The Lunar Geophysical Connection

The conjecture out of NASA JPL is that the moon has an impact on the climate greater than is currently understood:

Claire Perigaud (Caltech/JPL)
Has this research gone anywhere?  Looks as if has gone to this spin-off.
According to the current consensus, variability in wind is what contributes to forcing for behaviors such as the El Nino/Southern Oscillation (ENSO).
OK, but what forces the wind? No one can answer that apart from saying wind variability is just a part of the dynamic climate system.  And so we are lead to believe that a wind burst will cause an ENSO and then the ENSO event will create a significant disruptive transient to the climate much larger than the original wind stimulus. And that's all due to positive feedback of some sort.
I am only paraphrasing the current consensus.
A much more plausible and parsimonious explanation lies with external lunar forcing reinforced by seasonal cycles.

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ENSO and Noise

How do we determine confidence that we are not fitting to noise for the ENSO model ?  One way to do this is to compare the data against another model; in this case, a model that provides an instrumentally independent measure. One can judge data quality by comparing an index such as NINO34 against SOI, which are instrumentally independent measures (one based on temperature and one on atmospheric pressure).

If you look at a sliding correlation coefficient of these two indices along the complete interval, you will see certain years that are poorly correlated (see RED line below). Impressively, these are the same years that give poor agreement against the ENSO model (see BLUE line below). What this tells us is the poorly correlated years are ones with poor signal-to-noise ratio. But more importantly, it also indicates that the model is primarily fitting to the real ENSO signal (especially the peak values) and the noisy parts (closer to zero crossings or neutral ENSO conditions) are likely not contributing to the fit. And this is not a situation where the model will fit SOI better than NINO34 -- because it doesn't.

The tracking of SOI correlating to NINO34 matches that of Model to NINO34 across the range with the exception of some excursions during the 1950's, where SOI fit NINO34 better that the model fit NINO34. The average correlation coefficient of SOI to NINO34 across the entire range is 0.75 while the model against NINO34 is less but depending on the parameterization always above 0.6.

As a result of this finding, I started to use a modification of a correlation coefficient called a weighted correlation coefficient, whereby the third parameter set is a density function that remains near 1 when the signal-to-noise (SNR) ratio is high and closer to zero where the SNR is closer to zero. This allows the fit to concentrate on the intervals of strong SNR, thus reducing the possibility of over-fitting against noise.

Or is it really all noise?   (Added: 5/17/2017)

As I derived earlier, the solution to Laplace's tidal equations at the equator for a behavior such as QBO leads to a sin(k sin(f(t))) modulated time-series, where the inner sinusoid is essentially the forcing. This particular formulation (referred to as the sin-sin envelope) has interesting properties. For one, it has an amplitude limiting property due to the fact that a sinuosoid can't exceed an amplitude of unity. Besides this excursion-limiting behavior, this formulation can also show amplitude folding at the positive and negative extremes. In other words, if the amplitude is too large, the outer sin modulation starts to shrink the excursion, instead of just limiting it. So if there is a massive amplitude, what happens is that the folding will occur multiple times within the peak interval, thus resulting in a rapid up and down oscillation. This potentially can have the appearance of noise as the oscillations are so rapid that (1) they may blur the data record or (2) may be unsustainable and lead to some form of wave-breaking.  I am not sure if the latter is related to folding of geological strata.

So the question is: can this happen for ENSO? I have been feeding the solution to the delayed differential Mathieu equation as a forcing to the sin-sin envelope and find that it works effectively to match the "noisy" regions identified above.  In the figure below, the diamonds represent intervals with the poorest correlation between NINO34 and SOI and perhaps the noisiest in terms of SOI. In particular, the regions labelled 1 and 6 indicate rapid cyclic excursions.

By comparison, the model fit to ENSO shows the rapid oscillations near many of the same regions. In particular look at intervals indicated by diamonds 1 and 6 below, as well as the interval just before 1950.

Now, consider that these just happen to be the same regions that the ENSO model shows excessive amplitude folding.  The pattern isn't 100% but also doesn't appear to be coincidental, nor is it biased or forced (as the fitting procedure has no idea that these are considered the noisy intervals).  So the suggestion is that these are points in time that could have developed into massive El Nino or La Nina, but didn't because the forcing amplitude became folded. Thus they could not grow and instead the strong lunar gravitational forcing went into rapid oscillations which dissipated that energy. In fact, it's really the rate of change in the kinetic energy that scales with forcing, and the rapid oscillations identify that change. Connecting back to the theory, that's what the sin-sin envelope describes — its essentially a solution to a Hamiltonian that conserves the energy of the system. From the Sturm-Liouville equation that Laplace's tidal equations reduce to, this answer is analytically precise and provided in closed-form.

The caveat to this idea of course is that no one else in climate science is even close to considering such a sin-sin formulation.  Consider this:

... yet ...

An alternative model that matches ENSO does not exist, so there is nothing at the moment to refute.  And see above how it fits in balance with known physics.

ENSO Proxy Validation

This is a straightforward validation of the ENSO model presented at last December's AGU.

What I did was use the modern instrumental record of ENSO — the NINO34 data set — as a training interval, and then tested across the historical coral proxy record — the UEP data set.

The correlation coefficient in the out-of-band region of 1650 to 1880 is excellent, considering that only two RHS lunar periods (draconic and anomalistic month) are used for forcing. As a matter of fact, trying to get any kind of agreement with the UEP using an arbitrary set of sine waves is problematic as the time-series appears nearly chaotic and thus requires may Fourier components to fit. With the ENSO model in place, the alignment with the data is automatic. It predicts the strong El Nino in 1877-1878 and then nearly everything before that.

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ENSO and Tidal SLH - A Biennial Connection

It's becoming abundantly clear that ENSO is driven by lunisolar mechanisms, especially true considering that we use the preferred models describing sloshing of water volumes -- the Mathieu equation and the delay differential equation. What's more, given the fact that the ENSO model works so well, one can guess that a connection between ENSO and lunar tidal forces should carry over to models for historical sea-level height (SLH) tidal data.

From my preliminary work analyzing SLH tidal data in 2014, I found an intriguing pattern relating to the same biennial pattern observed in the ENSO model. An extract from that post is reproduced below:

"The idea, related to delay differential equations, is to determine if it is at all possible to model at least part of ENSO (through the SOI) with data from a point in time in the tidal record with a compensated point from the past. This essentially models the effect of the current wave being cancelled partially by the reflection of a previous wave.

This is essentially suggesting that SOI = k f(t) - k f(t-\Delta t) where f(t) is the tidal record and k is a constant.

After some experimenting, a good fit is obtained when the current tidal data is set to 3 months ago, and the prior data is taken from 26 months in the past. To model the negative of ENSO, the 3-month old data is subtracted from the 26-month old data, Figure 1:"

Fig 1: Model of ENSO uses Tidal Gauge readings from Sydney Harbor.

This observation suggests a biennial pattern whereby high correlations in the time series occur for events observed close to two-years apart, with the difference taken up by the ENSO signal. The latter is at least partly due to the inverted barometer effect on SLH.

The sensitivity of the biennial effect is shown in the following contour diagram, where the peak CC is indicated and a slanted line showing the 2-year differencing points (table of the correlation coefficients here)

Focussing on the mechanisms for the correlation, the predictor can be reformulated as

f(t) = f(t-\Delta t) + SOI / k

This brings up an interesting question as to why this biennial signal apparently can't be detected via conventional means. A likely guess is because it gets obscured by the ENSO signal. And since the ENSO signal has never been adequately modeled, no one has an inkling that a biennial signal coexists in the mix. Yet this unconventional delay differencing approach (incidentally discovered with the help of machine learning) is able to discern it with a strong statistical significance.

The gist of this discovery is that a biennial tidal SLH motion exists and likely contributes (or at least identifies) the nonlinear biennial modulation driving the ENSO model described in previous posts.


ENSO has elements of an annual teeter-totter. If you look through the research literature, you will find numerous references to a hypothesized behavior that one annual peak is followed by a lesser peak the next year. Yet, no evidence that this strict biennial cycle is evidence in the data — it's more of a hand-wavy physical argument that this can or should occur.

The way to model this teeter-totter behavior is via Mathieu equations and delay differential equations. Both of these provide a kind of non-linear modulation that can sustain a biennial feedback mechanism.

The other ingredient is a forcing mechanism. The current literature appears to agree that this is due to prevailing wind bursts, which to me seems intuitive but doesn't answer what forces the wind in the first place. As it turns out, only two parameters are needed to force the DiffEq and these align precisely with the primary lunar cycles that govern transverse and longitudinal directional momentum, the draconic and anomalistic months.

The ENSO model turns into a metrology tool

Note the following table and how the draconic and anomalistic values zone in on the true value when correlating model against data:

Fitting the DiffEq values progressively away from the true value for the lunar tidal cycle results in a smaller correlation coefficient.

Having one of these values align may be coincidence, but having both combine with that kind of resolution is telling. The biennial connection to SLH tidal measurements is substantiation that a biennial modulation is intrinsic to the physical process.

Canonical Solution of Mathieu Equation for ENSO

From a previous post, we were exploring possible solutions to the Mathieu equation given a pulsed stimulus.  This is a more straightforward decomposition of the differential equation using a spreadsheet.

The Mathieu equation:

f''(t) + \omega_0^2 (1 + \alpha \cos(\nu t)) f(t) = F(t)

can be approximated as a difference equation, where the second derivative f''(t) is ~ (f(t)-2f(t-dt)+f(t-2dt))/dt. But perhaps what we really want is a difference to the previous year and determine if that is enough to reinforce the biennial modulation that we are seeing in the ENSO behavior.

Setting up a spreadsheet with a lag term and a 1-year-prior feedback term, we apply both the biennial impulse-modulated lunar forcing stimulus and a yearly-modulated Mathieu term.

Fig. 1: Training (in shaded blue) and test for different intervals.

I was surprised by how remarkable the approximate fit was in the recent post, but this more canonical analysis is even more telling. The number of degrees of freedom in the dozen lunar amplitude terms apparently has no impact on over-fitting, even on the shortest interval in the third chart. There is noise in the ENSO data no doubt, but that noise seems to be secondary considering how the fit seems to mostly capture the real signal. The first two charts are complementary in that regard — the fit is arguably better in each of the training intervals yet the test interval results aren't really that much different from the direct fit looking at it by eye.

Just like in ocean tidal analysis, the strongest tidal cycles dominate;  in this case the Draconic and Anomalistic monthly, the Draconic and Anomalistic fortnightly, and a Draconic monthly+Anomalistic fortnightly cross term are the strongest (described here). Even though there is much room for weighting these factors differently on orthogonal intervals, the Excel Solver fit hones in on nearly the same weighted set on each interval.  As I said in a previous post, the number of degrees of freedom apparently do not lead to over-fitting issues.

One other feature of this fit was an application of a sin() function applied to the result. This is derived from the Sturm-Liouville solution to Laplace's tidal equation used in the QBO analysis  — which works effectively to normalize the model to the data, since the correlation coefficient optimizing metric does not scale the result automatically.

Pondering for a moment, perhaps the calculus is not so different to work out after all:


(from @rabaath on Twitter)

A bottom-line finding is that there is really not much complexity to this unique tidal formulation model of ENSO.  But because of the uniqueness of the seasonal modulation that we apply, it just doesn't look like the more-or-less regular cycles contributing to sea-level height tidal data.  Essentially similar algorithms are applied to find the right weighting of tidal factors, but whereas the SLH tidal data shows up in daily readings, the ENSO data is year-to-year.

Further, the algorithm does not take more than a minute or two to finish fitting the model to the data. Below is a time lapse of one such trial. Although this isn't an optimal fit, one can see how the training interval solver adjustment (in the shaded region) pulls the rest of the modeled time-series into alignment with the out-of-band test interval data.

Shortest Training Fit for ENSO

This is remarkable. Using the spreadsheet linked in the last post, the figure below is a model of ENSO derived completely by a training fit over the interval 1900 to 1920, using the Nino3.4 data series and applying the precisely phased Draconic and Anomalistic long-period tidal cycles.

Fig. 1 : The ENSO model in red. The blue BG region is used for training of the lunar tidal amplitudes against the Nino3.4 data in green. That data is square root compacted to convert it to an equivalent velocity.

Not much more to say. There is a major disturbance starting in the mid-1980's, but that is known from a Takens embedding analysis described in the first paper in this post.

Tidal Model of ENSO

The input forcing to the ENSO model includes combinations of the three major lunar months modulated by the seasonal solar cycle. This makes it conceptually similar to an ocean tidal analysis, but for ENSO we are more concerned about the long-period tides rather than the diurnal and semi-diurnal cycles used in conventional tidal analysis.

The three constituent lunar month factors are:

Month type Length in days
anomalistic 27.554549
tropical 27.321582
draconic 27.212220

So the essential cyclic terms are the following phased sinusoids

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Solution to Pulsed Mathieu Equation

Here is a bit of applied math that I have never seen described before.  It considers solving a variant of the Mathieu differential equation, an unwieldy beast that finds application in models of fluid sloshing (among others). Normally the Mathieu equation is only solvable as a stimulus forcing function convolved against the transcendental Mathieu function. Tools such as Mathematica include the Mathieu function in their library, but any other solution would require a full numerical DiffEq integration.

This is the typical Mathieu equation formulation:

f''(t) + ( \omega_0^2 + B cos (\nu_0 t) ) f(t) = 0

But instead of this nonlinear time-varying DiffEq, consider that we replace the sinusoidal modulation with a delta pulse. This is precisely our model of ENSO that we are trying to create from first principles. The pulse represents a forcing impulse from the alignment of a lunar and solar tidal cycle.

f''(t) + ( \omega_0^2 + B \delta (t) ) f(t) = 0

Taking the Laplace transform, we quickly arrive at what looks like an ordinary 2nd-order differential equation solution, albeit with an interesting initial condition:

s^2 F(s) + \omega_0^2 F(s) + B f(0) = 0

After taking the inverse Laplace transform, we have:

f(t) = I(t) \ast B f(0) \delta(t)

where I(t) represents the impulse response of the first two terms, which is then convolved with a delta function evaluated at a value of f(t) for t=0. This is the only remnant of the nonlinear nature of the classical Mathieu function, in that the initial condition has a scaling proportional to the value of the function at that time. On the other hand, the solution to an ordinary DiffEq would not be dependent on the value of the function.

Thereafter we can extend this to a general solution; by creating a pulse train of delta functions we get this final convolution:

f(t) = I(t) \ast B \sum_{n=0}^{\infty} f(t - n T) \delta(t - n T)

where T is the pulse train period.

This is straightforward to evaluate for any pulse train -- all we have to do is keep track of the changing value of f(t) as we come across each pulse.

Fig 1: A pulse train with both annual and biannual contributions.

In the above figure, why does the annual pulse have a two-year periodicity? That's due to the alignment of a non-congruent tidal period with a seasonal pulse in terms of a Fourier series as described here and mathematically refined here.

If that is not intuitive, we can still consider it more of an anzat and see where it takes us.

For an ENSO time series, we an convolve the above delta pulse train with a set of known tidal periods of arbitrary amplitude and phase. For a fit trained up to 1980, this is the extrapolation post-1980:

Fig 2: Fit of pulsed Mathieu equation using a set of tidal periods as a forcing function. The impulse response here is a simple year-long rectangular window, i.e. the response has a memory of only a calendar year. This can be further refined if necessary. Yet, the projected time-series evaluated out-of-band from the fitting interval does a good job of capturing the ENSO profile.

The projected waveform matches the last 16 years very effectively considering how noisy the ENSO series is, and is very close to the fit over the entire interval. This is apart from the possible perturbation around the 1982 El Nino. The fact that the monthly tidal periods differ by slight amounts dictates that long multi-decadal intervals should be used for fitting. (In contrast, the model of QBO is primarily Draconic so the fitting interval can be much shorter)

Fig 3: Fit over the entire interval. The set of tidal periods applied was limited to the 3 major months (draconic, anomalistic, and sidereal/tropical) and the multiplicative combinations creating the fortnightly tides. This may not be enough to get the details right but erred on the side of under-fitting to establish the physical mechanism.

This approach is the logical follow-on to the wave transformed fitting approach that I had been using, most recently here and here. Both of these approaches are equally clever, which is a necessary ingredient when one is dealing with the unwieldy Mathieu equations. Moreover they both pull out the obvious stationary aspects of the ergodic ENSO time series.

The ENSO Challenge

It's been quite a challenge decoding the physics of ENSO. Anything that makes the model more complex and with more degrees of freedom needs to be treated carefully. The period doubling bifurcation properties of wave sloshing has been an eye-opener for me. I experimented with adding a sub-harmonic period of 4 years to the 2-year Mathieu modulation and see if that improves the fit. By simply masking the odd behavior around 1981-1983, I came up with this breakdown of the RHS/LHS comparison.

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Lindzen doth protest too much

Incredible that Richard Lindzen was quoted as saying this:

Richard Lindzen, the Alfred P. Sloan Professor of Meteorology at MIT and a member of the National Academy of Sciences who has long questioned climate change orthodoxy, is skeptical that a sunnier outlook is upon us.

“I actually doubt that,” he said. Even if some of the roughly $2.5 billion in taxpayer dollars currently spent on climate research across 13 different federal agencies now shifts to scientists less invested in the calamitous narrative, Lindzen believes groupthink has so corrupted the field that funding should be sharply curtailed rather than redirected.

“They should probably cut the funding by 80 to 90 percent until the field cleans up,” he said. “Climate science has been set back two generations, and they have destroyed its intellectual foundations.”

Consider the psychological projection aspect of what Lindzen is asserting. The particularly galling part is this:

“Climate science has been set back two generations, and they have destroyed its intellectual foundations.”

It may actually be Lindzen that has set back generations of atmospheric science research with his deeply flawed model of the quasi-biennial oscillation of equatorial stratospheric winds — see my recent QBO presentation for this month's AGU meeting.   He missed a very simple derivation that he easily could have derived back in the 1960’s, and that could have set a nice “intellectual foundation” for the next 40+ years. Instead he has essentially "corrupted the field" of atmospheric sciences that could have been solved with the right application of Laplace's tidal equations — equations known since 1776 !

The "groupthink" that Lindzen set in motion on the causes behind QBO is still present in the current research papers, with many scientists trying to explain the main QBO cycle of 28 months via a relationship to an average pressure. See for example this paper I reviewed earlier this year.

To top it all off, he was probably within an eyelash of figuring out the nature of the forcing, given that he actually considered the real physics momentarily:

Alas, all those millions of taxpayer funds that Lindzen presumably received over the years didn't help, and he has been reduced to whining over what other climate scientists may receive in funding as he enters into retirement.

Methinks it's usually the case that the one that "doth protest too much" is the guilty party.

Added: here is a weird graphic of Lindzen I found on the cliscep blog. The guy missed the simple while focussing on the complex.


From climate scientist Dessler

From climate scientist Dessler