# Limits of Predictability?

A decade-old research article on modeling equatorial waves includes this introductory passage:

“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. “

https://doi.org/10.1029/2012JC007879

From my experience if it is relatively easy to generate a fit to data via a nonlinear model then it also may be easy to diverge from the fit with a small structural perturbation, or to come up with an alternative fit with a different set of parameters. This makes it difficult to establish an iron-clad cross-validation.

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-2π 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-2π 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-2π 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?

# Gerstner waves

An exact solution for equatorially trapped waves
Adrian Constantin, JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, C05029, doi:10.1029/2012JC007879, 2012

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. A good illustration of this feature is the fact that there is only one known explicit exact solution of the (nonlinear) governing equations for periodic two-dimensional traveling gravity water waves. This solution was first found 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 Niño phenomenon (see also the discussion in Cushman-Roisin and Beckers [2011]).

modulo-2π and Berry phase

# Darwin

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.

So which curve below is the time-series data of atmospheric pressure at Darwin and which is the Laplace’s Tidal Equation (LTE) model calibrated from dLOD measurements?

• (bottom, red) = ?
• (top, blue) = ??

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 =>

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

# Cross-validation

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.

# Lunar Eclipse Unremarkable

Neil deGrasse Tyson calls it

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.

# Response to the lunar cycle

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:

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 & 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 USA5, 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 :

1. Tidal factors are numerous so a measure such as dLOD is critical for calibrating the forcing.
2. Use the knowledge of a seasonal impulse, a la the spring predictability barrier, to advantage, while considering the temporal aliasing that it will cause.
3. 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

References

1. Lin, J., Qian, T. Switch Between El Nino and La Nina is Caused by Subsurface Ocean Waves Likely Driven by Lunar Tidal ForcingSci Rep 9, 13106 (2019).
2. Pukite, P., Coyne, D., Challou, D. (2018). Mathematical Geoenergy: Discovery, Depletion, and Renewal.  AGU/Wiley. DOI:10.1002/9781119434351
3. https://psl.noaa.gov/gcos_wgsp/Timeseries/Data/nino34.long.anom.data
4. Delplace, Pierre, J. B. Marston, and Antoine Venaille. “Topological origin of equatorial waves.” Science 358.6366 (2017): 1075-1077. https://arxiv.org/pdf/1702.07583.pdf
5. Shirley, James H. “Solar System Dynamics and Multiyear Droughts of the Western USA.” arXiv preprint arXiv:2112.02186 (2021).
6. Perigaud, Claire. “Importance of the Earth-Moon system for reducing uncertainties in climate modelling and monitoring.” (2009).
Perigaud, Claire. “ENSO Modulated by Lunar Cycles” (2009). https://www.aviso.altimetry.fr/fileadmin/documents/OSTST/2009/poster/Perigaud.pdf
7. Marcus, S. L. (2016). “Does an Intrinsic Source Generate a Shared Low-Frequency Signature in Earth’s Climate and Rotation Rate?,” Earth Interactions20(4), 1-14. Retrieved May 12, 2022, from https://journals.ametsoc.org/view/journals/eint/20/4/ei-d-15-0014.1.xml
8. Ray, Richard D., and Svetlana Y. Erofeeva. “Long‐period tidal variations in the length of day.” Journal of Geophysical Research: Solid Earth 119.2 (2014): 1498-1509. https://ir.library.oregonstate.edu/downloads/gx41mk60z
9. Ding, H., & Chao, B. F. (2018). Application of stabilized AR-z spectrum in harmonic analysis for geophysics. Journal of Geophysical Research: Solid Earth, 123, 8249– 8259. https://doi.org/10.1029/2018JB015890
10. Pukite, P. R., D. Coyne, and D. J. Challou. “Ephemeris calibration of Laplace’s tidal equation model for ENSO.” AGU Fall Meeting Abstracts. Vol. 2018. 2018. https://www.proquest.com/openview/1397f7741d3f81e0b26fb16b0c35da75/1

# Mf vs Mm

In an earlier post, the observation was that ENSO models may not be unique due to the numerous possibilities provided by nonlinear math. This was supported by the fact that a tidal forcing model based on the Mf (13.66 day) tidal factor worked equally as well as a Mm (27.55 day) factor. This was not surprising considering that the aliasing against an annual impulse gave a similar repeat cycle — 3.8 years versus 3.9 years. But I have also observed that mixing the two in a linear fashion did not improve the fit much at all, as the difference created a long interference cycle which isn’t observed in the ENSO time series data. But then thinking in terms of the nonlinear modulation required, it may be that the two factors can be combined after the LTE solution is applied.

# PDO is even, ENSO is odd

As the quality of the tidally-forced ENSO model improves, it’s instructive to evaluate its common-mode mechanism against other oceanic indices. So this is a re-evaluation of the Pacific Decadal Oscillation (PDO), in the context of non-autonomous solutions such as generated via LTE modulation. In particular, in this note we will clearly delineate the subtle distinction that arises when comparing ENSO and PDO. As background, it’s been frequently observed and reported that the PDO shows a resemblance to ENSO (a correlation coefficient between 0.5 and 0.6), but also demonstrates a longer multiyear behavior than the 3-7 year fluctuating period of ENSO, hence the decadal modifier.

A hypothesis based on LTE modulation is that decadal behavior arises from the shallowest modulation mode, and one that corresponds to even symmetry (i.e. cos not sin). So for a model that was originally fit to an ENSO time-series, it is anticipated that the modulation trending to a more even symmetry will reveal less rapid fluctuations — or in other words for an even f(x) = f(-x) symmetry there will be less difference between positive and negative excursions for a well-balanced symmetric input time-series. This should then exaggerate longer term fluctuations, such as in PDO. And for odd f(x) = -f(-x) symmetry it will exaggerate shorter term fluctuations leading to more spikiness, such as in ENSO.

# Moonfall and glacially slow geophysics advances

This blog is late to the game in commenting on the physics of the Hollywood film Moonfall — but does that really matter? Geophysics research and glacially slow progress seem synonymous at this point. In social media, unless one jumps on the event of the day within an hour, it’s considered forgotten. However, difficult problems aren’t unraveled quickly, and that’s what he have when we consider the Moon’s influence on the Earth’s geophysics. Yes, tides are easy to understand, but any other impact of the Moon is considered warily, perhaps over the course of decades, not as part of the daily news & entertainment cycle.

My premise: The movie Moonfall is a more pure climate-science-fiction film than Don’t Look Up. Discuss.

# Highly-Resolved Models of NAO and AO Indices

Revisiting earlier modeling of the North Atlantic Oscillation (NAO) and Arctic Oscillation (AO) indices with the benefit of updated analysis approaches such as negative entropy. These two indices in particular are intimidating because to the untrained eye they appear to be more noise than anything deterministically periodic. Whereas ENSO has periods that range from 3 to 7 years, both NAO and AO show rapid cycling often on a faster-than-annual pace. The trial ansatz in this case is to adopt a semi-annual forcing pattern and synchronize that to long-period lunar factors, fitted to a Laplace’s Tidal Equation (LTE) model.

Start with candidate forcing time-series as shown below, with a mix of semi-annual and annual impulses modulating the primarily synodic/tropical lunar factor. The two diverge slightly at earlier dates (starting at 1880) but the NAO and AO instrumental data only begins at the year 1950, so the two are tightly correlated over the range of interest.

The intensity spectrum is shown below for the semi-annual zone, noting the aliased tropical factors at 27.32 and 13.66 days standing out.

The NAO and AO pattern is not really that different, and once a strong LTE modulation is found for one index, it also works for the other. As shown below, the lowest modulation is sharply delineated, yet more rapid than that for ENSO, indicating a high-wavenumber standing wave mode in the upper latitudes.

The model fit for NAO (data source) is excellent as shown below. The training interval only extended to 2016, so the dotted lines provide an extrapolated fit to the most recent NAO data.

Same for the AO (data source), the fit is also excellent as shown below. There is virtually no difference in the lowest LTE modulation frequency between NAO and AO, but the higher/more rapid LTE modulations need to be tuned for each unique index. In both cases, the extrapolations beyond the year 2016 are very encouraging (though not perfect) cross-validating predictions. The LTE modulation is so strong that it is also structurally sensitive to the exact forcing.

Both NAO and AO time-series appear very busy and noisy, yet there is very likely a strong underlying order due to the fundamental 27.32/13.66 day tropical forcing modulating the semi-annual impulse, with the 18.6/9.3 year and 8.85/4.42 year providing the expected longer-range lunar variability. This is also consistent with the critical semi-annual impulses that impact the QBO and Chandler wobble periodicity, with the caveat that group symmetry of the global QBO and Chandler wobble forcings require those to be draconic/nodal factors and not the geographically isolated sidereal/tropical factor required of the North Atlantic.

It really is a highly-resolved model potentially useful at a finer resolution than monthly and that will only improve over time.

(as a sidenote, this is much better attempt at matching a lunar forcing to AO and jet-stream dynamics than the approach Clive Best tried a few years ago. He gave it a shot but without knowledge of the non-linear character of the LTE modulation required he wasn’t able to achieve a high correlation, achieving at best a 2.4% Spearman correlation coefficient for AO in his Figure 4 — whereas the models in this GeoenergyMath post extend beyond 80% for the interval 1950 to 2016! )