# Using Solar Eclipses to calibrate the ENSO Model

This is the forcing for the ENSO model, focusing on the non-mixed Draconic and Anomalistic cycles:

Note that the maximum excursions (perigee and declination excursion) align with the occurrence of total solar eclipses. These are the first three that I looked at, which includes the latest August 21 eclipse in the center chart.

There are about 90 more of these stretching back to 1880. The best way to fit the calibration is to take the negative excursions of the two lunar forcings and multiply these together, i.e. use the effective Draconic*Anomalistic amplitudes (also only take the fortnightly cycle of the Draconic, as eclipses occur during both the ascending and descending node crossings). The main fitting factors are the phases of the two lunar months.  To get the maximum alignment from the search solver, we maximize the sum of the effective amplitudes across the entire interval. This results in a phase difference between the two of about 0.74 radians based at the starting year of 1880 (i.e. year 0).

# The QBO anomaly of 2016 revisited

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

Quite a few papers were written on the topic

1. 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.
2. Randel, W. J., and M. Park. "Anomalous QBO Behavior in 2016 Observed in Tropical Stratospheric Temperatures and Ozone." AGU Fall Meeting Abstracts. 2016.
3. Dunkerton, Timothy J. "The quasi‐biennial oscillation of 2015–2016: Hiccup or death spiral?." Geophysical Research Letters 43.19 (2016).
4. Tweedy, O., et al. "Analysis of Trace Gases Response on the Anomalous Change in the QBO in 2015-2016." AGU Fall Meeting Abstracts. 2016.
5. 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.

# 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.

# The Hawkmoth Effect

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, A2, D2, A2D, AD2, A3, D3, A2D2, A3D, AD3, A4, D4, A2D3, A3D2, A4D1, A1D4, A5, D5 ].

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:

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

Switching between two models

# Recipe for ENSO model in one tweet

and for QBO

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

# Should you trust climate science? Maybe the eclipse is a clue

An example of a prediction:

"Looks like we're heading for La Nina going into Winter. That means I expect 2018 will not average much different from 2017, both close to 2015 level. Then a probable new record in 2019."

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.

# ENSO model for predicting El Nino and La Nina events

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.

# Millennium Prize Problem: Navier-Stokes

Watched the hokey movie Gifted on a plane ride. Turns out that the Millennium Prize for mathematically solving the Navier-Stokes problem plays into the plot.

I am interested in variations of the Navier–Stokes equations that describe hydrodynamical flow on the surface of a sphere.  The premise is that such a formulation can be used to perhaps model ENSO and QBO.

The so-called primitive equations are the starting point, as these create constraints for the volume geometry (i.e. vertical motion much smaller than horizontal motion and fluid layer depth small compared to Earth's radius). From that, we go to Laplace's tidal equations, which are a linearization of the primitive equations.

I give a solution here, which was originally motivated by QBO.

Of course the equations are under-determined, so the only hope I had of solving them is to provide this simplifying assumption:

${\frac{\partial\zeta}{\partial\varphi} = \frac{\partial\zeta}{\partial t}\frac{\partial t}{\partial\varphi}}$

If you don't believe that this partial differential coupling of a latitudinal forcing to a tidal response occurs, then don't go further. But if you do, then:

# Solar Eclipse 2017 : What else?

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).

Fig 1: Training interval 1880-1950 leads to extrapolated fit post-1950

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.

Fig 2: Sensitivity to selection of lunar periods.

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.