# Overturning Impulse

The only earth science class I took in college was limnology.

Of course I was a casual student of freshwater activities before that time, but certain behaviors of lakes were hammered home by taking this class. For example, the idea behind seasonal lake overturning. The overturning occurs as a singular event at a particular time of the year (monomictic once per year and dimictic twice per year, see figure to the right).

Saltwater doesn't show the same predilection for overturning as freshwater does, mainly because of the higher density differences above and below the thermocline, but the thermocline does vary, especially at latitudes located off the equator.

The ENSO model that appears so promising might be pointing to a partial overturning  showing a bimictic behavior.  The bi-prefix implies a biennial impulse that reverses direction every year, tied into a biennial Mathieu modulation associated with fluid sloshing dynamics. The Mathieu modulation is partially induced from the forcing conditions [1][2].

The following figure shows the modulation and impulse forcing that generates the best fit for the ENSO model. The BLUE is the Mathieu modulation while the ORANGE is the impulse.

The impulse may be related to a rapidly changing slope when the bimictic partial-overturning kicks in. The positive impulse as the Mathieu modulation moves from high to low, and the negative impulse as the modulation moves from low to high, both occurring at the same relative time late in the year.  Note that the polarity of this may be reversed, as the lunar tidal forcing at the impulse time provides the strength of this forcing through a multiplicative effect. This is shown below.

Whether to call it sloshing or a partial overturning remains to be determined. Yet overall this piece of the puzzle is the primary ansatz behind the entire model — since if we don't include it, the model does not generate the sharply distinct ENSO peaks and valleys. In other words, this is what causes the physical aliasing necessary to transform the monthly and fortnightly tidal cycles into a more erratic interannual cycling. This is in fact a simple model that shows more complex but still predictable dynamics.

## References

[1] T. B. Benjamin and F. Ursell, “The stability of the plane free surface of a liquid in vertical periodic motion,” presented at the Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 1954, vol. 225, pp. 505–515.

[2] S. Bale, K. Clavin, M. Sathe, A. S. Berrouk, F. C. Knopf, and K. Nandakumar, “Mixing in oscillating columns: Experimental and numerical studies,” Chemical Engineering Science, vol. 168, pp. 78–89, 2017.

# ENSO forcing - Validation via LOD data

If we don't have enough evidence that the forcing of ENSO is due to lunisolar cycles, this piece provides another independent validating analysis. What we will show is how well the forcing used in a model fit to an ENSO time series — that when isolated — agrees precisely with the forcing that generates the slight deviations in the earth's rotational speed, i.e. the earth's angular momentum. The latter as measured via precise measurements of the earth's length of day (LOD).  The implication is that the gravitational forcing that causes slight variations in the earth's rotation speed will also cause the sloshing in the Pacific ocean's thermocline, leading to the cyclic ENSO behavior.

# ENSO and Fourier analysis

Much of tidal analysis has been performed by Fourier analysis, whereby one can straightforwardly deduce the frequency components arising from the various lunar and solar orbital factors. In a perfectly linear world with only two ideal sinusoidal cycles, we would see the Fourier amplitude spectra of Figure 1.

Fig 1: Amplitude spectra for a signal with two sinusoidal Fourier components. To establish the phase, both a real value and imaginary value is plotted.

# ENSO model fit 1880-1980

This is an ENSO fit that only has knowledge of data prior to 1980. The data is 80% NINO34 and 20% SOI, with the latter providing finer structure.The lower fit includes a slight variation of the Draconic month according to this NASA page. It doesn't seem to do much.

A previous fit used values of A=0.785, M=0.3, K=-0.15 to 1920. This used A=0.866, M=0.207, K=-0.16.

# 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)
and
 07/05/2011 Dr. Claire Perigaud JPL Earth-Moon-Sun alignments influencing El Niños and water/air mass momentum
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.

# 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.

# 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.

Summary:

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:

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.