Lambert, Sébastien B., Steven L. Marcus, and Olivier de Viron. "Atmospheric Torques and Earth’s Rotation: What Drove the Millisecond-Level Length-of-Day Response to the 2015-16 El Niño?."

**RED**:

1

Lambert, Sébastien B., Steven L. Marcus, and Olivier de Viron. "Atmospheric Torques and Earth’s Rotation: What Drove the Millisecond-Level Length-of-Day Response to the 2015-16 El Niño?."

This figure contains the last three large El Ninos in dashed **RED**:

Remember the concern over the QBO anomaly/disruption during 2016?

Quite a few papers were written on the topic

- Newman, P. A., et al. "The anomalous change in the QBO in 2015–2016."
*Geophysical Research Letters*43.16 (2016): 8791-8797.Newman, P. A., et al. "The Anomalous Change in the QBO in 2015-16."*AGU Fall Meeting Abstracts*. 2016. -
Randel, W. J., and M. Park. "Anomalous QBO Behavior in 2016 Observed in Tropical Stratospheric Temperatures and Ozone."
*AGU Fall Meeting Abstracts*. 2016. -
Dunkerton, Timothy J. "The quasi‐biennial oscillation of 2015–2016: Hiccup or death spiral?."
*Geophysical Research Letters*43.19 (2016). -
Tweedy, O., et al. "Analysis of Trace Gases Response on the Anomalous Change in the QBO in 2015-2016."
*AGU Fall Meeting Abstracts*. 2016. -
Osprey, Scott M., et al. "An unexpected disruption of the atmospheric quasi-biennial oscillation."
*Science*353.6306 (2016): 1424-1427.

According to the lunar forcing model of QBO, which was also presented at AGU last year, the peak in acceleration should have occurred at the time pointed to by the **BLACK** downward arrow in the figure below. This was in April of this year. The **GREEN **is the QBO 30 hPa *acceleration* data and the **RED **is the QBO model.

Note that the training region for the model is highlighted in YELLOW and is in the interval from 1978 to 1990. This was well in the past, yet it was able to pinpoint the sharp peak 27 years later.

The disruption in 2015-2016 shown with shaded black may have been a temporary forcing stimulus. You can see that it obviously flipped the polarity with respect to the model. This will provoke a transient response in the DiffEq solution, which will then eventually die off.

The bottom-line is that the climate scientists who pointed out the anomaly were correct in that it was indeed a disruption, but this wasn't necessarily because they understood why it occurred — but only that it didn't fit a past pattern. It was good observational science, and so the papers were appropriate for publishing. However, if you look at the QBO model against the data, you will see many similar temporary disruptions in the historical record. So it was definitely not some cataclysmic event as some had suggested. I think most scientists took a less hysterical view and simply pointed out the reversal in stratospheric winds was unusual.

I like to use this next figure as an example of how this may occur (found in the comment from last year). A local hurricane will temporarily impact the tidal displacement via a sea swell. You can see that in the middle of the trace below. On both sides of this spike, the tidal model is still in phase and so the stimulus is indeed transient while the underlying forcing remains invariant. For QBO, instead of a hurricane, the disruption could be caused by a SSW event. It also could be an unaccounted-for lunar forcing pulse not captured in the model. That's probably worth more research.

As the QBO is still on a 28 month alignment, that means that the external stimulus — as with ENSO, likely the lunar tidal force — is providing the boundary condition synchronization.

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.

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.

[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.

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.

Contrasting to the well-known Butterfly Effect, there is another scientific modeling limitation known as the Hawkmoth Effect. Instead of simulation results being sensitive to initial conditions, which is the Butterfly Effect, the Hawkmoth Effect is sensitive to model structure. It's a more subtle argument for explaining why climate behavioral modeling is difficult to get right, and named after the hawkmoth because hawkmoths are "better camouflaged and less photogenic than butterflies".

Not everyone agrees that this is a real effect, or it just reveals shortcomings in correctly being able to model the behavior under study. So, if you have the wrong model or wrong parameters for the model, of course it may diverge from the data rather sharply.

In the context of the ENSO model, we already provided parameters for two orthogonal intervals of the data. Since there is some noise in the ENSO data — perfectly illustrated by the fact that SOI and NINO34 only have a correlation coefficient of 0.79 — it is difficult to determine how much of the parameter differences are due to over-fitting of that noise.

In the figure below, the middle panel shows the difference between the SOI and NINO34 data, with yellow showing where the main discrepancies or uncertainties in the true ENSO value lie. Above and below are the model fits for the earlier (1880-1950 shaded in a yellow background) and later (1950-2016) training intervals. In certain cases, a poorer model fit may be able to be ascribed to uncertainty in the ENSO measurement, such as near ~1909., ~1932, and ~1948, where the dotted red lines align with trained and/or tested model regions. The question mark at 1985 is a curiosity, as the SOI remains neutral, while the model fits to more La Nina conditions of NINO34.

There is certainly nothing related to the Butterfly Effect in any of this, since the ENSO model is not forced by initial conditions, but by the guiding influence of the lunisolar cycles. So we are left to determine how much of the slight divergence we see is due to non-stationary variation of the model parameters over time, or whether it is due to missing some other vital structural model parameters. In other words, the Hawkmoth Effect is our only concern.

In the model shown below, we employ significant over-fitting of the model parameters. The ENSO model only has two forcing parameters — the Draconic (D) and Anomalistic (A) lunar periods, but like in conventional ocean tidal analysis, to make accurate predictions many more of the nonlinear harmonics need to be considered [see Footnote 1]. So we start with A and D, and then create all combinations up to order 5, resulting in the set [ A, D, AD, A^{2}, D^{2}, A^{2}D, AD^{2}, A^{3}, D^{3}, A^{2}D^{2}, A^{3}D, AD^{3}, A^{4}, D^{4}, A^{2}D^{3}, A^{3}D^{2}, A^{4}D^{1}, A^{1}D^{4}, A^{5}, D^{5 }].

This looks like it has the potential for all the negative consequence of massive over-fitting, such as fast divergence in amplitude outside the training interval, yet the results don't show this at all. Harmonics in general will not cause a divergence, because they remain in phase with the fundamental frequencies both inside and outside the training interval. Besides that, the higher order harmonics start having a diminished impact, so this set is apparently about right to create an excellent correlation outside the training interval. The two other important constraints in the fit, are (1) the characteristic frequency modulation of the anomalistic period due to the synodic period (shown in the middle left inset) and (2) the calibrated lunar forcing based on LOD measurements (shown in the lower panel).

The resulting correlation of model to data is 0.75 inside the training interval (1880-1980) and 0.69 in the test interval (1980-2016). So this gets close to the best agreement we can expect given that SOI and NINO34 only reaches 0.79. Read this post for the structural model parameter variations for a reduced harmonic set to order 3 only.

Welcome to the stage of ENSO analysis where getting the rest of the details correct will provide only marginal benefits; yet these are still important, since as with tidal analysis and eclipse models, the details are important for fine-tuning predictions.

**Footnote:**

- For conventional tidal analysis, hundreds of resulting terms are the norm, so that commercial tidal prediction programs allow an unlimited number of components.

Full recipe for #ENSO. Seasonal phase lock. Biennial modulation. 1-year delay differential. Two lunar periods, Draconic(nodal) & Anomalistic

— Paul Pukite (@WHUT) August 23, 2017

and for QBO

Full recipe for #QBO. Take acceleration of data, via first derivative of velocity. Biannual phase lock. Draconic (nodal) lunar forcing.

— Paul Pukite (@WHUT) August 23, 2017

The common feature of the two is the application of Laplace's tidal equation and its closed-form solution.

An example of a prediction:

How does anyone know which way the ENSO behavior is heading if there is not a clear understanding of the underlying mechanism? [1]

For the prediction quoted above, the closer one gets to an peak or valley, the safer it is to make a dead reckoning guess. For example, I can say a low tide is coming if it is coming off a high tide — even if I have no idea what causes tides.

Yet, if we understand the mechanism behind ocean tides — that it is due to the gravitational pull of the sun and the moon — we can do a much better job of prediction.

The New York Times climate change reporter Justin Gillis suggests that climate science can make predictions as well as geophysicists can predict eclipses:

https://www.nytimes.com/2017/08/18/climate/should-you-trust-climate-science-maybe-the-eclipse-is-a-clue.html. And there is this:

Yet, if climate scientists can't figure out the mechanism behind a behavior such as ENSO, everyone is essentially in the same boat, fishing for a basic understanding.

So what happens if we can formulate the messy ENSO behavior into a basic geophysics problem, something on the complexity of tides? We are nowhere near that according to the current research literature, unless this finding — which has been a frequent topic here — turns out to be true.

In this case, the recent solar eclipse is in fact a clue. The precise orbit of the moon is vital to determining the cycles of ENSO. If this assertion is true, one day we will likely be able to predict when the next El Nino occurs, with the accuracy of predicting the next eclipse.

**Footnote:**

[1] Consider one common explanation invoking winds. In fact, shifts in the prevailing winds is not a mechanism because any shift or reversal requires a mechanism itself, see for example the QBO.

—Can capture vast majority of #ElNino and #LaNina events post-1950 by training only on pre-1950 data, with 1 lunar calibrating interval 🌛 pic.twitter.com/10PAirJAYD

— Paul Pukite (@WHUT) August 17, 2017

Applying the ENSO model to predict El Nino and La Nina events is automatic. There are no adjustable parameters apart from the calibrated tidal forcing amplitudes and phases used in the process of fitting over the training interval. Therefore the cross-validated interval from 1950 to present is untainted during the fitting process and so can be used as a completely independent and unbiased test.

The reason we can so accurately predict the solar eclipse of 2017 is because we have accurate knowledge of the moon's orbit around the earth and the earth's orbit around the sun.

Likewise, the reason that we could potentially understand the behavior of the El Nino Southern Oscillation (ENSO) is that we have knowledge of these same orbits. As we have shown and will report at this year's American Geophysical Union (AGU) meeting, the cyclic gravitational pull of the moon (lower panel in** Figure 1** below) interacting seasonally precisely controls the ENSO cycles (upper panel **Figure 1**).

**Figure 2** is how sensitive the fit is to the precise value of the lunar cycle periods. Compare the best ft values to the known lunar values here. This is an example of the science of metrology.

The implications of this research are far-ranging. Like knowing when a solar eclipse occurs helps engineers and scientists prepare power utilities and controlled climate experiments for the event, the same considerations apply to ENSO. Every future El Nino-induced heat-wave or monsoon could conceivably be predicted in advance, giving nations and organizations time to prepare for accompanying droughts, flooding, and temperature extremes.

Follow @whut on Twitter:

Because lunar & solar cycles so accurately known, we can predict #SolarEclipse2017 precisely. Same for #ENSO #ElNino https://t.co/M8xJ3DwOso

— Paul Pukite (@WHUT) August 13, 2017

If we split the modern ENSO data into two training intervals — one from 1880 to 1950 and one from 1950 to 2016, we get roughly equal-length time series for model evaluation.

As **Figure 1** shows, a forcing stimulus due to monthly-range LOD variations calibrated to the interval between 2000 to 2003 (lower panel) is used to train the ENSO model in the interval from 1880 to 1950. The extrapolated model fit in **RED **does a good job in capturing the ENSO data in the period beyond 1950.

Next, we reverse the training and verification fit, using the period from 1950 to 2016 as the training interval and then back extrapolating. **Figure 2** shows this works about as well.