpness is controlled by the power N. This can be approximated as a Fourier series with the same periods found previously: 2.37 years, 0.7 years, 0.41 years, etc.

Fig. 1: The sesonally modulated lunar cycle can be approximated by a sum of Fourier components

This gives the following fit to the QBO acceleration with a correlation coefficient of 0.35 (good, considering the data waveform has no filtering applied and very little optimization went into the fit):

Fig 2:  Modulation of Draconic lunar with seasonal period captures the acceleration of the QBO wind.

The excellent model fit makes sense as the acceleration of wind is simply a F=ma response to a gravitational forcing.

The bottom-line is that this acceleration version of the QBO is much more spiky than the more smoothly oscillating conventional wind speed description of QBO. Yet this spikiness is more closely representative of the seasonally modulated lunar forcing. So what was a caveat in the previous post is now a fundamental revelation which provides an excellent modeling path going forward.

Furthermore, a simple time integration of the acceleration model will recover the wind speed version of the QBO. And fortunately this will retain the same Fourier components that we used earlier to model the QBO, so a self-consistency with the prior model of the QBO is retained.

That's how a solution should ideally converge — unless one has all the pieces simultaneously, incremental changes to an initial model will provide the process to a solution.

Incidentally, I submitted this research to the AGU meeting for this fall and will find out if the abstract gets accepted in October.

8/28/2016: Additional information pulled from an Azimuth Project forum comment

Deriving the QBO from Laplace's tidal equations, it becomes apparent that the acceleration of wind and not the wind speed is the fundamental measure to characterize. With that premise, things seem to fall into place much more naturally.

First, multiply the nodal/draconic frequency by a sharply peaked yearly modulation and we get the following as a fit for QBO acceleration:

The fitting region is 1970 to 1982, and the rest is extrapolated. The correlation coefficient is not extremely high but note the amount of fine detail that gets exposed by the model.

To get back the wind speed QBO, the curves are integrated

Again the model fitting is only conducted on the interval from 1970 to 1982. Outside of that interval the model doesn't track every peak but enough of the fine detail is captured that it's obvious that the model has predictive power.

The supporting experiment involves running a symbolic regression machine learning trial on this same interval.

This image has been resized to fit in the page. Click to enlarge.

Amazingly, the ML finds all the same harmonics caused by multiplying the sharply peaked yearly signal (freq ~ 2π ) with the Draconic tide (aliased ~2.7 year). No other periods are detected.