Nonlinear Generation of Power Spectrum : ENSO

Something I learned early on in my research career is that complicated frequency spectra can be generated from simple repeating structures. Consider the spatial frequency spectra produced as a diffraction pattern produced from a crystal lattice. Below is a reflected electron diffraction pattern of a reconstructed hexagonally reconstructed surface of a silicon (Si) single crystal with a lead (Pb) adlayer ( (a) and (b) are different alignments of the beam direction with respect to the lattice). Suffice to say, there is enough information in the patterns to be able to reverse engineer the structure of the surface as (c).

from link

Now consider the ENSO pattern. At first glance, neither the time-series signal nor the Fourier series power spectra appear to be produced by anything periodically regular. Even so, let’s assume that the underlying pattern is tidally regular, being comprised of the expected fortnightly 13.66 day tropical/synodic cycle and the monthly 27.55 day anomalistic cycle synchronized by an annual impulse. Then the forcing power spectrum of f(t) looks like the RED trace on the left-side of the figure below, F(ω). Clearly that is not enough of a frequency spectra (a few delta spikes) necessary to make up the empirically calculated Fourier series for the ENSO data comprising ~40 intricately placed peaks between 0 and 1 cycles/year in BLUE.

click to expand

Yet, if we modulate that with an Laplace’s Tidal Equation solution functional g(f(t)) that has a G(ω) as in the yellow inset above — a cyclic modulation of amplitudes where g(x) is described by two distinct sine-waves — then the complete ENSO spectra is fleshed out in BLACK in the figure above. The effective g(x) is shown in the figure below, where a slower modulation is superimposed over a faster modulation.

So essentially what this is suggesting is that a few tidal factors modulated by two sinusoids produces enough spectral detail to easily account for the ~40 peaks in the ENSO power spectra. It can do this because a modulating sinusoid is an efficient harmonics and cross-harmonics generator, as the Taylor’s series of a sinusoid contains an effectively infinite number of power terms.

To see this process in action, consider the following three figures, which features a slider that allows one to get an intuitive feel for how the LTE modulation adds richness via harmonics in the power spectra.

  1. Start with a mild LTE modulation and start to increase it as in the figure below. A few harmonics begin to emerge as satellites surrounding the forcing harmonics in RED.
drag slider right for less modulation and to the left for more modulation

2. Next, increase the LTE modulation so that it models the slower sinusoid — more harmonics emerge

3. Then add the faster sinusoid, to fully populate the empirically observed ENSO spectral peaks (and matching the time series).

It appears as if by magic, but this is the power of non-linear harmonic generation. Note that the peak labeled AB amongst others is derived from the original A and B as complicated satellite-cross terms, which can be accounted for by expanding all of the terms in the Taylor’s series of the sinusoids. This can be done with some difficulty, or left as is when doing the fit via solver software.

To complete the circle, it’s likely that being exposed to mind-blowing Fourier series early on makes Fourier analysis of climate data less intimidating, as one can apply all the tricks-of-the-trade, which, alas, are considered routine in other disciplines.


Individual charts

https://imagizer.imageshack.com/img922/7013/VRro0m.png




 


Overfitting+Cross-Validation: ENSO→AMO

I presented at the 2018 AGU Fall meeting on the topic of cross-validation. From those early results, I updated a fitted model comparison between the Pacific ocean’s ENSO time-series and the Atlantic Ocean’s AMO time-series. The premise is that the tidal forcing is essentially the same in the two oceans, but that the standing-wave configuration differs. So the approach is to maintain a common-mode forcing in the two basins while only adjusting the Laplace’s tidal equation (LTE) modulation.

If you don’t know about these completely orthogonal time series, the thought that one can avoid overfitting the data — let alone two sets simultaneously — is unheard of (Michael Mann doesn’t even think that the AMO is a real oscillation based on reading his latest research article called “Absence of internal multidecadal and interdecadal oscillations in climate model simulations“).

This is the latest product (click to expand)

Read this backwards from H to A.

H = The two tidal forcing inputs for ENSO and AMO — differs really only by scale and a slight offset

G = The constituent tidal forcing spectrum comparison of the two — primarily the expected main constituents of the Mf fortnightly tide and the Mm monthly tide (and the Mt composite of Mf × Mm), amplified by an annual impulse train which creates a repeating Brillouin zone in frequency space.

E&F = The LTE modulation for AMO, essentially comprised of one strong high-wavenumber modulation as shown in F

C&D = The LTE modulation for ENSO, a strong low-wavenumber that follows the El Nino La Nina cycles and then a faster modulation

B = The AMO fitted model modulating H with E

A = The ENSO fitted model modulating the other H with C

Ordinarily, this would take eons worth of machine learning compute time to determine this non-linear mapping, but with knowledge of how to solve Navier-Stokes, it becomes a tractable problem.

Now, with that said, what does this have to do with cross-validation? By fitting only to the ENSO time-series, the model produced does indeed have many degrees of freedom (DOF), based on the number of tidal constituents shown in G. Yet, by constraining the AMO fit to require essentially the same constituent tidal forcing as for ENSO, the number of additional DOF introduced is minimal — note the strong spike value in F.

Since parsimony of a model fit is based on information criteria such as number of DOF, as that is exactly what is used as a metric characterizing order in the previous post, then it would be reasonable to assume that fitting a waveform as complex as B with only the additional information of F cross-validates the underlying common-mode model according to any information criteria metric.

For further guidance, this is an informative article on model selection in regards to complexity — “A Primer for Model Selection: The Decisive Role of Model Complexity

excerpt:

The Search for Order

Chap 10 Mathematical Geoenergy

For the LTE formulation along the equator, the analytical solution reduces to g(f(t)), where g(x) is a periodic function. Without knowing what g(x) is, we can use the frequency-domain entropy or spectral entropy of the Fourier series mapping an estimated x=f(t) forcing amplitude to a measured climate index time series such as ENSO. The frequency-domain entropy is the sum or integral of this mapping of x to g(x) in reciprocal space applying the Shannon entropy –I(f).ln(I(f)) normalized over the I(f) frequency range, which is the power spectral (frequency) density of the mapping from the modeled forcing to the time-series waveform sample.

This measures the entropy or degree of disorder of the mapping. So to maximize the degree of order, we minimize this entropy value.

This calculated entropy is a single scalar metric that eliminates the need for evaluating various cyclic g(x) patterns to achieve the best fit. Instead, what it does is point to a highly-ordered spectrum (top panel in the above figure), of which the delta spikes can then be reverse engineered to deduce the primary frequency components arising from the the LTE modulation factor g(x).

The approach works particularly well once the spectral spikes begin to emerge from the background. In terms of a physical picture, what is actually emerging are the principle standing wave solutions for particular wavenumbers. One can see this in the LTE modulation spectrum below where there is a spike at a wavenumber at 1.5 and one at around 10 in panel A (isolating the sin spectrum and cosine spectrum separately instead of the quadrature of the two giving the spectral intensity). This is then reverse engineered as a fit to the actual LTE modulation g(x) in panel B. Panel D is the tidal forcing x=f(t) that minimized the Shannon entropy, thus creating the final fit g(f(t)) in panel C when the LTE modulation is applied to the forcing.

The approach does work, which is quite a boon to the efficiency of iterative fitting towards a solution, reducing the number of DOF involved in the calculation. Prior to this, a guess for the LTE modulation was required and the iterative fit would need to evolve towards the optimal modulation periods. In other words, either approach works, but the entropy approach may provide a quicker and more efficient path to discovering the underlying standing-wave order.

I will eventually add this to the LTE fitting software distro available on GitHub. This may also be applicable to other measures of entropy such as Tallis, Renyi, multi-scale, and perhaps Bispectral entropy, and will add those to the conventional Shannon entropy measure as needed.

Gravitational Pull

In Chapter 12 of the book, we provide an empirical gravitational forcing term that can be applied to the Laplace’s Tidal Equation (LTE) solution for modeling ENSO. The inverse squared law is modified to a cubic law to take into account the differential pull from opposite sides of the earth.

excerpt from Mathematical Geoenergy (Wiley/2018)

The two main terms are the monthly anomalistic (Mm) cycle and the fortnightly tropical/draconic pair (Mf, Mf’ w/ a 18.6 year nodal modulation). Due to the inverse cube gravitational pull found in the denominator of F(t), faster harmonic periods are also created — with the 9-day (Mt) created from the monthly/fortnightly cross-term and the weekly (Mq) from the fortnightly crossed against itself. It’s amazing how few terms are needed to create a canonical fit to a tidally-forced ENSO model.

The recipe for the model is shown in the chart below (click to magnify), following sequentially steps (A) through (G) :

(A) Long-period fortnightly and anomalistic tidal terms as F(t) forcing
(B) The Fourier spectrum of F(t) revealing higher frequency cross terms
(C) An annual impulse modulates the forcing, reinforcing the amplitude
(D) The impulse is integrated producing a lagged quasi-periodic input
(E) Resulting Fourier spectrum is complex due to annual cycle aliasing
(F) Oceanic response is a Laplace’s Tidal Equation (LTE) modulation
(G) Final step is fit the LTE modulation to match the ENSO time-series

The tidal forcing is constrained by the known effects of the lunisolar gravitational torque on the earth’s length-of-day (LOD) variations. An essentially identical set of monthly, fortnightly, 9-day, and weekly terms are required for both a solid-body LOD model fit and a fluid-volume ENSO model fit.

Fitting tidal terms to the dLOD/dt data is only complicated by the aliasing of the annual cycle, making factors such as the weekly 7.095 and 6.83-day cycles difficult to distinguish.

If we apply the same tidal terms as forcing for matching dLOD data, we can use the fit below as a perturbed ENSO tidal forcing. Not a lot of difference here — the weekly harmonics are higher in magnitude.

Modified initial calibration of lunar terms for fitting ENSO

So the only real unknown in this process is guessing the LTE modulation of steps (F) and (G). That’s what differentiates the inertial response of a spinning solid such as the earth’s core and mantle from the response of a rotating liquid volume such as the equatorial Pacific ocean. The former is essentially linear, but the latter is non-linear, making it an infinitely harder problem to solve — as there are infinitely many non-linear transformations one can choose to apply. The only reason that I stumbled across this particular LTE modulation is that it comes directly from a clever solution of Laplace’s tidal equations.

for full derivation see Mathematical Geoenergy (Wiley/2018)

Reversing Traveling Waves

For the solution to Laplace’s Tidal Equation described in Chapter 12, the spatial and temporal results are separable, leading to a non-linear standing-wave time-series formulation:

sin(kx) sin(A sin(wt) )

By analogy to a linear standing-wave formulation, a solution such as

sin(kx) sin(wt)

with the following traveling wave solution (propagating in the +x direction):

sin(kx-wt)

becomes the following in the non-linear LTE solution mode:

sin(kxA sin(wt) )

This is also a traveling wave, but with the characteristic property of being able to periodically reverse direction from +x to –x depending on the value of A and w. As an intuitive aid, a standing wave can be considered as the superposition of two traveling waves traveling in opposite directions:

sin(kxA sin(wt) ) + sin(kx + A sin(wt) )

Here the cross terms cancel after applying the trig identity on sums, and the separable standing-wave result similar to the first equation results. But, whenever there is an imbalance of +x and -x travelling waves, a periodic reversing traveling-wave/standing-wave mix results. This is shown in the following animation, where a mix of nonlinear traveling-waves and standing-waves show the periodic reversal in direction quite clearly.

This reversal is actually observed in ocean measurements, as exemplified in this recent research article:

From their Figure 3, one can see this reversing process as the trajectory of a measured Argo float drift:

If that is not clear enough, the red arrows in the following annotated figure show the direction of the float motion. The drifting floats may not always exactly follow a trajectory as dictated by the velocity of a traveling wave, as this is partly a phase velocity with limited lateral volume displacement, but clearly a large wave-train such as a Tropical Instability Wave will certainly move a float. At least some of this is due to eddy behavior as the reversal is a natural consequence of a circular vortex motion of a large eddy.

Applying the LTE model to complete spatio-temporal data sets such as what Figure 3 is derived from would likely show an interesting match, adding value to the latest ENSO results, but this will require some digging into the data availability.

The SAO and Annual Disturbances

In Chapter 11 of the book Mathematical GeoEnergy, we model the QBO of equatorial stratospheric winds, but only touch on the related cycle at even higher altitudes, the semi-annual oscillation (SAO). The figure at the top of a recent post geometrically explains the difference between SAO and QBO — the basic idea is that the SAO follows the solar tide and not the lunar tide because of a lower atmospheric density at higher altitudes. Thus, the heat-based solar tide overrides the gravitational lunar+solar tide and the resulting oscillation is primarily a harmonic of the annual cycle.

Figure 1 : The SAO modeled with the GEM software fit to 1 hPa data along the equator
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Double-Sideband Suppressed-Carrier Modulation vs Triad

An intriguing yet under-reported finding concerning climate dipole cycles is the symmetry in power spectra observed. This was covered in a post on auto-correlations. The way that this symmetry reveals itself is easily explained by a mirror-folding about one-half some selected carrier frequency, as shown in Fig. 1 below.

Figure 1 : ENSO amplitude spectrum is mirror-folded about 1/2 the annual frequency.
Both the data and model align with their mirrored counterparts, as seen in highlighted box.
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Length of Day II

This is a continuation from the previous Length of Day post showing how closely the ENSO forcing aligns to the dLOD forcing.

Ding & Chao apply an AR-z technique as a supplement to Fourier and Max Entropy spectral techniques to isolate the tidal factors in dLOD

The red data points are the spectral values used in the ENSO model fit.

The top panel below is the LTE modulated tidal forcing fitted against the ENSO time series. The lower panel below is the tidal forcing model over a short interval overlaid on the dLOD/dt data.

That’s all there is to it — it’s all geophysical fluid dynamics. Essentially the same tidal forcing impacts both the rotating solid earth and the equatorial ocean, but the ocean shows a lagged nonlinear response as described in Chapter 12 of the book. In contrast, the solid earth shows an apparently direct linear inertial response. Bottom line is that if one doesn’t know how to do the proper GFD, one will never be able to fit ENSO to a known forcing.

Triad Waves

In Chapter 12 of the book, we discuss tropical instability waves (TIW) of the equatorial Pacific as the higher wavenumber (and higher frequency) companion to the lower wavenumber ENSO (El Nino /Southern Oscillation) behavior. Sutherland et al have already published several papers this year that appear to add some valuable insight to the mathematical underpinnings to the fluid-mechanical relationship.

“It is estimated that globally 1 TW of power is transferred from the lunisolar tides to internal tides[1]. The action of the barotropic tide over bottom topography can generate vertically propagating beams near the source. While some fraction of that energy is dissipated in the near field (as observed, for example, near the Hawaiian Ridge [2]), most of the energy becomes manifest as low-mode internal tides in the far field where they may then propagate thousands of kilometers from the source [3]. An outstanding question asks how the energy from these waves ultimately cascades from large to small scale where it may be dissipated, thus closing this branch of the oceanic energy budget. Several possibilities have been explored, including dissipation when the internal tide interacts with rough bottom topography, with the continental slopes and shelves, and with mean flows and eddies (for a recent review, see MacKinnon et al. [4]). It has also been suggested that, away from topography and background flows, internal modes may be dissipated due to nonlinear wave-wave interactions including the case of triadic resonant instability (hereafter TRI), in which a pair of “sibling” waves grow out of the background noise field through resonant interactions with the “parent” wave”

see reference [2]
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Australia Bushfire Causes

The Indian Ocean Dipole (IOD) and the El Nino Southern Oscillation (ENSO) are the primary natural climate variability drivers impacting Australia. Contrast that to AGW as the man-made driver. These two categories of natural and man-made causes form the basis of the bushfire attribution discussion, yet the naturally occurring dipoles are not well understood. Chapter 12 of the book describes a model for ENSO; and even though IOD has similarities to ENSO in terms of its dynamics (a CC of around 0.3) the fractional impact of the two indices is ultimately responsible for whether a temperature extreme will occur in a region such as Australia (not to mention other indices such as MJO and SAM).  

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