High Resolution ENSO Modeling

An intriguing discovery is that the higher-resolution aspects of the SOI time-series (as illustrated by the Australian BOM 30-day SOI moving average) may also have a tidal influence.  Note the fast noisy envelope that rides on top of the deep El Nino of 2015-2016 shown below:

For the standard monthly SOI as reported by NCAR and NOAA, this finer detail disappears.  BOM provides the daily SOI value for about the past ~ 3 years here.

Yet if we retain this in the 1880-present monthly ENSO model, by simultaneously isolating [1] the higher frequency fine structure from 2015-2017, the fine structure also emerges in the model. This is shown in the lower panel below.

This indicates that the differential equation being used currently can possibly be modified to include faster-responding derivative terms which will simultaneously show the multi-year fluctuations as well as what was thought to be a weekly-to-monthly-scale noise envelope. In fact, I had been convinced that this term was due to localized weather but a recent post suggested that this may indeed be a deterministic signal.

Lunisolar tidal effects likely do impact the ocean behavior at every known time-scale, from the well-characterized diurnal and semi-diurnal SLH tides to the long-term deep-ocean mixing proposed by Munk and Wunsch.  It's not surprising that tidal forces would have an impact on the intermediate time-scale ENSO dynamics, both at the conventional low resolution (used for El Nino predictions) and at the higher-resolution that emerges from SOI measurements (the 30-day moving average shown above).  Obviously, monthly and fortnightly oscillations observed in the SOI are commensurate with the standard lunar tides of periods 13-14 days and 27-28 days. And non-linear interactions may result in the 40-60 day oscillations observed in LOD.

from Earth Rotational Variations Excited by Geophysical Fluids, B.F. Chao, http://ivs.nict.go.jp/mirror/publications/gm2004/chao/

It's entirely possible that removing the 30-day moving average on the SOI measurements can reveal even more detail/

Footnote

[1] Isolation is accomplished by subtracting a 24-day average about the moving average value, which suppresses the longer-term SOI variation.

 

The ENSO Forcing Potential - Cheaper, Faster, and Better

Following up on the last post on the ENSO forcing, this note elaborates on the math.  The tidal gravitational forcing function used follows an inverse power-law dependence, where a(t) is the anomalistic lunar distance and d(t) is the draconic or nodal perturbation to the distance.

F(t) \propto \frac{1}{(R_0 + a(t) + d(t))^2}'

Note the prime indicating that the forcing applied is the derivative of the conventional inverse squared Newtonian attraction. This generates an inverse cubic formulation corresponding to the consensus analysis describing a differential tidal force:

F(t) \propto -\frac{a'(t)+d'(t)}{(R_0 + a(t) + d(t))^3}

For a combination of monthly and fortnightly sinusoidal terms for a(t) and d(t) (suitably modified for nonlinear nodal and perigean corrections due to the synodic/tropical cycle)   the search routine rapidly converges to an optimal ENSO fit.  It does this more quickly than the harmonic analysis, which requires at least double the unknowns for the additional higher-order factors needed to capture the tidally forced response waveform. One of the keys is to collect the chain rule terms a'(t) and d'(t) in the numerator; without these, the necessary mixed terms which multiply the anomalistic and draconic signals do not emerge strongly.

As before, a strictly biennial modulation needs to be applied to this forcing to capture the measured ENSO dynamics — this is a period-doubling pattern observed in hydrodynamic systems with a strong fundamental (in this case annual) and is climatologically explained by a persistent year-to-year regenerative feedback in the SLP and SST anomalies.

Here is the model fit for training from 1880-1980, with the extrapolated test region post-1980 showing a good correlation.

The geophysics is now canonically formulated, providing (1) a simpler and more concise expression, leading to (2) a more efficient computational solution, (3) less possibility of over-fitting, and (4) ultimately generating a much better correlation. Alternatively, stated in modeling terms, the resultant information metric is improved by reducing the complexity and improving the correlation -- the vaunted  cheaper, faster, and better solution. Or, in other words: get the physics right, and all else follows.

 

 

 

 

 

 

 

 

 

 

 

 

 

Approximating the ENSO Forcing Potential

From the last post, we tried to estimate the lunar tidal forcing potential from the fitted harmonics of the ENSO model. Two observations resulted from that exercise: (1) the possibility of over-fitting to the expanded Taylor series, and (2) the potential of fitting to the ENSO data directly from the inverse power law.

The Taylor's series of the forcing potential is a power-law polynomial corresponding to the lunar harmonic terms. The chief characteristic of the polynomial is the alternating sign for each successive power (see here), which has implications for convergence under certain regimes. What happens with the alternating sign is that each of the added harmonics will highly compensate the previous underlying harmonics, giving the impression that pulling one signal out will scramble the fit. This is conceptually no different than eliminating any one term from a sine or cosine Taylor's series, which are also all compensating with alternating sign.

The specific conditions that we need to be concerned with respect to series convergence is when r (perturbations to the lunar orbit) is a substantial fraction of R (distance from earth to moon) :

F(r) = \frac{1}{(R+r)^3}

Because we need to keep those terms for high precision modeling, we also need to be wary of possible over-fitting of these terms — as the solver does not realize that the values for those terms have the constraint that they derive from the original Taylor's series. It's not really a problem for conventional tidal analysis, as the signals are so clean, but for the noisy ENSO time-series, this is an issue.

Of course the solution to this predicament is not to do the Taylor series harmonic fitting at all, but leave it in the form of the inverse power law. That makes a lot of sense — and the only reason for not doing this until now is probably due to the inertia of conventional wisdom, in that it wasn't necessary for tidal analysis where harmonics work adequately.

So this alternate and more fundamental formulation is what we show here.

Continue reading

Reverse Engineering the Moon's Orbit from ENSO Behavior

With an ideal tidal analysis, one should be able to apply the gravitational forcing of the lunar orbit1 and use that as input to solve Laplace's tidal equations. This would generate tidal heights directly. But due to aleatory uncertainty with respect to other factors, it becomes much more practical to perform a harmonic analysis on the constituent tidal frequencies. This essentially allows an empirical fit to measured tidal heights over a training interval, which is then used to extrapolate the behavior over other intervals.  This works very well for conventional tidal analysis.

For ENSO, we need to make the same decision: Do we attempt to work the detailed lunar forcing into the formulation or do we resort to an empirical bottoms-up harmonic analysis? What we have being do so far is a variation of a harmonic analysis that we verified here. This is an expansion of the lunar long-period tidal periods into their harmonic factors. So that works well. But could a geophysical model work too?

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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.

Continue reading

CW

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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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.

Continue reading

AMO

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.

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ENSO tidal forcing validated by LOD data

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:

Taken from presentation for ref [2]

Calculating a spectral analysis of the best fit ENSO model forcing, note that all of the model peaks (in RED) match those found by Chao et al in their ΔLOD analysis :

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.

Tidal constituent coefficients taken from ref [3]

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.

 

References

[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