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.

 

 

 

 

 

 

 

 

 

 

 

 

 

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?

Continue reading

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.

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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

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.

q

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):

qf

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.
Continue reading

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(wa*t+pa+c_1*ABS(SIN(wm*t+k_1*ABS(SIN(wm*t+k_2))+c_2)))

\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.