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Deterministic and Stochastic Applied Physics

/ Paul Pukite (@whut) / Leave a comment

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

/ Paul Pukite (@whut) / 3 Comments

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.
① 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.
② 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.
③  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
④  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

/ Paul Pukite (@whut) / 2 Comments

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.” 

ENSO forcing – Validation via LOD data

/ Paul Pukite (@whut) / 10 Comments

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.

Continue reading →

ENSO and Fourier analysis

/ Paul Pukite (@whut) / 1 Comment

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.

Continue reading →

ENSO and Noise

/ Paul Pukite (@whut) / 7 Comments

How do we determine confidence that we are not fitting to noise for the ENSO model ?  One way to do this is to compare the data against another model; in this case, a model that provides an instrumentally independent measure. One can judge data quality by comparing an index such as NINO34 against SOI, which are instrumentally independent measures (one based on temperature and one on atmospheric pressure).

If you look at a sliding correlation coefficient of these two indices along the complete interval, you will see certain years that are poorly correlated (see RED line below). Impressively, these are the same years that give poor agreement against the ENSO model (see BLUE line below). What this tells us is the poorly correlated years are ones with poor signal-to-noise ratio. But more importantly, it also indicates that the model is primarily fitting to the real ENSO signal (especially the peak values) and the noisy parts (closer to zero crossings or neutral ENSO conditions) are likely not contributing to the fit. And this is not a situation where the model will fit SOI better than NINO34 — because it doesn’t.

The tracking of SOI correlating to NINO34 matches that of Model to NINO34 across the range with the exception of some excursions during the 1950’s, where SOI fit NINO34 better that the model fit NINO34. The average correlation coefficient of SOI to NINO34 across the entire range is 0.75 while the model against NINO34 is less but depending on the parameterization always above 0.6.

As a result of this finding, I started to use a modification of a correlation coefficient called a weighted correlation coefficient, whereby the third parameter set is a density function that remains near 1 when the signal-to-noise (SNR) ratio is high and closer to zero where the SNR is closer to zero. This allows the fit to concentrate on the intervals of strong SNR, thus reducing the possibility of over-fitting against noise.

Or is it really all noise?   (Added: 5/17/2017)

As I derived earlier, the solution to Laplace’s tidal equations at the equator for a behavior such as QBO leads to a sin(k sin(f(t))) modulated time-series, where the inner sinusoid is essentially the forcing. This particular formulation (referred to as the sin-sin envelope) has interesting properties. For one, it has an amplitude limiting property due to the fact that a sinuosoid can’t exceed an amplitude of unity. Besides this excursion-limiting behavior, this formulation can also show amplitude folding at the positive and negative extremes. In other words, if the amplitude is too large, the outer sin modulation starts to shrink the excursion, instead of just limiting it. So if there is a massive amplitude, what happens is that the folding will occur multiple times within the peak interval, thus resulting in a rapid up and down oscillation. This potentially can have the appearance of noise as the oscillations are so rapid that (1) they may blur the data record or (2) may be unsustainable and lead to some form of wave-breaking.  I am not sure if the latter is related to folding of geological strata.

So the question is: can this happen for ENSO? I have been feeding the solution to the delayed differential Mathieu equation as a forcing to the sin-sin envelope and find that it works effectively to match the “noisy” regions identified above.  In the figure below, the diamonds represent intervals with the poorest correlation between NINO34 and SOI and perhaps the noisiest in terms of SOI. In particular, the regions labelled 1 and 6 indicate rapid cyclic excursions.

By comparison, the model fit to ENSO shows the rapid oscillations near many of the same regions. In particular look at intervals indicated by diamonds 1 and 6 below, as well as the interval just before 1950.

Now, consider that these just happen to be the same regions that the ENSO model shows excessive amplitude folding.  The pattern isn’t 100% but also doesn’t appear to be coincidental, nor is it biased or forced (as the fitting procedure has no idea that these are considered the noisy intervals).  So the suggestion is that these are points in time that could have developed into massive El Nino or La Nina, but didn’t because the forcing amplitude became folded. Thus they could not grow and instead the strong lunar gravitational forcing went into rapid oscillations which dissipated that energy. In fact, it’s really the rate of change in the kinetic energy that scales with forcing, and the rapid oscillations identify that change. Connecting back to the theory, that’s what the sin-sin envelope describes — its essentially a solution to a Hamiltonian that conserves the energy of the system. From the Sturm-Liouville equation that Laplace’s tidal equations reduce to, this answer is analytically precise and provided in closed-form.

The caveat to this idea of course is that no one else in climate science is even close to considering such a sin-sin formulation.  Consider this:

… yet …

An alternative model that matches ENSO does not exist, so there is nothing at the moment to refute.  And see above how it fits in balance with known physics.

ENSO Proxy Validation

/ Paul Pukite (@whut) / 9 Comments

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.

Continue reading →

Canonical Solution of Mathieu Equation for ENSO

/ Paul Pukite (@whut) / 3 Comments

[mathjax]
From a previous post, we were exploring possible solutions to the Mathieu equation given a pulsed stimulus.  This is a more straightforward decomposition of the differential equation using a spreadsheet.

The Mathieu equation:

f''(t) + omega_0^2 (1 + alpha cos(nu t)) f(t) = F(t)

can be approximated as a difference equation, where the second derivative f”(t) is ~ (f(t)-2f(t-dt)+f(t-2dt))/dt. But perhaps what we really want is a difference to the previous year and determine if that is enough to reinforce the biennial modulation that we are seeing in the ENSO behavior.

Setting up a spreadsheet with a lag term and a 1-year-prior feedback term, we apply both the biennial impulse-modulated lunar forcing stimulus and a yearly-modulated Mathieu term.

Fig. 1: Training (in shaded blue) and test for different intervals.

I was surprised by how remarkable the approximate fit was in the recent post, but this more canonical analysis is even more telling. The number of degrees of freedom in the dozen lunar amplitude terms apparently has no impact on over-fitting, even on the shortest interval in the third chart. There is noise in the ENSO data no doubt, but that noise seems to be secondary considering how the fit seems to mostly capture the real signal. The first two charts are complementary in that regard — the fit is arguably better in each of the training intervals yet the test interval results aren’t really that much different from the direct fit looking at it by eye.

Just like in ocean tidal analysis, the strongest tidal cycles dominate;  in this case the Draconic and Anomalistic monthly, the Draconic and Anomalistic fortnightly, and a Draconic monthly+Anomalistic fortnightly cross term are the strongest (described here). Even though there is much room for weighting these factors differently on orthogonal intervals, the Excel Solver fit hones in on nearly the same weighted set on each interval.  As I said in a previous post, the number of degrees of freedom apparently do not lead to over-fitting issues.

One other feature of this fit was an application of a sin() function applied to the result. This is derived from the Sturm-Liouville solution to Laplace’s tidal equation used in the QBO analysis  — which works effectively to normalize the model to the data, since the correlation coefficient optimizing metric does not scale the result automatically.

Pondering for a moment, perhaps the calculus is not so different to work out after all:

C93Q9eVWsAA8N4Q[1]

(from @rabaath on Twitter)

A bottom-line finding is that there is really not much complexity to this unique tidal formulation model of ENSO.  But because of the uniqueness of the seasonal modulation that we apply, it just doesn’t look like the more-or-less regular cycles contributing to sea-level height tidal data.  Essentially similar algorithms are applied to find the right weighting of tidal factors, but whereas the SLH tidal data shows up in daily readings, the ENSO data is year-to-year.

Further, the algorithm does not take more than a minute or two to finish fitting the model to the data. Below is a time lapse of one such trial. Although this isn’t an optimal fit, one can see how the training interval solver adjustment (in the shaded region) pulls the rest of the modeled time-series into alignment with the out-of-band test interval data.

Tidal Model of ENSO

/ Paul Pukite (@whut) / 4 Comments

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

Continue reading →

Solution to Pulsed Mathieu Equation

/ Paul Pukite (@whut) / 2 Comments

Here is a bit of applied math that I have never seen described before.  It considers solving a variant of the Mathieu differential equation, an unwieldy beast that finds application in models of fluid sloshing (among others). Normally the Mathieu equation is only solvable as a stimulus forcing function convolved against the transcendental Mathieu function. Tools such as Mathematica include the Mathieu function in their library, but any other solution would require a full numerical DiffEq integration.

This is the typical Mathieu equation formulation:

f''(t) + ( omega_0^2 + B cos (nu_0 t) ) f(t) = 0

But instead of this nonlinear time-varying DiffEq, consider that we replace the sinusoidal modulation with a delta pulse. This is precisely our model of ENSO that we are trying to create from first principles. The pulse represents a forcing impulse from the alignment of a lunar and solar tidal cycle.

f''(t) + ( omega_0^2 + B delta (t) ) f(t) = 0

Taking the Laplace transform, we quickly arrive at what looks like an ordinary 2nd-order differential equation solution, albeit with an interesting initial condition:

s^2 F(s) + omega_0^2 F(s) + B f(0) = 0

After taking the inverse Laplace transform, we have:

f(t) = I(t) ast B f(0) delta(t)

where I(t) represents the impulse response of the first two terms, which is then convolved with a delta function evaluated at a value of f(t) for t=0. This is the only remnant of the nonlinear nature of the classical Mathieu function, in that the initial condition has a scaling proportional to the value of the function at that time. On the other hand, the solution to an ordinary DiffEq would not be dependent on the value of the function.

Thereafter we can extend this to a general solution; by creating a pulse train of delta functions we get this final convolution:

f(t) = I(t) ast B sum_{n=0}^{infty} f(t - n T) delta(t - n T)

where T is the pulse train period.

This is straightforward to evaluate for any pulse train — all we have to do is keep track of the changing value of f(t) as we come across each pulse.

Fig 1: A pulse train with both annual and biannual contributions.

In the above figure, why does the annual pulse have a two-year periodicity? That’s due to the alignment of a non-congruent tidal period with a seasonal pulse in terms of a Fourier series as described here and mathematically refined here.

If that is not intuitive, we can still consider it more of an anzat and see where it takes us.

For an ENSO time series, we an convolve the above delta pulse train with a set of known tidal periods of arbitrary amplitude and phase. For a fit trained up to 1980, this is the extrapolation post-1980:

Fig 2: Fit of pulsed Mathieu equation using a set of tidal periods as a forcing function. The impulse response here is a simple year-long rectangular window, i.e. the response has a memory of only a calendar year. This can be further refined if necessary. Yet, the projected time-series evaluated out-of-band from the fitting interval does a good job of capturing the ENSO profile.

The projected waveform matches the last 16 years very effectively considering how noisy the ENSO series is, and is very close to the fit over the entire interval. This is apart from the possible perturbation around the 1982 El Nino. The fact that the monthly tidal periods differ by slight amounts dictates that long multi-decadal intervals should be used for fitting. (In contrast, the model of QBO is primarily Draconic so the fitting interval can be much shorter)

Fig 3: Fit over the entire interval. The set of tidal periods applied was limited to the 3 major months (draconic, anomalistic, and sidereal/tropical) and the multiplicative combinations creating the fortnightly tides. This may not be enough to get the details right but erred on the side of under-fitting to establish the physical mechanism.

This approach is the logical follow-on to the wave transformed fitting approach that I had been using, most recently here and here. Both of these approaches are equally clever, which is a necessary ingredient when one is dealing with the unwieldy Mathieu equations. Moreover they both pull out the obvious stationary aspects of the ergodic ENSO time series.