Mathematical GeoEnergy

Book will be out next year published by Wiley

This blog will be ramped up for the book, but contains all the research leading up to the book.

Two papers at AGU 2017:



Dynamic Context Server

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.


  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.

Lunisolar Forcing of Earthquakes

I have previously cited two research groups from last year that linked lunar cycles to the triggering of earthquakes.

  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.

Yet, there is another research paper that is mind-boggling in terms of establishing correlation that predated this work by 5 years:

Kolvankar, Vinayak G. “Sun, Moon and Earthquakes.” New Concepts in Global Tectonics Newsletter 60 (2011): 50-66.  PDF

This paper is an obscure newsletter with no citations according to Google Scholar, but makes strong claims with a comprehensive spatio-temporal data analysis to back it up:

“During this study we also noticed that the Sun position in terms of universal time (GMT) has some links to the Earthquake-Moon distance (EMD) together with Sun-Earth-Moon (SEM) angle. In this paper we explored all the features of this relationship. It is astonishing to see that over 98% of worldwide earthquakes faithfully follow the straight-line relationship between the Sun position or GMT timings with (EMD+SE
M). This proves beyond any doubt that the vast majority (98% ) of worldwide earthquakes are governed by the Sun and Moon. Even the smaller earthquakes in the magnitude range of 2-3 faithfully follow this relationship. It is also seen that numerous aftershocks, which follow any major earthquake, faithfully follow straight-line curves, generated by the plot for (EMD+SEM) vs GMT timings. For all the plots providing earthquake data, for 00 hours GMT, the earthquake commences at the mean position of the longitude range (of the area under study) on the X axis (EMD+SEM).
It is seen that for plots for longitude ranges close to +/- 1800, the earthquake plot occupies the central strip running diagonally from the origin of the plot. For all other plots, for 00 hours GMT, the earthquake commences at the mean longitude of the area under study, for the Sun’s position opposite the earthquake region (1800 out of phase).
Basically it is seen that all these earthquakes are triggered by the Earth tides caused by the positions of the Sun and the Moon and this process seems to be the primary triggering mechanism for all worldwide earthquakes. This includes deep-focus earthquakes, which are affected by Earth tides as shown by the earlier study (Kolvankar et al., 2010).”

Kolvankar compiled sets of what he calls “earthquake plots” which isolate the timing of the lunar+solar orbital path with frequencies of earthquakes (via a scatter density plot) in specific locations.
There’s nothing really mathematically complex about this result. It’s simply the result of data mining and then plotting a pairwise relationship in a discovered pattern.  I am not going to check his results because I don’t have access to the data he analyzed, but I can plot something similar based on another well-known physical behavior. Below is the locii of points that tracks the zenith point of the sun in the sky.

Perhaps too obvious, but the zenith occurs at GMT high-noon (12H) only at the longitude corresponding to Greenwich, UK, and is offset elsewhere.  This has physical significance in that this is the point at which sunlight is strongest overhead. There is no deviation in the points from a straight line, as the earth’s rotation is deterministic (the data is artificially created for the sake of argument).

For earthquakes, the relationship is different, but can be interpreted in the similar geographic fashion — only instead involving the sun and the moon.

Kolvankar stumbled on to the slightly more complex spatio-temporal relationship as he describes in the Acknowledgements section of the paper:

“… the author had some discussion about the access database management of over five hundred thousand earthquakes. Mr Rahul Kesarkar demonstrated various aspects of the access database software. During this time, we became aware of the relationship between the GMT timings and the EMD + SEM. I wholeheartedly thank Mr Rahul Kesarkar for his efforts, which brought the relationship between these two parameters to our notice.”

The mechanism behind this is straightforward to understand. Earthquakes are triggered after stress builds up over time and the strain is released at a fault.  All that is happening with the lunar tidal force is that a small excursion is provided at a time prior to that at which it would have randomly occurred.  That happens to coincide with a specific lunisolar alignment at a set latitude-longitude. Kolvankar has a figure describing this configuration:

From the later PNAS article, the following figure shows how the stress is elevated temporarily at a fortnightly cycle:

The PNAS people (van der Els et al) somehow missed the Kolvankar paper.  I had found it a while ago when I discussed the topic here and later here at the Azimuth Project forum.  Since that time I lost track of it, but easily found it again with a Google search.

Now, it may be true that the lunar trigger to earthquakes is not enough to have any practicality for prediction.  There is still a significant stochastic element in determining the range of time that an earthquake will get triggered.  That is still largely unknowable, except in a probabilistic sense.

Yet, there is no threshold trigger in the ocean as it responds gradually to lunar and solar tidal forcing. That means that ocean tides are highly predictable and can be predicted far in the advance, just like the recent solar eclipse.  That’s a true deterministic behavior, in contrast to the stochastic nature of earthquakes.

So now you can understand why the finding of a lunar forcing for ENSO and El Nino events is also a breakthrough. Similar to diurnal and semidurnal ocean tides, the gradual fortnightly and monthly cycling of the moon’s orbit creates a deterministic outcome suitable for prediction years and decades in advance of the ENSO event.


What’s sad about reporting on this kind of research is the amount of push-back one gets. As I described at Azimuth, I was banned from commenting at Physics Forums in 2015 because I had the gall to suggest that the moon may be causing more physical behaviors than the familiar ocean tidal effect and solar eclipses. The former high-school student that runs Physics Forums is anti-research and mostly runs the site so he can do homework for kids that are too lazy to do it themselves. My mistake for believing that something labelled Physics Forums might actually be interested in discussing real physics.

But it does point out an interesting situation. That is: millions of people will flock to see the moon causing a solar eclipse, yet a single guy in India named Kolvankar figured out the pattern behind the moon triggering earthquakes and he gets roundly ignored. Such is the wacky world of cutting-edge scientific research with its own unwritten rules of discourse. Perhaps not this bad, but as George Monbiot said years ago: “Tell people something they know already, and they will thank you for it. Tell them something new, and they will hate you for it.”

More likely is that the new information will simply be ignored because it did not follow the established protocol for disseminating results. Apparently it’s better to wait for someone with credentials to report the findings and then jump on the bandwagon when it becomes consensus. Everything else is conspiracy talk — which perhaps is not hard to understand given the popularity of the crackpots at WUWT and Tallbloke’s Talkshop.

To preface his earlier quote, Monbiot also gave this advice:

“So the task of the environmental journalist is not just to highlight damage to the environment. It is not just to challenge some of humankind’s most fundamental perceptions. It is to challenge humanity itself. I hope I am not putting you off.”

Excepting Monbiot, lots of these science journalists can’t be bothered.  They too only follow the consensus. Better to just read their posts on recycled science topics than to get them to read something interesting.

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:  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.


[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{partialzeta}{partialvarphi} = frac{partialzeta}{partial t}frac{partial t}{partialvarphi}}

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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