From the https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html website, there is this figure:

I am guessing that the major ticks on the left vertical axis are days, and that the centered major tick (opposite to 180) is the mean.

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From the https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html website, there is this figure:

I am guessing that the major ticks on the left vertical axis are days, and that the centered major tick (opposite to 180) is the mean.

This paper that a couple of people alerted me to is likely one of the most radical research findings that has been published in the climate science field for quite a while:

Topological origin of equatorial waves

Delplace, Pierre, J. B. Marston, and Antoine Venaille. Science (2017): eaan8819.

An earlier version on ARXIV was titled Topological Origin of Geophysical Waves, which is less targeted to the equator.

The scientific press releases are all interesting

**Science Magazine**: Waves that drive global weather patterns finally explained, thanks to inspiration from bagel-shaped quantum matter**Science Daily**: What Earth's climate system and topological insulators have in common**Physics World**: Do topological waves occur in the oceans?

What the science writers make of the research is clearly subjective and filtered through what they understand.

This is puzzling:

“This temperature and zonal wind structure resembles those of Earth’s quasi-biennial oscillation (QBO) and Jupiter’s quasiquadrennial oscillation (QQO), in which temperature anomalies and eastward/westward winds alternate in altitude”

Fouchet, T., et al. “An equatorial oscillation in Saturn’s middle atmosphere.” Nature 453.7192 (2008): 200.

And recently the final results of the Cassini spacecraft mission were in the news:

“The density wave is generated by the gravitational pull of Saturn’s moon Janus.”

Wild! Cassini Probe Spots Weird Waves in Saturn's RingsSeptember 11, 2017 https://www.space.com/38114-weird-waves-saturn-rings-cassini-photo.html

But no one in the research literature has made the connection of the moon's orbit to the dynamics of the QBO.

From Fouchet *et al*, again

On Earth, the alternating wind regimes repeat at intervals that vary from 22 to 34 months, with an average period of about 28 months. On

Jupiter, the equatorial stratospheric temperature exhibits a 4.4-year period and the equatorial zonal winds in the upper troposphere oscillate with a4.5-year period. Long-term ground-based monitoring reveals a period of14.7±0.9 terrestrial years on Saturn. The observational similarities between Saturn’s oscillation and the QBO and QQO are the strong equatorial confinement of temperature minima and maxima and associated shear layers, a stronger eastward than westward shear layer, and the bounding of the equatorial oscillation at latitudes 15–20° north and south. Temperatures near these latitudes are relatively high when equatorial temperatures are relatively low, and vice versa.

On Earth and Jupiter, the quasi-periodic oscillations are triggered by the interaction between upwardly propagating waves and the mean zonal flow.

Both Jupiter and Saturn have 4 significant moons,making the collective lunar orbit difficult to describe. It's possible that Saturn's and Jupiter's "QBO" are more like the Earth's upper stratosphere oscillations, which align to the semiannual period (0.5 year period). This is suggestive as the values for Jupiter and Saturn's "QBO" period are closer to 1/2 the planet's full calendar year period, as show below

Planet | "year" length | "QBO" period | "QBO upper" period |
---|---|---|---|

Earth | 1 year | 2.37 years | 0.5 year |

Saturn | 29.46 years | ||

Jupiter | 11.86 years |

Now that we have strong evidence that AMO and PDO follows the biennial modulated lunar forcing found for ENSO, we can try modeling the Chandler wobble in detail. Most geophysicists argue that the Chandler wobble frequency is a resonant mode with a high-Q factor, and that random perturbations drive the wobble into its characteristic oscillation. This then interferes against the yearly wobble, generating the CW beat pattern.

But it has really not been clearly established that the measure CW period is a resonant frequency. I have a detailed rationale for a lunar forcing of CW in this post, and Robert Grumbine of NASA has a related view here.

The key to applying a lunar forcing is to multiply it by a extremely regular seasonal pulse, which introduces enough of a non-linearity to create a physically-aliased modulation of the lunar monthly signal (similar as what is done for ENSO, QBO, AMO, and PDO).

After spending several years on formulating a model of ENSO (then and now) and then spending a day or two on the AMO model, it's obvious to try the other well-known standing wave oscillation — specifically, the Pacific Decadal Oscillation (PDO). Again, all the optimization infrastructure was in place, with the tidal factors fully parameterized for automated model fitting.

This fit is for the entire PDO interval:

What's interesting about the PDO fit is that I used the AMO forcing ** directly as a seeding input**. I didn't expect this to work very well since the AMO waveform is not similar to the PDO shape except for a vague sense with respect to a decadal fluctuation (whereas ENSO has no decadal variation to speak of).

Yet, by applying the AMO seed, the convergence to a more-than-adequate fit was rapid. And when we look at the primary lunar tidal parameters, they all match up closely. In fact, only a few of the secondary parameters don't align and these are related to the synodic/tropical/nodal related 18.6 year modulation and the Ms* series indexed tidal factors, in particular the Msf factor (the long-period lunisolar synodic fortnightly). This is rationalized by the fact that the Pacific and Atlantic will experience maximum nodal declination at *different times* in the 18.6 year cycle.

After spending several years (*edit:* part-time) on formulating a model of ENSO (then and now), I decided to test out the formulation on another standing wave oscillation — specifically, the Atlantic Multidecadal Oscillation (AMO). All the optimization infrastructure was in place, with the tidal factors fully parameterized for automated model fitting.

This fit is for a training interval 1900-1980:

The ~60 year oscillation is a hallmark of AMO, and according to the results, this arises primarily from the anomalistic lunar forcing cycle modulated by a biennial seasonal modulation. Because of the spiked biennial modulation, we do not get a single long-period cycle but one that is also modulated by the forcing monthly tidal periods. As with ENSO, second-order effects in the anomalistic cycle described by lunar evection and variation is critical.

Outside of the training interval, the cross-validated test interval matches the AMO data arguably well. Since AMO is based on SST anomalies, it's possible that strong ENSO episodes and volcanic perturbations (e.g. post 1991 Pinatubo eruption) can have an impact on the AMO measure.

This is a typical fit over the entire interval.

This is the day after I started working on the AMO model, so these results are preliminary but also promising. AMO has a completely different character than ENSO and is more of an upper latitude phenomenon, which means that the tidal forces have a different impact than the equatorial ENSO cycle. Some more work may reveal whether the volcanic or ENSO forcing overrides the tidal forcing in certain intervals.

For the ENSO model, there is an ambiguity in simultaneously identifying the lunar month duration (draconic, anomalistic, and tropical) ** and **the duration of a year. The physical aliasing is such that the following

So that during the fitting process, if you allow the duration of the individual months and the year to co-vary, then the two should scale approximately by the number of lunar months in a year ~13.3 = 1/0.075. And sure enough, that's what is found, a set of year/month pairs that provide a maximized fit along a ridge line of possible solutions, but *only one* that is ultimately correct for the average year duration over the entire range:

By regressing on the combination of linear slopes, the value of the year that minimizes the error to each of the known lunar month values is 365.244 days. This lies within the interval defined by the value of the calendar year = 365.25 days — which includes a leap day every 4 years, and the more refined leap year calculation = 365.242 days — which includes the 100 and 400 year corrections (there are additional leap second corrections).

This analysis provides further confidence that the ENSO model is approaching the status of a metrology tool for gauging lunisolar cycles. The tropical month is estimated slow by about 1/2 a minute, while the draconic month is fast by a 1/2 a minute, and the average anomalistic month is spot on to within a second.

This is what the fit looks like for a 365.242 day long calendar year trained over the entire interval. It is the accumulation of the sharply matching peaks and valleys which allow the solver function to zone in so precisely to the known tidal factors.

About the only issue that hobbles our ability to achieve fits as good as ocean tidal analysis is the amount of noise near neutral ENSO conditions in the time-series data. The highlighted yellow regions in the comparison between NINO34 and SOI time-series data shown below indicate intervals whereby a sliding correlation coefficient drops closer to zero. (*The only odd comparison is the blue highlighted region around 1985, where SOI is extremely neutral while NINO34 appears La Nina-like. Is SOI pressure related to a second derivative of NINO34 temperature?*).

Those same yellow regions are also observed as discrepancies between the NINO34 data and the ENSO best model fit.

.

This is a straightforward validation of the forcing used on the lunar-driven ENSO model.

The paper by Chao et al [1] provides a comprehensive spectral analysis of the earth's length of day (LOD) variations using both a wavelet analysis and a power spectrum analysis. The wavelet analysis provides insight into the richness of the LOD cyclic variations (c.f. the Chao ref 6 in a recent post) :

Both the wavelet and the power spectrum (below) show the 6-year Fourier component that appears in the ENSO model as a mixed tidal forcing.

The original premise is that the change in LOD via the equivalent angular momentum change will impart a forcing on the Pacific ocean thermocline as per a *reduced-gravity model*:

There are additional peaks not found by Chao but those are reduced in magnitude, as can be inferred from the log (i.e. dB) scale. If these actually exist in the Chao spectrum, they may be buried in the background noise. Also, the missing Sa and Ssa peaks are the seasonal LOD variations that are taken into account separately by the model, as most ENSO data sets are typically filtered to remove seasonal data.

The tidal constituents shown above in the Chao power spectra are defined in the following Doodson table [3]. Chao likely is unable to discriminate the tropical values from the draconic and anomalistic values, being so close in value. On the other hand, the ENSO model needs to know these values precisely. Each of the primary Mm, Mf, Mtm, and Mqm and satellite Msm, Msf, Mstm, Msqm factors align with the first 4 harmonics of the mixed nonlinear ENSO model with the 2nd order satellites arising from the anomalistic correction.

This is an excellent validation test because this particular LOD power spectrum has not been used previously in the ENSO model fitting process. If the peaks did not match up, then the original premise for LOD forcing would need to be reconsidered.

[1] B. F. Chao, W. Chung, Z. Shih, and Y. Hsieh, “Earth’s rotation variations: a wavelet analysis,” Terra Nova, vol. 26, no. 4, pp. 260–264, 2014.

[2] A. Capotondi, “El Niño–Southern Oscillation ocean dynamics: Simulation by coupled general circulation models,” *Climate Dynamics: Why Does Climate Vary?*, pp. 105–122, 2013.

[3] D. D. McCarthy (ed.): IERS Conventions (1996) (IERS Technical Note No. 21) :

Chapter 6

I've been wanting to try this for awhile — to see if the solver setup used for fitting to ENSO would work for conventional tidal analysis. The following post demonstrates that if you give it the recommended tidal parameters and let the solver will grind away, it will eventually find the best fitting amplitudes and phases for each parameter.

The context for this analysis is an excellent survey paper on tsunami detection and how it relates to tidal prediction:

S. Consoli, D. R. Recupero, and V. Zavarella, “A survey on tidal analysis and forecasting methods for Tsunami detection,” J. Tsunami Soc. Int.

33 (1), 1–56.

The question the survey paper addresses is whether we can one use knowledge of tides to deconvolute and isolate a tsunami signal from the underlying tidal sea-level-height (SLH) signal. The practical application paper cites the survey paper above:

Percival, Donald B., et al. "Detiding DART® buoy data for real-time extraction of source coefficients for operational tsunami forecasting." Pure and Applied Geophysics 172.6 (2015): 1653-1678.

This is what the raw buoy SLH signal looks like, with the tsunami impulse shown at the end as a bolded line:

After removing the tidal signals with various approaches described by Consoli *et al*, the isolated tsunami impulse response (due to the 2011 Tohoku earthquake) appears as:

As noted in the caption, the simplest harmonic analysis was done with 6 constituent tidal parameters.

As a comparison, the ENSO solver was loaded with the same tidal waveform (after digitizing the plot) along with 4 major tidal parameters and 4 minor parameters to be optimized. The solver's goal was to maximize the correlation coefficient between the model and the tidal data.

The yellow region is training which reached almost a 0.99 correlation coefficient, with the validation region to the right reaching 0.92.

This is the complex Fourier spectrum (which is much less busy than the ENSO spectra):

The set of constituent coefficients we use is from the Wikipedia page where we need the periods only. Of the following 5 principal tidal constituents, only N_{2} is a minor factor in this case study.

In practice, multiple linear regression would provide faster results for tidal analysis as the constituents add linearly (see the CSALT model). In contrast, for ENSO there are several nonlinear steps required that precludes a quick regression solution. Yet, this tidal analysis test shows how effective and precise a solution the solver supplies.

The entire analysis only took an evening of work, in comparison to the difficulty of dealing with ENSO data, which is much more noisy than the clean tidal data. Moreover, the parameters for conventional tidal analysis stress the synodic/tropical/sidereal constituents — unfortunately, these are of little consequence for ENSO analysis, which requires the draconic and anomalistic parameters and the essential correction factors. The synodic tide is the red herring for the unwary when dealing with global phenomena such as ENSO, QBO, and LOD. The best way to think about it is that the synodic cycle impacts locally and most immediately, whereas the anomalistic and draconic cycles have global and more cumulative impacts.

A somewhat hidden cyclic variation in the length-of-day (LOD) in the earth's rotation, of between 6 and 7 years, was first reported in Ref [1] and analyzed in Ref [2]. Later studies further refined this period [3,4,5] closer to 6 years.

It's well known that the moon's gravitational pull contributes to changes in LOD [6]. Here is the set of lunar cycles that are applied as a forcing to the ENSO model using LOD as calibration.

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