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 !

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.

Strictly Biennial Cycles in ENSO

Continuing from a previous post describing the historical evolution of ocean dynamics and tidal theory, this paper gives an early history of ENSO [1].

The El Niño–Southern Oscillation (ENSO) is among the most pervasive natural climate oscillations on earth, affecting the web of life from plankton to people. During mature El Niño (La Niña) events, the sea surface temperature (SST) in the eastern equatorial Pacific warms (cools), leading to global-scale responses in the terrestrial biosphere transmitted through modifications of large-scale atmospheric circulation. The dynamics of—and global responses to—ENSO have been studied for nearly eight decades (Walker and Bliss 1932; Ropelewski and Halpert 1989; Kiladis and Diaz 1989; Yulaeva and Wallace 1994). Cyclic patterns in climate events have also been connected to something resembling ENSO as early as the mid-nineteenth century. Reminiscing on his 1832 visit to Argentina during his expedition on the H.M.S. Beagle, British naturalist Charles Darwin notes “[t]hese droughts to a certain degree seem to be almost periodical; I was told the dates of several others, and the intervals were about fifteen years” (Darwin 1839). Nearly 60 years later, Darwin enters into his journal “. . . variations in climate sometimes appear to be the effect of the operation of some very general cause” (Darwin 1896). Some believe this “very general cause” was actually an early piecing together of ENSO and its now notorious impact on extreme weather events in South America (Cerveny 2005). It is only a coincidence that Darwin may have been among the first to point out the cyclic nature of ENSO, and the focus of this paper is the association between ENSO and the Galápagos Islands, which also owe their fame to Darwin.

Beyond this history, the purpose of this particular paper is to investigate the mechanics behind ENSO and to isolate the “very general cause” that Darwin first hypothesized (and isn’t it always the case how the most intellectually curious are at the root of scientific investigations?). According to this same paper[1], a “strictly biennial” cycle is routinely observed in ENSO when run with an ocean general circulation model (OGCM). Yet they observe correctly as quoted below that “Such strictly biennial regularity is not realistic, as ENSO in nature at present is neither perfectly regular nor significantly biennial.”

Note how strong the biennial Fourier factor is in their simulation (along with the perfectly acceptable harmonic at 2/3 year which will shape the biennial into anything from a triangle to a square wave). With our ENSO model, I can easily reproduce a strictly biennial cycle just by changing the forcing from a lunar monthly cycle (incongruent with a yearly cycle) to anything that is a harmonic with the yearly cycle. So it’s our claim that it’s the lunar cycle that remains the key factor that changes the ENSO cycle into something that is “neither perfectly regular nor significantly biennial” in the words of the cited paper. The biennial factor is still there but it gets modified and split by the lunar cycle to the extent that no biennial factor remains in the Fourier spectra.

Yet if we look into the GCM’s that researchers have developed and you will find that none have any capabilities for introducing a lunar tidal factor as a forcing.  Why is that?  Probably because someone long ago simply asserted that the lunar gravitational pull wasn’t important for ENSO, contrary to its critical importance for understanding ocean tides.   So is this lunar effect really the “very general cause” that Darwin was thinking of to explain ENSO?

As a result of some intellectual curiosity to actually test the tidal forcing against a biennial modulation, I think the answer is a definitive yes. This is how sensitive the fitting of the model is to selection of the two forcing cycles

By adjusting the values progressively away from the true value for the lunar tidal cycle (27.2122 days for the Draconic cycle and 27.55455 days for the Anomalistic cycle), it will result in a smaller correlation coefficient. This doesn’t happen by accident. Fitting this same model to 200 years of ENSO coral proxy data also doesn’t happen by accident. And extracting precisely phased and correlated lunar cycles to the actual forcing applied to the earth’s rotation also doesn’t happen by accident. I think it’s time for the GCM’s to revisit the role of lunar forcing, just as NASA JPL was about to before they decided to pull the plug on their own lunar research initiative [2].

References

[1] K. B. Karnauskas, R. Murtugudde, and A. J. Busalacchi, “The effect of the Galápagos Islands on ENSO in forced ocean and hybrid coupled models,” Journal of Physical Oceanography, vol. 38, no. 11, pp. 2519–2534, 2008.

[2]  From a post-mortem —  “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.” 

Overturning Impulse

The only earth science class I took in college was limnology.

Of course I was a casual student of freshwater activities before that time, but certain behaviors of lakes were hammered home by taking this class. For example, the idea behind seasonal lake overturning. The overturning occurs as a singular event at a particular time of the year (monomictic once per year and dimictic twice per year, see figure to the right).

Saltwater doesn’t show the same predilection for overturning as freshwater does, mainly because of the higher density differences above and below the thermocline, but the thermocline does vary, especially at latitudes located off the equator.

The ENSO model that appears so promising might be pointing to a partial overturning  showing a bimictic behavior.  The bi-prefix implies a biennial impulse that reverses direction every year, tied into a biennial Mathieu modulation associated with fluid sloshing dynamics. The Mathieu modulation is partially induced from the forcing conditions [1][2].

The following figure shows the modulation and impulse forcing that generates the best fit for the ENSO model. The BLUE is the Mathieu modulation while the ORANGE is the impulse.

The impulse may be related to a rapidly changing slope when the bimictic partial-overturning kicks in. The positive impulse as the Mathieu modulation moves from high to low, and the negative impulse as the modulation moves from low to high, both occurring at the same relative time late in the year.  Note that the polarity of this may be reversed, as the lunar tidal forcing at the impulse time provides the strength of this forcing through a multiplicative effect. This is shown below.

Whether to call it sloshing or a partial overturning remains to be determined. Yet overall this piece of the puzzle is the primary ansatz behind the entire model — since if we don’t include it, the model does not generate the sharply distinct ENSO peaks and valleys. In other words, this is what causes the physical aliasing necessary to transform the monthly and fortnightly tidal cycles into a more erratic interannual cycling. This is in fact a simple model that shows more complex but still predictable dynamics.

References

[1] T. B. Benjamin and F. Ursell, “The stability of the plane free surface of a liquid in vertical periodic motion,” presented at the Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 1954, vol. 225, pp. 505–515.

[2] S. Bale, K. Clavin, M. Sathe, A. S. Berrouk, F. C. Knopf, and K. Nandakumar, “Mixing in oscillating columns: Experimental and numerical studies,” Chemical Engineering Science, vol. 168, pp. 78–89, 2017.

ENSO forcing – Validation via LOD data

If we don’t have enough evidence that the forcing of ENSO is due to lunisolar cycles, this piece provides another independent validating analysis. What we will show is how well the forcing used in a model fit to an ENSO time series — that when isolated — agrees precisely with the forcing that generates the slight deviations in the earth’s rotational speed, i.e. the earth’s angular momentum. The latter as measured via precise measurements of the earth’s length of day (LOD).  The implication is that the gravitational forcing that causes slight variations in the earth’s rotation speed will also cause the sloshing in the Pacific ocean’s thermocline, leading to the cyclic ENSO behavior.

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ENSO and Fourier analysis

Much of tidal analysis has been performed by Fourier analysis, whereby one can straightforwardly deduce the frequency components arising from the various lunar and solar orbital factors. In a perfectly linear world with only two ideal sinusoidal cycles, we would see the Fourier amplitude spectra of Figure 1.

Fig 1: Amplitude spectra for a signal with two sinusoidal Fourier components. To establish the phase, both a real value and imaginary value is plotted.

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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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ENSO and Tidal SLH – A Biennial Connection

[mathjax]It’s becoming abundantly clear that ENSO is driven by lunisolar mechanisms, especially true considering that we use the preferred models describing sloshing of water volumes — the Mathieu equation and the delay differential equation. What’s more, given the fact that the ENSO model works so well, one can guess that a connection between ENSO and lunar tidal forces should carry over to models for historical sea-level height (SLH) tidal data.

From my preliminary work analyzing SLH tidal data in 2014, I found an intriguing pattern relating to the same biennial pattern observed in the ENSO model. An extract from that post is reproduced below:

“The idea, related to delay differential equations, is to determine if it is at all possible to model at least part of ENSO (through the SOI) with data from a point in time in the tidal record with a compensated point from the past. This essentially models the effect of the current wave being cancelled partially by the reflection of a previous wave.

This is essentially suggesting that $$SOI = k f(t) – k f(t-Delta t)$$ where f(t) is the tidal record and k is a constant.

After some experimenting, a good fit is obtained when the current tidal data is set to 3 months ago, and the prior data is taken from 26 months in the past. To model the negative of ENSO, the 3-month old data is subtracted from the 26-month old data, Figure 1:”

Fig 1: Model of ENSO uses Tidal Gauge readings from Sydney Harbor.

This observation suggests a biennial pattern whereby high correlations in the time series occur for events observed close to two-years apart, with the difference taken up by the ENSO signal. The latter is at least partly due to the inverted barometer effect on SLH.

The sensitivity of the biennial effect is shown in the following contour diagram, where the peak CC is indicated and a slanted line showing the 2-year differencing points (table of the correlation coefficients here)

Focussing on the mechanisms for the correlation, the predictor can be reformulated as

f(t) = f(t-Delta t) + SOI / k


or that the 2-year prior SLH reading, with the addition of the SOI signal, will accurately represent the current SLH reading subject to a ~3 month lag.

My thinking is that this correlation can only occur if there is an actual biennial signal in the SLH itself. We can mathematically conjecture what is happening with the two year differencing by assuming that the SLH has the same height offset whenever pairs of points are separated by two years. You can see this in the following figure:

This brings up an interesting question as to why this biennial signal apparently can’t be detected via conventional means. A likely guess is because it gets obscured by the ENSO signal. And since the ENSO signal has never been adequately modeled, no one has an inkling that a biennial signal coexists in the mix. Yet this unconventional delay differencing approach (incidentally discovered with the help of machine learning) is able to discern it with a strong statistical significance.

The gist of this discovery is that a biennial tidal SLH motion exists and likely contributes (or at least identifies) the nonlinear biennial modulation driving the ENSO model described in previous posts.

Summary:

ENSO has elements of an annual teeter-totter. If you look through the research literature, you will find numerous references to a hypothesized behavior that one annual peak is followed by a lesser peak the next year. Yet, no evidence that this strict biennial cycle is evidence in the data — it’s more of a hand-wavy physical argument that this can or should occur.

The way to model this teeter-totter behavior is via Mathieu equations and delay differential equations. Both of these provide a kind of non-linear modulation that can sustain a biennial feedback mechanism.

The other ingredient is a forcing mechanism. The current literature appears to agree that this is due to prevailing wind bursts, which to me seems intuitive but doesn’t answer what forces the wind in the first place. As it turns out, only two parameters are needed to force the DiffEq and these align precisely with the primary lunar cycles that govern transverse and longitudinal directional momentum, the draconic and anomalistic months.

The ENSO model turns into a metrology tool

Note the following table and how the draconic and anomalistic values zone in on the true value when correlating model against data:

Fitting the DiffEq values progressively away from the true value for the lunar tidal cycle results in a smaller correlation coefficient.

Having one of these values align may be coincidence, but having both combine with that kind of resolution is telling. The biennial connection to SLH tidal measurements is substantiation that a biennial modulation is intrinsic to the physical process.

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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