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 !

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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Scaling El Nino

Recently, the rock climber Alex Honnold took a route up El Capitan without ropes.There's no room to fail at that. I prefer a challenge that one can fail at, and then keep trying.  This is the ascent to conquering El Nino:

The Free-thought Route*

Χ  Base camp:  ENSO (El Nino/Southern Oscillation) is a sloshing behavior, mainly in the thermocline where the effective gravity makes it sensitive to angular momentum changes.
Χ  Faster forcing cycles reinforce against the yearly cycle, creating aliased periods. How?
Χ  Monthly lunar tidal cycles provide the aliased factors: Numbers match up perfectly.
This aliasing also works for QBO, an atmospheric analog of ENSO.
Χ  A biennial meta-stability appears to be active. Cycles reinforce on alternating years.
Χ  The well-known Mathieu modulation used for sloshing simulations also shows a biennial character.
Machine learning experiments help ferret out these patterns.
Χ  The delay differential equation formulation matches up with the biennial Mathieu modulation with a delay of one-year.  That's the intuitive yearly see-saw that is often suggested to occur.
  The Chandler wobble also shows a tidal forcing tendency, as does clearly the earth's LOD (length-of-day) variations.
Χ  Integrating the DiffEq model provides a good fit, including long-term coral proxy records
Χ  Solving the Laplace tidal equation via a Sturm-Liouville expression along the equator helps explain details of QBO and ENSO
  Close inspection of sea-level height (SLH) tidal records show evidence of both biennial and ENSO characteristics
Δ Summit: Final validation of the geophysics comparing ENSO forcing against LOD forcing.

Model fits to ENSO using a training interval

The route encountered several dead-ends with no toe-holds or hand-holds along the way (e.g. the slippery biennial phase reversal, the early attempts at applying Mathieu equation). In retrospect many of these excursions were misguided or overly complex, but eventually other observations pointed to the obvious route.

This is a magnification of the fitting contour around the best forcing period values for ENSO. These pair of peak values are each found to be less than a minute apart from the known values of the Draconic cycle (27.2122 days) and Anomalistic cycle (27.5545 days).

The forcing comes directly from the angular momentum variations in the Earth's rotation. The comparison between what the ENSO model uses and what is measured via monitoring the length-of-day (LOD) is shown below:



*  This is not the precise route I took, but how I wish it was in hindsight.

ENSO model fit 1880-1980

This is an ENSO fit that only has knowledge of data prior to 1980. The data is 80% NINO34 and 20% SOI, with the latter providing finer structure.The lower fit includes a slight variation of the Draconic month according to this NASA page. It doesn't seem to do much.

A previous fit used values of A=0.785, M=0.3, K=-0.15 to 1920. This used A=0.866, M=0.207, K=-0.16.

The Lunar Geophysical Connection

The conjecture out of NASA JPL is that the moon has an impact on the climate greater than is currently understood:

Claire Perigaud (Caltech/JPL)
Has this research gone anywhere?  Looks as if has gone to this spin-off.
According to the current consensus, variability in wind is what contributes to forcing for behaviors such as the El Nino/Southern Oscillation (ENSO).
OK, but what forces the wind? No one can answer that apart from saying wind variability is just a part of the dynamic climate system.  And so we are lead to believe that a wind burst will cause an ENSO and then the ENSO event will create a significant disruptive transient to the climate much larger than the original wind stimulus. And that's all due to positive feedback of some sort.
I am only paraphrasing the current consensus.
A much more plausible and parsimonious explanation lies with external lunar forcing reinforced by seasonal cycles.

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ENSO Proxy Validation

This is a straightforward validation of the ENSO model presented at last December's AGU.

What I did was use the modern instrumental record of ENSO — the NINO34 data set — as a training interval, and then tested across the historical coral proxy record — the UEP data set.

The correlation coefficient in the out-of-band region of 1650 to 1880 is excellent, considering that only two RHS lunar periods (draconic and anomalistic month) are used for forcing. As a matter of fact, trying to get any kind of agreement with the UEP using an arbitrary set of sine waves is problematic as the time-series appears nearly chaotic and thus requires may Fourier components to fit. With the ENSO model in place, the alignment with the data is automatic. It predicts the strong El Nino in 1877-1878 and then nearly everything before that.

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Shortest Training Fit for ENSO

This is remarkable. Using the spreadsheet linked in the last post, the figure below is a model of ENSO derived completely by a training fit over the interval 1900 to 1920, using the Nino3.4 data series and applying the precisely phased Draconic and Anomalistic long-period tidal cycles.

Fig. 1 : The ENSO model in red. The blue BG region is used for training of the lunar tidal amplitudes against the Nino3.4 data in green. That data is square root compacted to convert it to an equivalent velocity.

Not much more to say. There is a major disturbance starting in the mid-1980's, but that is known from a Takens embedding analysis described in the first paper in this post.

Tidal Model of ENSO

The input forcing to the ENSO model includes combinations of the three major lunar months modulated by the seasonal solar cycle. This makes it conceptually similar to an ocean tidal analysis, but for ENSO we are more concerned about the long-period tides rather than the diurnal and semi-diurnal cycles used in conventional tidal analysis.

The three constituent lunar month factors are:

Month type Length in days
anomalistic 27.554549
tropical 27.321582
draconic 27.212220

So the essential cyclic terms are the following phased sinusoids

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