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.

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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.

Follow @whut on Twitter:

ENSO Split Training for Cross-Validation

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.

Fig. 1: Training 1880 to 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.

Fig. 2: Training interval 1950 to 2016

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Deterministic and Stochastic Applied Physics

Pierre-Simon Laplace was one of the first mathematicians who took an interest in problems of probability and determinism.  It's surprising how much of the math and applied physics that Laplace developed gets used in day-to-day analysis. For example, while working on the ENSO and QBO analysis, I have invoked the following topics at some point:

  1. Laplace's tidal equations
  2. Laplace's equation
  3. Laplacian differential operator
  4. Laplace transform
  5. Difference equation
  6. Planetary and lunar orbital perturbations
  7. Probability methods and problems
    1. Inductive probability
    2. Bayesian analysis, e.g. the Sunrise problem
  8. Statistical methods and applications
    1. Central limit theorem
    2. Least squares
  9. Filling in holes of Newton's differential calculus
  10. Others here

Apparently he did so much and was so comprehensive that in some of his longer treatises he often didn't cite the work of others, making it difficult to pin down everything he was responsible for (evidently he did have character flaws).

In any case, I recall applying each of the above in working out some aspect of a problem. Missing was that Laplace didn't invent Fourier analysis but the Laplace transform is close in approach and utility.

When Laplace did all this research, he must have possessed insight into what constituted deterministic processes:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

— Pierre Simon Laplace,
A Philosophical Essay on Probabilities[wikipedia]
This is summed up as:

He also seemed to be a very applied mathematician, as per a quote I have used before  “Probability theory is nothing but common sense reduced to calculation.”  Really nothing the least bit esoteric about any of Laplace's math, as it seemed always motivated by solving some physics problem or scientific observation. It appears that he wanted to explain all these astronomic and tidal problems in as simple a form as possible. Back then it may have been esoteric, but not today as his techniques have become part of the essential engineering toolbox. I have to wonder if Laplace were alive now whether he would agree that geophysical processes such as ENSO and QBO were equally as deterministic as the sun rising every morning or of the steady cyclic nature of the planetary and lunar orbits. And it wasn't as if Laplace possessed confirmation bias that behaviors were immediately deterministic; as otherwise he wouldn't have spent so much effort in devising the rules of probability and statistics that are still in use today, such as the central limit theorem and least squares.

Perhaps he would have glanced at the ENSO problem for a few moments, noticed that in no way that it was random, and then casually remarked with one his frequent idiomatic phrases:

"Il est aisé à voir que..."  ... or ..  ("It is easy to see that...").

It may have been so obvious that it wasn't important to give the details at the moment, only to fill in the chain of reasoning later.  Much like the contextEarth model for QBO, deriving from Laplace's tidal equations.

Where are the Laplace's of today that are willing to push the basic math and physics of climate variability as far as it will take them? It has seemingly jumped from Laplace to Lorenz and then to chaotic uncertainty ala Tsonis or mystifying complexity ala Lindzen. Probably can do much better than to punt like that ... on first down even !

Confirmation Bias

Someone long ago must have stated that the El Nino/Southern Oscillation (ENSO) phenomenon was not related to lunisolar (lunar+solar) tidal forcing. This negative result (or null result) is not documented anywhere (AFAICT) but is likely considered conventional wisdom by climate scientists. The most direct evidence that climate scientists don't consider lunisolar forcing is that it appears nowhere in the parameterization of general circulation model (GCM) source code.

As a general rule, negative findings are rarely reported in research journals:

"As it stands now, researchers are typically rewarded (tenure, grants, better jobs, etc.) for publishing a quantity of publications in prestigious journals. They do this by

  • Running small and statistically weak studies (they are easy to do) that produce only positive results, since journals tend to not publish negative findings.
  • Ignoring negative findings.
  • Publishing only new and exciting findings that journals are looking for.
  • Never checking old findings for accuracy and replicability.
  • Changing methodologies in mid-stream to assure positive results."

I imagine that if a budding graduate student devised a hypothetical ENSO/lunar tidal connection as a potential thesis topic, it would be rejected by his advisor. The advisor would not want to risk his reputation or track record by going down a potential dead end. The same is perhaps true of the recent case of NASA JPL rejecting the proposal of one of their research teams who suggested funding for this actual topic.  Read an excerpt from this footnote:

"None of the peer-reviewers nor collaborators in 2006 had anticipated that the most remarkable large-scale process that we were going to find comes from ocean circulations fueled by Luni-Geo-Solar gravitational energy. We found evidence of the existence of this energy in the data produced by satellites like QuikSCAT and ASCAT. Following the standard
from the 1970’s of using these satellite data as winds in numerical modeling of oceans and climate has created and continues to create significant errors in the simulated ocean temperature, salinity, and currents as well as in the atmosphere. Together with our co-workers, we chose not to publish the errors until a solution to appropriately use

satellite data in numerical modeling was found. However, over the following years, proposed solutions were not considered because of various factors including economic and scientific pressure to publish and continue the standard agenda."


This is a clear example of confirmation bias stalling promising research. Yet, apparently there are no issues with pushing iffy models of ENSO based on nebulous chaos theory by climate change deniers such as Anastasios Tsonis.
Hmmm ... something is not right with this picture.

So if this lunisolar model of ENSO pans out, it is an excellent example of how confirmation bias impeded scientific progress, but with the scientific method eventually winning out.

Fig 1: Top is the ENSO model trained on NINO3.4 data from 1880 to 1920, with the lower curve providing a calibration of the lunar forcing based on sensitive LOD measurements of the earth's rotation. ENSO is not a chaotic process if it can be stimulated directly by the known lunisolar forcing.

And we can do the same confirmation bias exercise with the quasibiennial oscillation (QBO) phenomenon, substituting the climate change denier Richard Lindzen for Tsonis as the impediment to progress.  Lindzen couldn't find the lunar connection (even though there is plenty of evidence he tried), so just assumed it wasn't there.  Everyone that followed Lindzen's original model essentially confirmed his bias and so no progress was made, until the bias was removed and the lunisolar forcing re-evaluated.



The difference here is that I am not preparing a thesis or working for NASA. This is one way of inoculating oneself from historical confirmation biases -- by not being part of an inside consensus, there is no one to suggest to "not go there".  By the same token, I now possess an apparent confirmation bias that a lunisolar forcing plays a primary role in certain climate phenomena.  Yet, it's a weak confirmation bias because I didn't start with this view, but it gathered steam based on all the evidence accrued over the past few years. It is now up to others to use the scientific method to reject this model. And, of course, I will be the first to abandon this model if I come across strong evidence to reject it. After all, I don't have any particular allegiance to the moon gods, only in the learned view that oscillations of this nature do not occur via spontaneous resonance.

As an important footnote to this post, consider the recent admission that lunar forces play a significant role in triggering earthquakes. Up to the last year, the confirmation bias was that the lunar gravitational forcing was too weak to trigger earthquakes, and so the onset was historically described in statistical terms. The earthquake itself triggered by the passage and time and the slow creep of a fault. But the tide turned in 2016 when two independent groups found significant correlations with lunar cycles -- a Japanese group led by Ide [1] and a US Geological Survey group led by van der Elst [2]. These are the same fortnightly lunar cycles (see Figure 2 below) that are used in the ENSO model described above (compare to lower chart in Figure 1).  So the new thinking is that indeed the gravitational pull of the moon will trigger the slipping of a fault, and this happens enough that future predictions of earthquakes (for example along the San Andreas fault [3]) can use tidal tables to aid the analysis.

Fig 2: Lunar forced earthquake analysis by van der Elst et al [2]. Note the fortnightly cycles similar to Fig 1 above.

The bottom-line is that we need to monitor the earth sciences consensus regarding lunar forcing in the next few years, both in terms of ENSO and QBO climate behavior and with regard to earthquake analysis.   Scientific theories are not binding, unlike sporting events -- "World cup matches cannot be replayed, but science can be corrected afterwards.". Thus, the confirmation bias of "no lunar forcing" is not necessarily set in stone.

References

  1. Ide, Satoshi, Suguru Yabe, and Yoshiyuki Tanaka. "Earthquake potential revealed by tidal influence on earthquake size-frequency statistics." Nature Geoscience 9.11 (2016): 834-837.
  2. van der Elst, Nicholas J., et al. "Fortnightly modulation of San Andreas tremor and low-frequency earthquakes." Proceedings of the National Academy of Sciences (2016): 201524316.
  3. Delorey, Andrew A., Nicholas J. van der Elst, and Paul A. Johnson. "Tidal triggering of earthquakes suggests poroelastic behavior on the San Andreas Fault." Earth and Planetary Science Letters 460 (2017): 164-170.

ENSO+QBO Elevator Pitch

Most papers on climate science take pages and pages of exposition before they try to make any kind of point. The excessive verbiage exists to rationalize their limited understanding of the physics, typically by explaining how complex it all is.

Conversely, think how easy it is to explain sunrise and sunset. From a deterministic point of view [1] and from our understanding of a rotating earth and an illuminating sun, it's trivial to explain that a sunrise and sunset will happen once each per day. That and perhaps another sentence would be all that would be necessary to write a research paper on the topic ...  if it wasn't already common knowledge. Any padding to this would be unnecessary to the basic understanding. For example, going further and explaining why the earth rotates amounts to answering the wrong question. Thus the topic is essentially an elevator pitch.

If sunset/sunrise is too elementary an example, one could explain ocean tides. This is a bit more advanced because the causal connection is not visible to the eye. Yet all that is needed here is to explain the pull of gravity and the orbital rate of the moon with respect to the earth, and the earth to the sun. A precise correlation between the lunisolar cycles is then applied to verify causality. One could add another paragraph to explain how mixed tidal effects occur, but that should be enough for an expository paper.

We could also be at such a point in our understanding with respect to ENSO and QBO. Most of the past exposition was lengthy because the causal factors could not be easily isolated or were rationalized as random or chaotic. Yet, if we take as a premise that the behavior was governed by the same orbital factors as what governs the ocean tides, we can make quick work of an explanation.

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