To see how startingly aligned this mapping is, consider the SLH readings from Ft. Denison in Sydney Harbor. The interval from 1980 to 2012 is shown below, along with a fit used recently to model ENSO.

*(click to expand chart)*

I chose a shorter interval to somewhat isolate the trend from a secular sea-level rise due to AGW. The last point is 2012 because tide gauge data collection ended then.

As cross-validation, this fit is extrapolated backwards to show how it matches the historic SOI cycles

Much of the fine structure aligns well, indicating that *intrinsically *the dynamics behind sea-level-height at this scale are due to ENSO changes, associated with the inverted barometer effect. The SOI is essentially the pressure differential between Darwin and Tahiti, so the prevailing atmospheric pressure occurring during varying ENSO conditions follows the rising or lowering Sydney Harbor sea-level in a synchronized fashion. The change is 1 cm for a 1 mBar change in pressure, so that with the SOI extremes showing 14 mBar variation at the Darwin location, this accounts for a 14 cm change in sea-level, roughly matching that shown in the first chart. Note that being a differential measurement, SOI does not suffer from long-term secular changes in trend.

Yet, the unsaid implication in all this is that not only are the daily variations in SLH due to lunar and solar cyclic tidal forces, but so are these monthly to decadal variations. The longstanding impediment is that oceanographers have not been able to solve Laplace’s Tidal Equations that reflected the non-linear character of the ocean’s response to the long-period lunisolar forcing. Once that’s been analytically demonstrated, we can observe that both SLH and ENSO share essentially identical lunisolar forcing (see chart below), arising from that same common-mode linked mechanism.

Many geographically located tidal gauge readings are available from the Permanent Service for Mean Sea Level (PSMSL) repository so I can imagine much can be done to improve the characterization of ENSO via SLH readings.

**REFERENCES**

[1] F. Zou, R. Tenzer, H. S. Fok, G. Meng and Q. Zhao, “The Sea-Level Changes in Hong Kong From Tide-Gauge Records and Remote Sensing Observations Over the Last Seven Decades,” in *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 14, pp. 6777-6791, 2021, doi: 10.1109/JSTARS.2021.3087263.

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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! )*

Cryptography in its common use applies a *key *to enable a user to decode a scrambled data stream according to the instruction pattern embedded within the key. If diffraction-based crystallography required a complex unknown key to decode from reciprocal space, it would seem hopeless, but that’s exactly what we are dealing with when trying to decipher climate dipole time-series -— we don’t know what the decoding key is. If that’s the case, no wonder climate science has never made any progress in modeling ENSO, as it’s an existentially difficult problem.

The breakthrough is in identifying that an analytical solution to Laplace’s tidal equations (LTE) provides a crystallography+cryptography analog in which we can make some headway. The challenge is in identifying the decoding key (an unknown forcing) that would make the reciprocal-space inversion process (required for LTE demodulation) straightforward.

According to the LTE model, the forcing has to be a combination of tidal factors mixed with a seasonal cycle (stages 1 & 2 in the figure above) that would enable the last stage (Fourier series a la diffraction inversion) to be matched to empirical observations of a climate dipole such as ENSO.

The forcing key used in an ENSO model was described in the last post as a predominately *Mm*-based lunar tidal factorization as shown below, leading to an excellent match to the NINO34 time series after a minimally-complex LTE modulation is applied.

Critics might say and justifiably so, that this is potentially an over-fit to achieve that good a model-to-data correlation. There are too many degrees of freedom (DOF) in a tidal factorization which would allow a spuriously good fit depending on the computational effort applied (see **Reference 1** at the end of this post).

Yet, if the forcing key used in the ENSO model was **reused as is **in fitting an independent climate dipole, such as the AMO, and this same key required little effort in modeling AMO, then the over-fitting criticism is invalidated. What’s left to perform is finding a distinct low-DOF LTE modulation to match the AMO time-series as shown below.

This is an example of a *common-mode cross-validation* of an LTE model that I originally suggested in an AGU paper from 2018. Invalidating this kind of analysis is exceedingly difficult as it requires one to show that the erratic cycling of AMO can be randomly created by a few DOF. In fact, a few DOFs of sinusoidal factors to reproduce the dozens of AMO peaks and valleys shown is virtually impossible to achieve. I leave it to others to debunk via an independent analysis.

addendum: LTE modulation comparisons, essentially the wavenumber of the diffraction signal:

This is the *forcing *power spectrum showing the principal * Mm *tidal factor term at period 3.9 years, with nearly identical spectral profiles for both ENSO and AMO.

According to the precepts of cryptography, decoding becomes straightforward once one knows the key. Similarly, nature often closely guards its secrets, and until the key is known, for example as with DNA, climate scientists will continue to flounder.

** References **

- Chao, B. F., & Chung, C. H. (2019). On Estimating the Cross Correlation and Least Squares Fit of One Data Set to Another With Time Shift.
*Earth and Space Science*, 6, 1409–1415. https://doi.org/10.1029/2018EA000548

“*For example, two time series with predominant linear trends (very low DOF) can have a very high ρ (positive or negative), which can hardly be construed as an evidence for meaningful physical relationship. Similarly, two smooth time series with merely a few undulations of similar timescale (hence low DOF) can easily have a high apparent ρ just by fortuity especially if a time shift is allowed. On the other hand, two very “erratic” or, say, white time series (hence high DOF) can prove to be significantly correlated even though their apparent ρ value is only moderate. The key parameter of relevance here is the DOF: A relatively high ρ for low DOF may be less significant than a relatively low ρ at high DOF and vice versa.*“

From the discussion section of a paper: *“Attributing correlation skill of dynamical GCM precipitation forecasts to statistical ENSO teleconnection using a set-theory-based approach”*

Is this a joke?

]]>“Nonlinear long-period tidal forcing with application to ENSO, QBO, and Chandler wobble”, EGU General Assembly Conference Abstracts, 2021, EGU21-10515

ui.adsabs.harvard.edu/abs/2021EGUGA..2310515P/abstract

“Nonlinear Differential Equations with External Forcing”, ICLR 2020 Workshop DeepDiffEq

https://openreview.net/forum?id=XqOseg0L9Q

“Mathematical Geoenergy: Discovery, Depletion, and Renewal”, John Wiley & Sons, 2019, chapter 12: “Wave Energy”

https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch12

“Ephemeris calibration of Laplace’s tidal equation model for ENSO”, AGU Fall Meeting 2018,

https://www.essoar.org/doi/abs/10.1002/essoar.10500568.1

“Biennial-Aligned Lunisolar-Forcing of ENSO: Implications for Simplified Climate Models”,

AGU Fall Meeting 2017, https://www.essoar.org/doi/abs/10.1002/essoar.b1c62a3df907a1fa.b18572c23dc245c9.1

“Analytical Formulation of Equatorial Standing Wave Phenomena: Application to QBO and ENSO”, AGU Fall Meeting Abstracts 2016, OS11B-04,

ui.adsabs.harvard.edu/abs/2016AGUFMOS11B..04P/abstract

Given that I have worked on this topic persistently over this span of time, I have gained considerable insight into how straightforward it has become to generate relatively decent fits to climate dipoles such as ENSO. Paradoxically, this is both good and bad. It’s good because the model’s recipe is algorithmically simply described in terms of plausibility and parsimony. That’s largely because it’s a straightforward non-linear extension of a conventional tidal analysis model. However that non-linearity opens up the possibility for many similar model fits that are equally good, yet difficult to discriminate between. So it’s bad in the sense that I can come to an impasse in selecting the “best” model.

This is oversimplifying a bit but the framing issue is if you knew the answer was 72, but have a hard time determining whether the question being posed was one of 2×36, 3×24, 4×18, 6×12, 8×9, 9×8, 12×6, 18×4, 24×3, or 36×2. Each gives the right answer, but potentially not the right mechanism. This is a fundamental problem with non-linear analysis.

A conventional tidal analysis by itself is just a few fundamental tidal factors (exactly 4) but made devastatingly accurate by the introduction of 2nd-order harmonics and cross-harmonics. All these harmonics are generated by non-linear effects but the frequency spectrum is so clean and distinct for a sea-level-height (SLH) time-series that the equivalent solution to *k × F* = 72 is essentially a scaling identification problem where the *k* is the scale factor for the corresponding cyclic tidal factors *F*.

Yet, by applying the non-linear LTE solution to the problem of modeling ENSO, we quickly discover that the algorithm is a wickedly effective harmonics and cross-harmonics generator. Any number of combinations of harmonics can develop an adequate fit depending on the variable LTE modulation applied. So it could be a small LTE modulation mixed with a wide assortment of tidal factors (the 2×36 case) or it could be a large LTE modulation mixed with a minimum of tidal factors (the 18×4 case). Or it could be something in between (e.g. the 8×9 case). This is all a result of the sine-of-a-sine non-linearity of the LTE formulation, related to the Mach-Zehnder modulation used in optical cryptography applications. The latter bit is the hint that things may not be unambiguously decoded given the fact that M-Z has been discovered to be nature’s own built-in encryption device.

However, there remains lots of light at the end of this tunnel, as I have also discovered that the tidal factor spread is likely largely governed by a single lunar tidal constituent, the 27.55 day anomalistic **Mm** cycle interfering with an annual impulse. That’s essentially 2 of the 4 tidal factors, with the other 2 lunar factors providing a 2nd-order correction. For the longest time I had been focused on the 13.66 day tropical **Mf **cycle as that also lead to a decent fit over the years, specifically since the first beat harmonic of the **Mf **cycle with the annual impulse is 3.8 years while the **Mm** cycle is 3.9 years. These two terms are close enough that they only go out-of-phase after ~130 years, which is the extent of the ENSO time-series. Only when you try to simplify a model fit by iterating over the space of factor combinations will you discover the difference between 3.8 and 3.9.

In terms of geophysics, the **Mf **factor is a tractional tidal forcing operating parallel to the ocean’s surface influenced by the moon’s latitudinal declination, while the **Mm **factor is a largely perpendicular gravitational forcing influenced by the perigean cycle of the Moon-to-Earth distance. The latter may be the “right mechanism” as each can give close to the “right answer”.

So the gist of the fitting observations is that far fewer harmonic factors are required for a decent **Mm**-based model than for a **Mf**-based model. This is slightly at the expense of a stronger LTE modulation, but the parsimony of an **Mm**-based model can’t be beat, as I will show via the following analysis charts…

This is a good model fit based on a slightly modified **Mm**-based factorization, with a sample-and-hold placed on a strong annual impulse

The comparison of the modified **Mm **tidal factorization to the pure **Mm **is below (the reason the 27.55 day periodicity doesn’t appear is because of the monthly aliasing used in plotting).

The slight pattern on top of pure signal is due to a 6-year beat of the Mm period with the 27.212 day lunar draconic pattern indicating the time between equatorial nodal crossings of the moon. This is the strongest factor of the ascension cycle described in the solar and lunar ephemeris published recently by Sung-Ho Na. As highlighted above by numbered cycles, ~20 occur in the span of 120 years.

Below, an expanded look showing how slight a correction is applied

The integrated forcing after the annual impulse is shown below. The sample-and-hold integration exaggerates low-frequency differences so the distinction between the pure **Mm** forcing and **Mm**+harmonics is more apparent. The 6-year periodicity is obscured by longer term variations.

The log-scaled power spectra of the integrated tidal forcing is shown below. Note the overwhelmingly strong peak near the 0.25/year frequency (3.9 year cycle). The rest of the peaks are readily matched to periodicities in the Na ephemerides [1] according to their strength.

The LTE modulation is quite strong for this factorization. As shown below, the forcing levels need to sinusoidally fold several times over to match the observed ENSO behavior. See the recent post Inverting non-autonomous functions for a recipe to aid in iterating for the underlying LTE modulation.

The parsimony of this model can’t be emphasized enough. It’s equivalent to the agreement of a conventional tidal forcing analysis to a SLH time-series in that only a single lunar tidal factor accounts for a majority of the modulation. Only the challenge of finding the correct LTE modulation stands in the way of producing an unambiguously correct model for the underlying ENSO behavioral dynamics.

[1] Sung-Ho Na, **Chapter 19 – Prediction of Earth tide**, Editor(s): Pijush Samui, Barnali Dixon, Dieu Tien Bui, Basics of Computational Geophysics, Elsevier, 2021, Pages 351-372, ISBN 9780128205136,

https://doi.org/10.1016/B978-0-12-820513-6.00022-9. (note: the ephemerides for the Earth-Moon-Sun system matches closely the online NASA JPL ephemerides generator available at https://ssd.jpl.nasa.gov/horizons, but this paper is more useful in that it algorithmically states the contributions of the various tidal factors in the source code supplied. Source code also available at https://github.com/pukpr/GeoEnergyMath/tree/master/src)

The above is from an informative OSU press release from last year titled Solving climate’s toughest questions, one challenge at a time. The following quotes are from that page, bold emphasis mine.

Jialin Lin, associate professor of geography, has spent the last two decades tackling those challenges, and in the past two years, he’s had breakthroughs in answering two of forecasting’s most pernicious questions: predicting theshift between El Niño and La Niñaand predicting which hurricanes will rapidly intensify.

Now, he’s turning his attention to creating more accurate models predicting global warming and its impacts, leading an international team of 40 climate experts to create

a new bookidentifying the highest-priority research questions for the next 30-50 years.

Lin set out to create a model that could

accurately identify ENSO shiftsby testing — and subsequently ruling out — all the theories and possibilities earlier researchers had proposed. Then, Lin realized current models only considered surface temperatures, and he decided to dive deeper.

He downloaded 140 years of deep-ocean temperature data, analyzed them and

made a breakthrough discovery.

“After 20 years of research, I finally found that the shift was caused by an ocean wave 100 to 200 meters down in the deep ocean,” Lin said, whose research was published in a

Naturejournal. “The propagation of this wave from the western Pacific to the eastern Pacificgenerates the switch from La Niña to El Niño.”

The wave repeatedly appeared two years before an El Niño event developed, but Lin went one step further to explain what generated the wave and discovered it was caused by the

moon’s tidal gravitational force.

“The tidal force is even

easier to predict,” Lin said. “That will widen the possibility for an even longer lead of prediction. Now you can predict not only for two years before, but 10 years before.”

Essentially, the idea is that these subsurface waves can in no way be caused by surface wind as the latter only are observed later (likely as an after-effect of the sub-surface thermocline nearing the surface and thus modifying the atmospheric pressure gradient). This counters the long-standing belief that ENSO transitions occur as a result of prevailing wind shifts.

The other part of the article concerns correlating hurricane intensification is also interesting.

p.s. It’s all tides : Climatic Drivers of Extreme Sea Level Events Along the

Coastline of Western Australia

- Introduction
- Hypothesis Testing
- Causality, Interaction, and Feedback
- Emerging concepts and pathways of information physics
- Sharper Predictions Using Occam’s Digital Razor

By anticipating all these ideas, you can find plenty of examples and derivations (with many centered on the ideas of Maximum Entropy) in our book Mathematical Geoenergy.

Here is an excerpt from the “Emerging concepts” entry, which indirectly addresses negative entropy:

That is, there are likely many earth system behaviors that are highly ordered, but the complexity and non-linearity of their mechanisms makes them appear stochastic or chaotic (high positive entropy) yet the reality is that they are just a complicated deterministic model (negative entropy). We just aren’t looking hard enough to discover the underlying patterns on most of this stuff.

An excerpt from the Occam’s Razor entry, lifts from my cite of Gell-Mann

Parsimony of models is a measure of negative entropy

]]>The consistency of interdecadal changes in the Earth’s rotation variations

On the ~ 7 year periodic signal in length of day from a frequency domain stepwise regression method

These cycles may be related to aliased tidal periods with the annual cycle, as in modeling ENSO.

A paper describing new satellite measurements for precision LOD measurements.

“BeiDou satellite radiation force models for precise orbit

determination and geodetic applications” from TechRxiv

Note the detail on the 13.6 day fortnightly tidal period

]]>This NASA press release has received mainstream news attention.

The 18.6 year nodal cycle will generate higher tides that will exaggerate sea-level rise due to climate change.

Yahoo news item:

https://news.yahoo.com/lunar-orbit-apos-wobble-apos-173042717.html

So this is more-or-less a known behavior, but hopefully it raises awareness to the other work relating lunar forcing to ENSO, QBO, and the Chandler wobble.

**Cited paper**

Thompson, P.R., Widlansky, M.J., Hamlington, B.D. *et al.* Rapid increases and extreme months in projections of United States high-tide flooding. *Nat. Clim. Chang.* **11, **584–590 (2021). https://doi.org/10.1038/s41558-021-01077-8

The following is recent research on mobility dispersion, contrast to something I blogged on years ago.

https://phys.org/news/2021-05-mobility-reveals-universal-law-cities.html

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