Interface-Inflection Geophysics

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

  1. Science Magazine: Waves that drive global weather patterns finally explained, thanks to inspiration from bagel-shaped quantum matter
  2. Science Daily: What Earth's climate system and topological insulators have in common
  3. 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.

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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).

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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.

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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.

Identification of Lunar Parameters and Noise

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 f = frequency will give approximately equivalent fits for a range of year/month pairs (see this as well).

f = Year/LunarMonth - 13

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.

Yellow shading at intervals around 1930, 1936, 1948  indicate discrepancies between the NINO34 data in green and the ENSO model in red.


Earthquakes, tides, and tsunami prediction

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 N2 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.

The 6-year oscillation in Length-of-Day

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.

Change in detected LOD follows a ~6-yr cycle, from Ref [3]

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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Limits to Goodness of Fit

Based on a comparison of local interval correlations between the NINO34 and SOI indices, there probably is a limit to how well a model can be fit to ENSO.  The lower chart displays a 4-year-windowed correlation coefficient (in RED) between the two indices (shown in upper chart):

Note that in the interval starting at 1930, the correlation is poor for about 7 years.

Next note that the ENSO model fit shows a poor correlation to the NINO34 data in nearly the same intervals (shown as dotted GREEN). This is an odd situation but potentially revealing. The fact that both the ENSO model and SOI don't match the NINO34 index over the same intervals, suggests that the model may match SOI better than it does NINO34.  Yet, because of the excessive noise in SOI, this is difficult to verify.

But more fundamentally, why would NINO34 not match SOI in these particular intervals? These regions do seem to be ENSO-neutral, not close to El Nino or La Nina episodes.  Some also seem to occupy regions of faster, noisy fluctuations in the index.

It could be that the ENSO lunar tidal model is revealing the true nature of the ENSO dynamics, and these noisier, neutral regions are reflecting some other behavior (such as amplitude folding) — but since they also appear to be obscured by noise, it makes it difficult to unearth.


The paper by Zajączkowska[1] also applies a local correlation to compare the lunar tidal cycles to plant growth dynamics. There's a treasure trove of recent research on this topic.


Second-Order Effects in the ENSO Model

For ocean tidal predictions, once an agreement is reached on the essential lunisolar terms, then the second-order terms are refined. Early in the last century Doodson catalogued most of these terms:

"Since the mid-twentieth century further analysis has generated many more terms than Doodson's 388. About 62 constituents are of sufficient size to be considered for possible use in marine tide prediction, but sometimes many fewer can predict tides to useful accuracy."

That's possibly the stage we have reached in the ENSO model.  There are two primary terms for lunar forcing (the Draconic and Anomalistic) cycles, that when mixed with the annual and biannual cycles, will reproduce the essential ENSO behavior.  The second-order effects are the  modulation of these two lunar cycles with the Tropical/Synodic cycle.  This is most apparent in the modification of the Anomalistic cycle. Although not as important as in the calculation of the Total Solar Eclipse times, the perturbation is critical to validating the ENSO model and to eventually using it to make predictions.

The variation in the Anomalistic period is described at the NASA Goddard eclipse page. They provide two views of the variation, a time-domain view and a histogram view.

Time domain view
Histogram view

Since NASA Goddard doesn't provide an analytical form for this variation, we can see if the ENSO Model solver can effectively match it via a best-fit search to the ENSO data. This is truly an indirect method.

First we start with a parametric approximation to the variation, described by a pair of successive frequency modulated (and full-wave rectified) terms that incorporate the Tropical-modified term, wm. The Anomalistic term is wa.


\cos(\omega_a t+\phi_a+c_1 \cdot |\sin(\omega_m t+k_1 \cdot |\sin(\omega_m t+k_2)|+c_2)|)

This can generate the cusped behavior observed, but the terms pa, c_1, c_2, k_1, and k_2 need to be adjusted to align to the NASA model. The solver will try to do this indirectly by fitting to the 1880-1950 ENSO interval.

Plotting in RED the Anomalistic time series and the histogram of frequencies embedded in the ENSO waveform, we get:

Time domain view of model
Histogram view of model

This captures the histogram view quite well, and the time-domain view roughly (in other cases it gives a better cusped fit).  The histogram view is arguably more important as it describes the frequency variation over a much wider interval than the 3-year interval shown.

What would be even more effective is to find the correct analytical representation of the Anomalistic frequency variation and then plug that directly into the ENSO model. That would provide another constraint to the solver, as it wouldn't need to spend time optimizing for a known effect.

Yet as a validation step, the fact that the solver detects the shape required to match the variation is remarkable in itself. The solver is obviously searching for the forcing needed to produce the ENSO waveform observed, and happens to use the precise parameters that also describe the second-order Anomalistic behavior.  That could happen by accident but in that case there have been too many happy accidents already, i.e. period match, LOD match, Eclipse match, QBO match, etc.

Using Solar Eclipses to calibrate the ENSO Model

This is the forcing for the ENSO model, focusing on the non-mixed Draconic and Anomalistic cycles:

Note that the maximum excursions (perigee and declination excursion) align with the occurrence of total solar eclipses. These are the first three that I looked at, which includes the latest August 21 eclipse in the center chart.

There are about 90 more of these stretching back to 1880. The best way to fit the calibration is to take the negative excursions of the two lunar forcings and multiply these together, i.e. use the effective Draconic*Anomalistic amplitudes (also only take the fortnightly cycle of the Draconic, as eclipses occur during both the ascending and descending node crossings). The main fitting factors are the phases of the two lunar months.  To get the maximum alignment from the search solver, we maximize the sum of the effective amplitudes across the entire interval. This results in a phase difference between the two of about 0.74 radians based at the starting year of 1880 (i.e. year 0).

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