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 180° 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 180° longitudinal difference between the Atlantic and Pacific.
ENSO ~ sin (k F(t))
AMO ~sin (k F(t) + π + ϕ)
Any additional phase shift ϕ 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:
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 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.
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
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.
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.
Experimenting with linking to slide presentations instead of a trad blog post. The PDF linked below is an eye-opener as the NINO34 fit is the most parsimonious ever, at the expense of a higher LTE modulation (explained here). The cross-validation involves far fewer tidal factors than dealt with earlier, the two factors used (Mf and Mm tidal factors) rivaling the one factor used in QBO (described here).
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.
The forcing spectrum like this, with the aliased draconic (27.212d) factor circled:
For QBO, we remove all the lunar factors except for the draconic, as this is the only declination factor with the same spherical group symmetry as the semi-annual solar declination.
And after modifying the annual (ENSO spring-barrier) impulse into a semi-annual impulse with equal and opposite excursions, the resultant model matches well (to first order) the QBO time series.
Although the alignment isn’t perfect, there are indications in the structure that the fit has a deeper significance. For example, note how many of the shoulders in the structure align, as highlighted below in yellow
The peaks and valleys do wander about a bit and might be a result of the sensitivity to the semi-annual impulse and the fact that this is only a monthly resolution. The chart below is a detailed fit of the QBO using data with a much finer daily resolution. As you can see, slight changes in the seasonal timing of the semi-annual pulse are needed to individually align the 70 and 30 hBar QBO time-series data.
The underlying forcing of the ENSO model shows both an 18-year Saros cycle (which is an eclipse alignment cycle of all the tidal periods), along with a 6-year anomalistic/draconic interference cycle. This modulation of the main anomalistic cycle appears in both the underlying daily and monthly profile, shown below before applying an annual impulse. The 6-year is clearly evident as it aligns with the x-axis grid 1880, 1886, 1892, 1898, etc.
The 6-year cycle in the LOD is not aligned as strictly as the tidal model and it tends to wander, but it seems a more plausible and parsimonious explanation of the modulation than for example in this paper (where the 6-year LOD cycle is “similarly detected in the variations of C22 and S22, the degree-2 order-2 Stokes coefficients of the Earth’s gravitational field”).
Cross-validation confidence improves as the number of mutually agreeing alignments increase. Given the fact that controlled experiments are impossible to perform, this category of analyses is the best way to validate the geophysical models.
The underlying structure of the solution shouldn’t be surprising, since as with Mach-Zehnder, it’s fundamentally related to a path integral formulation known from mathematical physics. As derived via quantum mechanics (originally by Feynman), one temporally integrates an energy Hamiltonian over a path allowing the wave function to interfere with itself over all possible wavenumber (k) and spatial states (x).
Because of the imaginary value i in the exponential, the result is a sinusoidal modulation of some (potentially complicated) function. Of course, the collective behavior of the ocean is not a quantum mechanical result applied to fluid dynamics, yet the topology of the equatorial waveguide can drive it to appear as one, see the breakthrough paper “Topological Origin of Equatorial Waves” for a rationale. (The curvature of the spherical earth can also provide a sinusoidal basis due to a trigonometric projection of tidal forces, but this is rather weak — not expanding far beyond a first-order expansion in the Taylor’s series)
Moreover, the rather strong interference may have a physical interpretation beyond the derived mathematical interpretation. In the past, I have described the modulation as wave breaking, in that the maximum excursions of the inner function f(t) are folded non-linearly into itself via the limiting sinusoidal wrapper. This is shown in the figure below for progressively increasing modulation.
In the figure above, I added an extra dimension (roughly implying a toroidal waveguide) which allows one to visualize the wave breaking, which otherwise would show as a progressively more rapid up-and-down oscillation in one dimension.
Perhaps coincidentally (or perhaps not) this kind of sinusoidal modulation also occurs in heuristic models of the double-gyre structure that often appears in fluid mechanics. In the excerpt below, note the sin(f(t)) formulation.
The interesting characteristic of the structure lies in the more rapid cyclic variations near the edge of the gyre, which can be seen in an animation (Jupyter notebook code here).
Whether the equivalent of a double-gyre is occurring via the model of the LTE 1-D equatorial waveguide is not clear, but the evidence of double-gyre wavetrains (Lagrangian coherent structures, Kelvin–Helmholtz instabilities), occurring along the equatorial Pacific is abundantly clear through the appearance of tropical instability (TIW) wavetrains.
These so-called coherent structures may be difficult to isolate for the time being, especially if they involve subtle interfaces such as thermocline boundaries :
Mercator analysis does show higher levels of waveguide modulation, so perhaps this will be better discriminated over time (see figure below with the long wavelength ENSO dipole superimposed along with the faster TIW wavenumbers in dashed line, with the double-gyre pairing in green + dark purple), and something akin to a 1-D gyre structure will become a valid description of what’s happening along the thermocline. In other words, the wave-breaking modulation due to the LTE modulation is essentially the same as the vortex gyre mapped into a 1-D waveguide.
Sea-level height has several scales. At the daily scale it represents the well-known lunisolar tidal cycle. At a multi-decadal, long-term scale it represents behaviors such as global warming. In between these two scales is what often appears to be noisy fluctuations to the untrained eye. Yet it’s fairly well-accepted  that much of this fluctuation is due to the side-effects of alternating La Nina and El Nino cycles (aka ENSO, the El Nino Southern Oscillation), as represented by measures such as NINO34 and SOI.
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.
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.
 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.