The SubStack authoring interface has good math equation mark-up, convenient graphics embedding, and an excellent footnoting system. On first pass, it only lacks control over font color.
The articles are focused on applying neural network cross-validation to ENSO and AMO modeling, as suggested previously. I haven’t completely explored the configuration space but one aspect that may becoming clear is the value of wavelet neural networks (WNN) for time-series analysis. The WNN approach seems much more amenable to extracting sinusoidal modulation of the input-to-output mapping — trained on a rather short interval and then cross-validated out-of-band. The Mexican hat wavelet (2nd derivative of a Gaussian) as an activation function in particular locks in quickly to an LTE modulation that took longer to find with the custom search software I have developed at GitHub. I think the reason for the efficiency is that it’s optimizing to a Taylor’s series expansion of the input terms, a classic nonlinear expansion that NN’s excel at.
The following training run using the Mexican hat activation and ADAM optimizer is an eye-opener, as it achieved an admirable fit within a minute of computation.
The GREEN on BLUE is training on NINO4 data over two end-point intervals, with the RED cross-validation over the out-of-band region. The correlation coefficient is 0.34, which is impressive considering the nature of the waveform. Clearly there is similarity.
Moreover, if we compare the model fit to data via the WNN against the LTE harmonics approach, you can also see where the two fare equally poorly. Below in the outer frame is the NINO4 LTE fit with the YELLOW arrow pointing downward at a discrepancy (a peak in the data not resolved in the fit). In comparison the yellow-bordered inset shows the same discrepancy on the WNN training run. So the fingerprints essentially match with no coaching.
The neural net chain is somewhat deep with 6 layers, but I think this is needed to expand to the higher-order terms in the Taylor’s series. In the directed graph below, L01 is the input tidal forcing and L02 is the time axis (with an initial very low weighting).
It also appears temporally stationary across the entire time-span, so that the WNN temporal contribution appears minimal.
In a previous fit the horizontal striations (indicating modulation factor at a forcing level) matched with the LTE model, providing further evidence that the the WNN was mapping to an optimal modulation.
Sorry to have to point this out, but it’s not my fault that geophysicists and climatologists can’t perform controlled experiments to test out various hypotheses. It’s not their fault either. It’s all nature’s decision to make gravitational forces so weak and planetary objects so massive to prevent anyone from scaling the effect to laboratory size to enable a carefully controlled experiment. One can always create roughly-equivalent emulations, such as a magnetic field experiment (described in the previous blog post) and validate a hypothesized behavior as a controlled lab experiment. Yet, I suspect that this would not get sufficient buy-in, as it’s not considered the actual real thing.
And that’s the dilemma. By the same token that analog emulators will not be trusted by geophysicists and climatologists, so too scientists from other disciplines will remain skeptical of untestable claims made by earth scientists. If nothing definitive comes out of a thought experiment that can’t be reproduced by others in a lab, they remain suspicious, as per their education and training.
It should therefore work both ways. As featured in the previous blog post, the model of the Chandler wobble forced by lunar torque needs to be treated fairly — either clearly debunked or considered as an alternative to the hazy consensus. ChatGPT remains open about the model, not the least bit swayed by colleagues or tribal bias. As the value of the Chandler wobble predicted by the lunar nodal model (432.7 days) is so close to the cited value of 433 days, as a bottom-line it should be difficult to ignore.
There are other indicators in the observational data to further substantiate this, see Chandler Wobble Forcing. It also makes sense in the context of the annual wobble.
As it stands, the lack of an experiment means a more equal footing for the alternatives, as they are all under equal amounts of suspicion.
Same goes for QBO. No controlled experiment is possible to test out the consensus QBO models, despite the fact that the Plumb and McEwan experiment is claimed to do just that. Sorry, but that experiment is not even close to the topology of a rotating sphere with a radial gravitational force operating on a gas. It also never predicted the QBO period. In contrast, the value of the QBO predicted by the lunar nodal model (28.4 months) is also too close to the cited value of 28 to 29 months to ignore. This also makes sense in the context of the semi-annual oscillation (SAO) located above the QBO .
Both the Chandler wobble and the QBO have the symmetry of a global wavenumber=0 phenomena so therefore only nodal cycles allowed — both for lunar and solar.
Next to ENSO. As with LOD modeling, this is not wavenumber=0 symmetry, as it must correspond to the longitude of a specific region. No controlled experiment is possible to test out the currently accepted models, premised as being triggered by wind shifts (an iffy cause vs. effect in any case). The mean value of the ENSO predicted by the tidal LOD-caibrated model (3.80 years modulated by 18.6 years) is too close to the cited value of 3.8 years with ~200 years of paleo and direct measurement to ignore.
In BLUE below is the LOD-calibrated tidal forcing, with linear amplification
In BLUE again below is a non-linear modulation of the tidal forcing according to the Laplace’s Tidal Equation solution, and trained on an early historical interval. This is something that a neural network should be able to do, as it excels at fitting to non-linear mappings that have a simple (i.e. low complexity) encoding — in this case it may be able to construct a Taylor series expansion of a sinusoidal modulating function.
The neural network’s ability to accurately represent a behavior is explained as a simplicity bias — a confounding aspect of machine learning tools such as ChatGPT and neural networks. The YouTube video below explains the counter-intuitive notion of how a NN with a deep set of possibilities tends to find the simplest solution and doing this without over-fitting the final mapping.
So that deep neural networks are claimed to have a built-in Occam’s Razor propensity, finding the most parsimonious input-output mappings when applied to training data. This is spot on with what I am doing with the LTE mapping, but bypassing the NN with a nonlinear sinusoidal modulation optimally fit on training data by a random search function.
I am tempted to try a NN on the ENSO training set as an experiment and see what it finds.
April 2, 2023
“I am tempted to try a NN on the ENSO training set as an experiment and see what it finds.”
“Nonlinear aspects plays a major role in the understanding of fluid flows. The distinctive fact that in nonlinear problems cause and effect are not proportional opens up the possibility that a small variation in an input quantity causes a considerable change in the response of the system. Often this type of complication causes nonlinear problems to elude exact treatment. “
This doesn’t mean we don’t keep trying. Applying the dLOD calibration approach to an applied forcing, we can model ENSO via the NINO34 climate index across the available data range (in YELLOW) in the figure below (parameters here)
The lower right box is a modulo-2ฯ reduction of the tidal forcing as an input to the sinusoidal LTE modulation, using the decline rate (per month) as the divisor. Why this works so well per month in contrast to per year (where an annual cycle would make sense) is not clear. It is also fascinating in that this is a form of amplitude aliasing analogous to the frequency aliasing that also applies a modulo-2ฯ folding reduction to the tidal periods less than the Nyquist monthly sampling criteria. There may be a time-amplitude duality or Lagrangian particle-relabeling in operation that has at its central core the trivial solutions of Navier-Stokes or Euler differential equations when all segments of forcing are flat or have a linear slope. Trivial in the sense that when a forcing is flat or has a 1st-order slope, the 2nd derivatives due to divergence in the differential equations vanish (quasi-static). This means that only the discontinuities, which occur concurrently with the annual ENSO predictability barrier, need to be treated carefully (the modulo-2ฯ folding could be a topological Berry phase jump?). Yet, if these transitions are enhanced by metastable interface instabilities as during thermocline turn-over then the differential equation conditions could be transiently relaxed via a vanishing density difference. Much happens during a turn-over, but it doesn’t last long, perhaps indicating a geometric phase. MV Berry also discusses phase changes in the context of amphidromic tidal singularities here.
Suffice to say that the topological properties of reduced dimension volumes and at interfaces remain mysterious. The main takeaway is that a working NINO34-fitted ENSO model is produced, and if not here then somewhere else a machine-learning algorithm will discover it.
The key next step is to apply the same tidal forcing to an AMO model, taking care not to change the tidal factors enough to produce a highly sensitive nonlinear response in the LTE model. So we retain an excluded interval from training (in YELLOW below) and only adjust the LTE parameters for the region surrounding this zone during the fitting process (parameters here).
The cross-validation agreement is breathtakingly good in the excluded (out-of-band) training interval. There is zero cross-correlation between the NINO34 and AMO time-series to begin with so that this is likely revealing the true emergent characteristics of a tidally forced mechanism.
As usual all the introductory work is covered in Mathematical Geoenergy
A community peer-review contributed to a recent QBO article is here and PDF here. The same question applies to QBO as ENSO or AMO: is it possible to predict future behavior? Is the QBO model less sensitive to input since the nonlinear aspect is weaker?
Nonlinear aspects plays a major role in the understanding of fluid flows. The distinctive fact that in nonlinearproblems cause and effect are not proportional opens up thepossibility that a small variation in an input quantity causes aconsiderable change in the response of the system. Often thistype of complication causes nonlinear problems to eludeexact treatment. A good illustration of this feature is the factthat there is only one known explicit exact solution of the(nonlinear) governing equations for periodic two-dimensional traveling gravity water waves. This solution was firstfound in a homogeneous fluid by Gerstner
These are trochoidal waves
Even within the context of gravity waves explored in the references mentioned above, a vertical wall is not allowable. This drawback is of special relevance in a geophysical context since [cf. Fedorov and Brown, 2009] the Equator works like a natural boundary and equatorially trapped waves, eastward propagating and symmetric about the Equator, are known to exist. By the 1980s, the scientific community came to realize that these waves are one of the key factors in explaining the El Niรฑo phenomenon (see also the discussion in Cushman-Roisin and Beckers [2011]).
It turns out that the Darwin location of the Southern Oscillation Index (SOI) dipole is brilliantly easy to behaviorally model on it’s own.
The input forcing is calibrated to the differential length-of-day (LOD) with a correlation coefficient of 0.9997, and only a few terms are required to capture the standing-wave modes corresponding to the ENSO dipole.
As a bonus, the couple of years outside of the training interval are extrapolated from the model. This shouldn’t be hard for climate scientists, …. or is it still too difficult?
If that isn’t enough to discriminate between the two, the power spectra of the LTE mapping to model and to data is shown below. This identifies a couple of the lower frequency modulations as strong peaks and a few weaker higher harmonic peaks that sharpen the model’s detail. This shows that the data’s behavior possesses a high amount of order not apparent in the time series.
Poll on Twitter =>
So which curve is the time-series data of atmospheric pressure at Darwin in Australia and which is the LTE model calibrated from length-of-day (LOD) measurements?
Why isn’t the Tahiti time-series included since that would provide additional signal discrimination via a differential measurement as one should be the complement of the other? It should accentuate the signal and remove noise (and any common-mode behavior) if the Darwin and Tahiti are perfect anti-nodes for all standing-wave modes. However, it appears that only the main ENSO standing-wave mode is balanced in all modes.
In that case, the Darwin set alone works well. Mastodon
I doubt many climate scientists have taken a class in limnology, the study of freshwater lakes. I have as an elective science course in college. They likely have missed the insight of thinking about the thermocline and how in dimictic upper-latitude lakes the entire lake overturns twice a year as the imbalance of densities due to differential heating or cooling causes a buoyancy instability.
Cross-validation is essentially the ability to predict the characteristics of an unexplored region based on a model of an explored region. The explored region is often used as a training interval to test or validate model applicability on the unexplored interval. If some fraction of the expected characteristics appears in the unexplored region when the model is extrapolated to that interval, some degree of validation is granted to the model.
This is a powerful technique on its own as it is used frequently (and depended on) in machine learning models to eliminate poorly performing trials. But it gains even more importance when new data for validation will take years to collect. In particular, consider the arduous process of collecting fresh data for El Nino Southern Oscillation, which will take decades to generate sufficient statistical significance for validation.
So, what’s necessary in the short term is substantiation of a model’s potential validity. Nothing else will work as a substitute, as controlled experiments are not possible for domains as large as the Earth’s climate. Cross-validation remains the best bet.
Using as few independent parameters as possible, the difference in characterizing the temporal behavior of ENSO and AMO may amount to a standing-wave phase change. Noted earlier that ENSO and AMO can be derived from a common lunisolar forcing — and have now found that the LTE modulation is not that fundamentally different between the two.
The (nearly) common forcing
with the applied LTE of a 180ยฐ phase difference
leads to adequately fitted models to the respective time series
The fact that the fundamental (and 7th harmonic) are aligned between ENSO and AMO strongly suggest that the standing-wave wavenumbers are not governed by the basin geometry but are more of a global characteristic that remains coherent across the land masses. The Atlantic basin has a smaller width than the Pacific so intuitively one might have predicted unique wavenumbers that would fit within the bounding coastlines, but this is perhaps not the case.
Instead, the LTE modulation wraps around the earth and produces an anti-phase relationship in keeping with the approximately 180ยฐ longitudinal difference between the Atlantic and Pacific.
ENSO ~ sin (k F(t))
AMO ~sin (k F(t) +ย ฯ + ฯ)
Any additional phase shift ฯ can also easily produce the anomalously large multidecadal variations in the AMO due to the biasing properties of the sinusoidal LTE modulation.
Just a matter of time until machine-learning algorithms start discovering these patterns. But, alas, they may not know how to deal with the findings
NdGT has a point — you do see the earth’s shadow moving across the moon, but once covered, a #lunarEclipse just looks like a duller moon (similar “new moons” are also observed like clockwork and thus take the excitement out of it). Yet the alignment of tidal forces does a number on the Earth’s climate that is totally cryptic and thus overlooked. Perhaps old Dr. Neil would find more interesting tying lunar cycles to climate indices such as ENSO and the Indian Ocean Dipole? It’s all based on geophysical fluid dynamics. Oh, and a bonus — discriminate on the variability of IOD and there’s the underlying AGW trend!
BTW, a key to this IOD model fit is to apply dual annual impulses, one for each monsoon season, summer and winter. Whereas, ENSO only has the spring predictability barrier.
The premise of the paper is that the ocean will show modulation of mixing with a cycle of ~18 years corresponding to the 18.6-year lunar declination cycle. That may indeed be the case, but it likely pales in comparison to the other so-called long-period tidal cycles. In particular, every ~2 weeks the moon makes a complete north-south-north declination cycle that likely has a huge impact on the climate as it sloshes the subsurface thermocline (cite the paper by Lin & Qian1). Unfortunately, this much shorter cycle is not directly observed in the observational data, making it a challenge to determine how the pattern manifests itself. In the following, I will describe how this is accomplished, referring to the complete derivation found in Chapter 12 of Mathematical Geoenergy2.
Consider that the 2-week lunar declination cycle is observed very clearly in the Earth’s rotational speed, measured in terms of small transient changes in the length of day (LOD). From the IERS site, we can plot the differential LOD (dLOD) and fit to the known tidal factors, leaving a clean closed-form signal that one can use as a forcing function to evaluate the ocean response, in this case comparing it to the well-defined ENSO climate index.
The 18.6-year nodal cycle can be seen in the modulation of the cyclic dLOD data. At a higher resolution, the comparison is as follows:
As the high-resolution dLOD measurement only goes back to 1962, but the ENSO NINO34 time-series goes back to 1870 3, the closed form model fit can be used across the entire duration of a climate index (assuming that tidal factors are stationary).
To do that, we first make the assumption that the tidal cycle is modulated on an annual cycle, corresponding to the well-known “spring predictability barrier”. So, by integrating a sequence of May impulses against the value of the tidal forcing at that point, the following time series is generated.
Obviously, this does not match the ENSO NINO34 signal, but assuming that the subsurface response is non-linear (derivation in cite #2 below) and creates standing wave-modes based on the geometry of the ocean basin, then one can use a suitable transformation to potentially extract the pattern. The best approach based on the solution to the shallow-water wave model (i.e. Laplace’s Tidal Equations) is to map the input forcing (graph above) to the output corresponding to the NINO34 index, using a Fourier series expansion.
The result is the Laplace’s Tidal Equation (LTE) modulation spectra, shown below in a particular cross-validation configuration. Here, the NINO34 data is split into 2 halves, one time-series taken from 1870-1945 and the second from 1945-2020. The spectra were calculated individually and then multiplied point-by-point to identify long-lived stationary standing-wave nodes in the modulation. Thus, it isolates modulations that are common to each interval.
This is a log-plot, so the peak excursions shown are statistically significant and so can be modeled by a handful of quantifiable standing-wave modulations. The lowest wavenumber modulations are associated with the ENSO dipole modes and the higher wavenumber modulations are potentially associated with tropical instability waves (TIW)2.
As a final step, by applying this set of modulations to the lunisolar forcing (the blue chart above), a fit to the NINO34 time-series results. The chart shown below is a very good fit and can be cross-validated via several approaches10.
The mix of incommensurate tidal factors, the annual impulse, and a nonlinear response function is what causes the highly erratic nature of the ENSO waveform. It is neither chaotic nor random, as some researchers claim but instead is deterministically tied to the tidal and annual cycles, much like conventional tidal cycles have proven over the course of time.
To further quantify the decomposition of the tidal factors that force both the dLOD and the sloshing ENSO response, the paper by Ray and Erofeeva is vital8. When trying to understand the assignment of frequencies, note that after the annual impulse is applied, the known tidal factors corresponding to such tidal factors labelled Mf, Mm, etc get shifted from normal positions due to signal aliasing (see chart below in gray). This is a confusing factor to those who have not encountered aliasing before. As an example, the long-term modulation (>100 years) displayed in the blue chart above is due to the aliased 9.133 day Mt tidal factor, which almost synchronizes with the annual cycle, but the amount it is off leads to a gradual modulation in the forcing — so overall confusing in that a 9 day cycle could cause multidecadal changes.
Ding &ย Chao9 provide an independent analysis of LOD that provides a good cross-check to the non-aliased cross-factors. It may be possible to use lunar ephemeris data to calibrate the forcing but that adds degrees-of-freedom that could lead to over-fitting 10.
The reason that Lin & Qian were not able to further substantiate their claim of tidal forcing lies in that they could not associate the seasonal aliasing and a nonlinear mapping against their observations, only able to demonstrate the cause and effect of tidal forcing on the thermocline and thereby ruling out wind forcing. Other sources to cite are “Topological origin of equatorial waves” 4 and “Solar System Dynamics and Multiyear Droughts of the Western USA” 5, the latter discussing the impact of axial torques on the climate. Researchers at NASA JPL including J.H. Shirley, C. Perigaud6, and S.L. Marcus7 have touched on the LOD, lunar, ENSO connection over the years.
Bottom-line take aways :
Tidal factors are numerous so a measure such as dLOD is critical for calibrating the forcing.
Use the knowledge of a seasonal impulse, a la the spring predictability barrier, to advantage, while considering the temporal aliasing that it will cause.
The solution to the geophysical fluid dynamics produces a non-linear response, so clever transform techniques such as Fourier series are useful to isolate the pattern.
A recent citation to use: Pukite, Paul. “Nonlinear long-period tidal forcing with application to ENSO, QBO, and Chandler wobble.” EGU General Assembly Conference Abstracts. 2021. https://ui.adsabs.harvard.edu/abs/2021EGUGA..2310515P
Delplace, Pierre, J. B. Marston, and Antoine Venaille. “Topological origin of equatorial waves.”ย Scienceย 358.6366 (2017): 1075-1077. https://arxiv.org/pdf/1702.07583.pdf
Shirley, James H. “Solar System Dynamics and Multiyear Droughts of the Western USA.”ย arXiv preprint arXiv:2112.02186ย (2021).
Ray, Richard D., and Svetlana Y. Erofeeva. “Longโperiod tidal variations in the length of day.”ย Journal of Geophysical Research: Solid Earthย 119.2 (2014): 1498-1509. https://ir.library.oregonstate.edu/downloads/gx41mk60z
Ding, H., &ย Chao, B. F.ย (2018).ย Application of stabilized AR-zย spectrum in harmonic analysis for geophysics.ย Journal of Geophysical Research: Solid Earth,ย 123,ย 8249โย 8259.ย https://doi.org/10.1029/2018JB015890
