Search for El Nino

The model for ENSO includes a nonlinear search feature that finds the best-fit tidal forcing parameters.  This is similar to what a conventional ocean tidal analysis program performs — finding the best-fitting lunar tidal parameters based on a measured historic interval of hundreds of cycles. Since tidal cycles are abundant — occurring at least once per day — it doesn't take much data collected over a course of time to do an analysis.  In contrast, the ENSO model cycles over the course of years, so we have to use as much data as we can, yet still allow test intervals.

What follows is the recipe (more involved than the short recipe) that will guarantee a deterministic best-fit from a clean slate each time. Very little initial condition information is needed to start with, so that the final result can be confidently recovered each time, independent of training interval.

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Variation in the Length of the Anomalistic Month

For the ENSO model, we use two constraints for the fitting process. One of the constraints is to maximize the correlation coefficient for the model over the ENSO training interval selected. The other constraint is to maximize the correlation of the selected lunar tidal forces over a measured Length-of-Day (LOD) interval. The latter constrains the lunar tidal forcing to known values that will actually change the angular momentum of the earth's rotation. This in turn drives the sloshing of the Pacific ocean's thermocline leading to the ENSO cycle. The two constraints are simultaneously met by heuristically maximizing the average of the correlation coefficients.

In addition, there are the fixed constraints of the primary lunar periods corresponding to the Draconic/nodal cycle and the Anomalistic cycle.

This combination gives a fairly effective fit over the entire training cycle, but there is an important additional constraint that needs to be applied to the Anomalistic cycle. The NASA eclispse and moon's orbit page describes the situation :

"The anomalistic month is defined as the revolution of the Moon around its elliptical orbit as measured from perigee to perigee. The length of this period can vary by several days from its mean value of 27.55455 days (27d 13h 18m 33s). Figure 4-4 plots the difference of the anomalistic month from the mean value for the 3-year interval 2008 through 2010. ... the eccentricity reaches a maximum when the major axis of the lunar orbit is pointed directly towards or directly away from the Sun (angles of 0° and 180°, respectively). This occurs at a mean interval of 205.9 days, which is somewhat longer than half a year because of the eastward shift of the major axis. "

This is a significant variation in the anomalistic cycle over the course of a year. We don't use this variation as a constraint but we can use it as a fitting parameter and then compare the variation obtained over that shown above.

Using the 205.9 day value ~365/(2-2/8.85), we break this into Fourier components of half this value and twice this value. The mean Anomalistic period of 27.5545 days is then frequency modulated by the slower periods by the standard engineering procedure. We then allow the amplitude and phase of each factor to vary during the training to obtain the best fit (this is slightly different than the concise form used previously).

If we zoom in on the anomalistic period variation, we get this match to the NASA Goddard model:

There is no reason to believe that this match would spontaneously occur given that there are 3 amplitudes and 3 phase factors involved. Yet it matches precisely to the (1) peak positions, (2) relative amplitudes, and to the (3) cusped shape via the Fourier series summation. An even better fit is obtained if we use abs(sin(π time/205.9+Φ)) as the fitting function as it naturally creates more of a cusp shape due to the full-wave rectification of the sine wave.

Conventional tidal analysis is renowned for being an exacting procedure [1], where the known tidal periods are broken down into equivalently similar sets of harmonic factors, yet applied on a diurnal or semidurnal basis. The only difference here is that ENSO responds to the monthly and fortnightly long-period tides and not the short-period ones.


→ This model fit gives further validation to the lunar tidal mechanism for forcing ENSO.  The exacting process of generating the correct lunar tidal variations (along with the subtle biennial modulation and the tricky aliasing) have likely contributed to the fact that the pattern has remained hidden for so long.  This is actually not so different a situation as the long hidden connection between triggering of earthquakes and the dynamic  moon-sun-earth alignment. That pattern is also hidden, only exposed recently. Alas, not everything can be quite as obvious as the pattern matching of ocean surface tides to the lunisolar cycles.

References

[1] S. Consoli, D. R. Recupero, and V. Zavarella, “A survey on tidal analysis and forecasting methods for Tsunami detection,” arXiv preprint arXiv:1403.0135, 2014.

UPDATE

The reason for the peculiar shape of the Anomalistic frequency variations is due to a different slope (i.e. velocity) on one lobe of the elliptical orbit than on the other. You can get this by generating another sinusoidal modulation on top of the average elliptical sinusoid. This generates an asymmetric sawtooth in the phase angle (see blue line below) and the characteristic spiked or cusped profile in the effective frequency or derivative of this value (see red line below).
I am starting to use this formulation in the ENSO model as it is quite concise.

Annual and Biennial Modulations

Here are a couple of items I had in comments in this blog or recovered from elsewhere.


(1) There is an interesting highly nonlinear spiked peak around November in chlorophyl phytoplankton at higher latitudes, which matches the position of the forcing modulation used in the ENSO model (-0.123 years).
chlorophyl

Phenological Responses to ENSO in the Global Oceans
Surveys in Geophysics, 2017, Volume 38, Number 1, Page 277
M.-F. Racault, S. Sathyendranath, N. Menon

Is this a response or a nodal forcing mechanism?

(2) In another paper, scientists analyzing historical ENSO records used a CGM to support some of their observations. What they do not adequately explain is the strength of the biennial period in their simulation results.

TRACE

Evolution and forcing mechanisms of El Niño over the past 21,000 years
Zhengyu Liu, Zhengyao Lu, Xinyu Wen, B. L. Otto-Bliesner, A. Timmermann, K. M. Cobb, Nature 515, 550–553 (27 November 2014) doi:10.1038/nature13963

Intro section

“To understand ENSO’s evolution during the past 21 kyr, we analyse the baseline transient simulation (TRACE) conducted with the Community Climate System model version 3 (CCSM3). This simulation uses the complete set of realistic climate forcings: orbital, greenhouse gases, continental ice sheets and meltwater discharge (Fig. 1a, d and Methods). TRACE has been shown to replicate many key features of the global climate evolution”

Methods section

“Model ENSO.ENSO simulated by the model for the present day shows many realistic features, although the ENSO period tends to be biased towards quasi-biannual, as opposed to a broader 2–7-year peak in the observation38. The ENSO mode resembles the SST mode29 and propagates westwards as in many CGCMs. In the past 21 kyr, the preferred period of model ENSO remains at quasi-biannual, with the power spectrum changing only modestly with time (Extended Data Fig. 1).”

This is interesting in that they mischaracterize the periodicity as biannual, which also means semiannual, or twice a year, but that it is clearly biennial in the chart, which is defined as once every two years. That could be just a typo not caught during proof-reading. Yet the peak is sharply centered around a two-year fundamental period, which is the interesting aspect.

I replot that curve below to show the symmetry around the 2-year period. Drawing a Lorentzian curve around that frequency and linearizing the axis makes it symmetric.

symm

A Lorentzian or Cauchy often results as the frequency response of a driven damped harmonic oscillator. If that is the case, what the simulation shows may be the result of forcing comprised of different stimuli frequencies collected over time and the accompanying frequency response (e.g. a Bode plot). Yet this implies that the characteristic frequency or eigenvalue would need to be 2 years. But why would a characteristic frequency just happen to align with a period so close to 2 years? The 2 year period is likely not an eigenvalue of the properties of a damped harmonic oscillator, e.g. a spring constant and a damper, but more than likely connected to a period doubling based on the annual cycle. This is what Faraday and Rayleigh observed and then explained in the earliest liquid sloshing experiments [1]

Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe,Springer Science & Business Media, 2006

This is a spectrum of the biennial modulated ENSO data of the last 130 years. Pairs show up +/- around the 2 year central period. These are all related to aliased lunar cycles, when associated with a sharp biennial modulation.  This is a more detailed view showing the phase response as well.

folded

The strongest factors above are shown as pluses on the historical model below, these are not to scale in the vertical but positioned along the horizontal to highlight the symmetry:

spikes

So the actual situation is that the measured ENSO power spectrum over the modern instrumental record is not a smooth Lorentizian but a discrete set of frequencies that are balanced +/- around the central biennial frequency. A question is whether these discrete frequencies are fixed over time. Based on what I found with ENSO proxy data it appears that it may indeed be fixed, at least for spans of 100's of years -- http://contextearth.com/2017/05/12/enso-proxy-validation/

References

[1] Rayleigh, Lord. "XXXIII. On maintained vibrations." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 15.94 (1883): 229-235.