QBO Split Training

As with ENSO, we can train QBO on separate intervals and compare the fit on each interval.  The QBO 30 hPa data runs from 1953 to the present.  So we take a pair of intervals — one from 1953-1983 (i.e. lower) and one from 1983-2013 (i.e. higher) — and compare the two.

The primary forcing factor is the seasonally aliased nodal or Draconic tide which is shown in the upper left on the figure.  The lower interval fit in BLUE matches extremely well to the higher interval fit in RED, with a correlation coefficient above 0.8.

These two intervals have no inherent correlation other than what can be deduced from the physical behavior generating the time-series.  The other factors are the most common long-period tidal cycles, along with the seasonal factor.  All have good correlations — even the aliased anomalistic tide (lower left), which features a pair of closely separated harmonics, clearly shows strong phase coherence over the two intervals.

That's what my AGU presentation was about — demonstrating how QBO and ENSO are simply derived from known geophysical forcing mechanisms applied to the fundamental mathematical geophysical fluid dynamics models. Anybody can reproduce the model fit with nothing more than an Excel spreadsheet and a Solver plugin.

Here are the PowerPoint slides from the presentation.

Compact QBO Derivation

I created a QBO page that is a concise derivation of the theory behind the oscillations:

http://contextEarth.com/compact-qbo-derivation/

Four key observations allow this derivation to work

  1. Coriolis effect cancels at the equator and use a small angle (in latitude) approximation to capture any differential effect.
  2. Identification of wind acceleration and not wind speed as the measure of QBO.
  3. Associating a latitudinal displacement with a tidal elevation via a partial derivative expansion to eliminate an otherwise indeterminate parameter.
  4. Applying a seasonal aliasing to the lunar tractive forces which ends up perfectly matching the observed QBO period.

These are obscure premises but all are necessary to derive the equations and match to the observations.

This model should have been derived long ago .... that's what has me stumped. Years ago I spent hours working on transport equations for semiconductor devices so have a good feel for how to handle these kinds of DiffEq's. You literally had to do this otherwise you would never develop the intuition on how a transistor or some other device works. The QBO for some reason reminds me quite a bit of solving the Hall effect. And of course it has geometric resemblance to the physics behind an electric motor or generator. Maybe I am just using a different lens in solving these kinds of problems.

Validating ENSO cyclostationary deterministic behavior

I tend to write a more thorough analysis of research results, but this one is too interesting not to archive in real-time.

First, recall that the behavior of ENSO is a cyclostationary yet metastable standing-wave process, that is forced primarily by angular momentum changes. That describes essentially the physics of liquid sloshing. Setting input forcings to the periods corresponding to the known angular momentum changes from the Chandler wobble and the long-period lunisolar cycles, it appears trivial to capture the seeming quasi-periodic nature of ENSO effectively.

The key to this is identifying the strictly biennial yet metastable modulation that underlies the forcing. The biennial factor arises from the period doubling of the seasonal cycle, and since the biennial alignment (even versus odd years)  is arbitrary, the process is by nature metastable (not ergodic in the strictest sense).  By identifying where a biennial phase reversal occurs, the truly cyclostationary arguments can be isolated.

The results below demonstrate multiple regression training on 30 year intervals, applying only known factors of the Chandler and lunisolar forcing (no filtering applied to the ENSO data, an average of NINO3.4 and SOI indices). The 30-year interval slides across the 1880-2013 time series in 10-year steps, while the out-of-band  fit maintains a significant amount of coherence with the data:

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Common Origins of Climate Behaviors

I have been on a path to understand ENSO via its relationship to QBO and the Chandler wobble (along with possible TSI contributions, which is fading) for awhile now. Factors such as QBO and CW have all been considered as possible forcing mechanisms, or at least as correlations, to ENSO in the research literature.

Over time, I got sidetracked into trying to figure out the causes of QBO and the Chandler wobble hoping that it might shed light into how they could be driving ENSO.

But now that we see how the QBO and the Chandler wobble both derive from the seasonally aliased lunar Draconic cycle, it may not take as long to piece the individual bits of evidence together.

I am optimistic based on how simple these precursor models are. As far as both QBO and the Chandler wobble are concerned, one can't ask for a simpler explanation than applying the moon's Draconic orbital cycle as a common forcing mechanism.

As a possible avenue to pursue, the post on Biennial Connection from QBO to ENSO seems to be the most promising direction, as it allows for a plausible phase reversal mechanism in the ENSO standing wave. I'll keep on kicking the rocks to see if anything else pops out.

 

Pukite's Model of the Quasi-Biennial Oscillation

I decided to name this model after myself because there are no free parameters and so is locked into place. There's nowhere to hide if it is invalidated, but it is so concise and precise that it's likely worth the risk of attaching my name to it.

The idea is as described earlier : Find the lunar forcing on the earth and then alias the forcing to a seasonal (yearly) period. This becomes the forcing for the QBO. The rationale is that the faster lunar cycles will not cause the stratospheric winds to change direction, but if these cycles are provoked with a seasonal peak in energy, then a longer-term multiyear period will emerge. This is a well-known mechanism that occurs in many different natural phenomena.

There are two steps to the model. (1) Determine the lunar gravitational potential as a function of time, and (2) plot the potential in units of 1 month or 1 year. The last part is critical, as that emulates the aliasing required to remove the sub-monthly cycles in the lunar forcing.

If one then matches this plot against the QBO time-series, you will find a high correlation coefficient. If the lunar potential is tweaked away from its stationary set of parameters, the fit degrades rapidly. So it becomes essentially a binary match. If it didn't fit, then the lunar gravitational potential hypothesis would be invalidated. But since it does fit precisely, then it remains a highly plausible model.

Figure 1 shows the mean-square potential of the lunar gravitational pull, also known as the tidal generating potential (applied in the context of predicting tides). On the left, the scale is expanded.

Fig. 1 : Mean-square potential of the lunar gravitational pull (from [1]). On the left, the scale is expanded.

As background, I originally discovered the connection of QBO to the lunar potential via machine learning (Eureqa), see Figure 2.

Fig 2: The original connection from QBO to a tide generating potential was discovered by machine learning -- upper right, panel A. The fitted signal was unaliased, squared and shown to align in panel B. Panel C shows the details with the fully unaliased signal at a finer scale.

This fit worked remarkably well considering that it is very difficult to dig out the aliased periods. Letting the machine learning run for a day helped considerably.

Yet it is also useful to reverse the direction of the fitting process. Instead of deducing the model from a sinusoidal decomposition, let us estimate the tidal generating potential as shown in Figure 1 and described by Ray [1]. We then inductively proceed  forward and see how well it fits to the QBO time-series.

Fig 3 : Empirical fit to the tidal generating potential of Figure 1.

This empirical fit uses only three factors -- the lunar cycles corresponding to the Draconic month, the Anomalistic month, and the Tropical month. Those are known to a high precision, along with a value for the Tropical year. The composition of these factors is then squared to generate the empirical model of the potential.

If we lay the empirical model on top of Ray's diagram, it looks like Figure 4.

Fig 4 : Alignment of empirical model with Ray chart. Note the long-term 18.6 year (diurnal) beat period and the shorter 4.425 year (semidiurnal) beat period. Also a rapid bi-annual component.

On the expanded scale, the sub-monthly periods appear, as shown in Figure 5.

Fig 5 : Expanded scale showing the sub-monthly variations in the tidal generating potential.

These higher frequency components disappear when the alias is introduced.

I did not do a complete ephemeris-based empirical model for the tidal generating potential as Ray did, since the basic pattern is fairly easy to deduce from the three lunar cycles.

The final step is to un-square and then alias the tidal generating potential and compare to the QBO time-series. This is shown in Figure 6.

Fig 6: Fit of the unaliased tidal generating potential to the QBO

There is nothing at all complicated about the recipe for fitting the tidal generating potential to the QBO. It is a mechanical process since none of the lunar cycles parameters can be changed.

As a next step I will submit this finding to Physical Review Letters.  From what I have seen in the literature search, there is no consideration of applying a straightforward forcing of the lunar gravitational pull to model QBO.  It appears that most QBO models derive from what Richard Lindzen originally proposed some 40+ years ago -- but since many mainstream climate scientists do not consider Lindzen (an AGW denier) very trustworthy or even competent (e.g. a trail of retracted papers and debunked theories), it's likely that his original model was simply wrong, or at best, incomplete. What the new model does is provide a concise recipe and a highly plausible geophysical context for understanding the origin of QBO.

The further significance of all this is that the same lunar forcing that applies to QBO also likely applies to the phenomena of El Nino and modeling the ENSO time-series, see the ENSO sloshing paper and some more recent work.

References

[1] R. D. Ray, “Decadal climate variability: Is there a tidal connection?,” Journal of climate, vol. 20, no. 14, pp. 3542–3560, 2007.

ENSO redux

I've been getting push-back on the ENSO sloshing model that I have devised over the last year.  The push-back revolves mainly about my reluctance to use it for projection, as in immediately.  I know all the pitfalls of forecasting -- the main one being that if you initially make a wrong prediction, even with the usual caveats, you essentially don't get a second chance.   The other problem with forecasting is that it is not timely; in other words, one will have to wait around for years to prove the validity of a model.   Who has time for that ? 🙂

Yet, there are ways around forecasting into the future. One of which primarily involves using prior data as a training interval, and then using other data in the timeline (out-of-band data) as a check.

I will give an example of using training data of SOI from 1880 - 1913 (400 months of data points) to predict the SOI profile up to 1980 (800 months of data points). We know and other researchers [1] have confirmed that ENSO undergoes a transition around 1980, which obviously can't be forecast.   Other than that, this is a very aggressive training set, which relies on ancient historical data that some consider not the highest quality. The results are encouraging to say the least.

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Paper on Sloshing Model for ENSO

I recently archived the paper to ARXIV and submitted to PRL.

Get the paper here from ARXIV as a PDF.

The nearly year-long investigation is time-lined and outlined here.

The final model fit:

Fig 1: Figure from the paper. When the correlation reached 0.8, I thought it might be the right time to stop. The noise in the Darwin or Tahiti time-series was the limiting factor in how good the fit could eventually become.

Thanks for the good comments!

One application of the model described below

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An ENSO Predictor Based on a Tide Gauge Data Model

Earlier this year, I decided to see how far I could get in characterizing the El Nino / Southern Oscillation through a simple model, which I referred to as the Southern Oscillation Index Model, or SOIM for short (of course pronounced with a Brooklyn accent). At the time, I was coming off a research project where the task was to come up with simple environmental models, or what are coined as context models, and consequently simple patterns were on my mind.

So early on I began working from the premise that a simple nonlinear effect was responsible for the erratic oscillations of the ENSO. The main candidate, considering that the ENSO index of SOI was clearly an oscillating time-series, was the Mathieu equation formulation. This is well known as a generator of highly erratic yet oscillating waveforms.  Only later did I find out that the Mathieu equation was directly used in modeling sloshing volumes of liquids [1][2]  --  which makes eminent sense as the term "sloshing" is often used to describe the ENSO phenomena as it applies to the equatorial Pacific Ocean (see here for an example).

Over the course of the year I have had intermittent success in modeling ENSO with a Mathieu formulation for sloshing, but was not completely satisfied,  largely due to the overt complexity of the result.

However, in the last week I was motivated to look at a measure that was closer to the concept of sloshing, namely that of sea surface height. The SOI is an atmospheric pressure measure so has a more tenuous connection to the vertical movement of water that is involved in sloshing. Based on the fact that tidal gauge data was available for Sydney harbor (Fort Denison here)  and that this was a long unbroken record spanning the same interval as the SOI records, I did an initial analysis posted here.

The main result was that the tidal gauge data could be mapped to the SOI data through a simple transformation and so could be used as a proxy for the ENSO behavior. The excellent correlation after a delay differential of 24 months is applied  is shown in Figure 1 below.

Fig 1:  The first step is to map a proxy (tide gauge data) to the SOI data

That was the first part of the exercise, as we still need to be able to quantify the tidal sea surface height oscillations in terms of a Mathieu type of model. Only then can we make predictions on future ENSO behavior.

As it turns out the model appears to greatly simplify, as the forcing, F(t), for the right hand side (RHS) of the Mathieu formulation consists of annual, biannual (twice a year), and biennial (once every two years) factors.

 \frac{d^2x(t)}{dt^2}+[a-2q\cos(2\omega t)]x(t)=F(t)

The last biennial factor, though not well known outside of narrow climate science circles [3], is critical to the model's success.

Although the Mathieu differential equation is simple, the solution requires numerical computation. I (along with members of the Azimuth Project) like to use Mathematica as a solver.

The complete solution over a 85-year span is shown in Figure 2 below

Fig 2: The second step is to model the tidal data in terms of a sloshing formulation. The biennial factor shows a phase reversal around 1953, switching from an even to odd year periodicity. The yellow highlighted area is one of the few regions that a correlation is clearly negative. Otherwise the fit models the behavioral details quite effectively.

This required an optimization of essentially three Mathieu factors, the a and q amplitudes, and the ω modulation (along with its phase). These are all fixed and constitute the LHS of the differential equation.  The RHS of the differential equation essentially comprises the amplitudes of the annual, biannual, and biennial sinusoids, along with phase angles to synchronize to the time of the year. And as with any 2nd-order differential equation, the initial conditions for y(t) and y'(t) are provided.

As I began the computation with a training interval starting from 1953 (aligning with the advent of QBO records), I was able to use the years prior to that for a validation.  As it turns out, the year 1953 marked a change in the biennial phase, switching from odd-to-even years (or vice versa depending on how it is defined).  Thus the validation step only required a one-year delay in the biennial forcing (see the If [ ] condition in the equation of Figure 2).

The third step is to project the model formulation into the future. Or further back into the past using ENSO proxies. The Azimuth folks including Dara and company are helping with this, along with two go-to guys at the U of MN who shall remain nameless at the present time, but they know who they are.

Ultimately, since the model fitting of the tide data works as well as it does, with the peaks and values of the sloshing waters effectively identified at the correct dates in the time series, it should be straightforward to transform this to an ENSO index such as SOI and then extrapolate to the future. The only unknown is when the metastable biennial factor will switch odd/even year parity.  There is some indication that this happened shortly after the year 2000, as I stopped the time series at this point.  It is best to apply the initial conditions y and y' at this transition to avoid a discontinuity in slope, and since we already applied the initial conditions at the year 1953, this analysis will have to wait.

The previous entries in this series are best observed by walking backwards from this post, and by visiting the Azimuth Forum.   Science is messy and nonlinear as practiced, but the results are often amazing.  We will see how this turns out.

References

[1] Faltinsen, Odd Magnus, and Alexander N Timokha. Sloshing. Cambridge University Press, 2009.
[2] Frandsen, Jannette B. “Sloshing Motions in Excited Tanks.” Journal of Computational Physics 196, no. 1 (2004): 53–87.
[3] Kim, Kwang-Yul, James J O’Brien, and Albert I Barcilon. “The Principal Physical Modes of Variability over the Tropical Pacific.” Earth Interactions 7, no. 3 (2003): 1–32.

 


Keep it Lit

Good luck to the People's Climate Marchers.  I read Bill McKibben's book Long Distance several years ago, and realize that persistence and endurance pays off. I also realize that there are no leaders in the movement, and that we all have to pull together to get off of fossil fuel.  If we each do our share, the outcome will tend more toward the good than to the bad.

 

 

The SOIM: substantiating the Chandler Wobble and tidal connection to ENSO

(see later posts here)

My previous posts on modeling the Southern Oscillation Index as a periodically modulated wave equation -- in particular via the Mathieu equation -- are listed below:

  1. The Southern Oscillation Index Model
  2. SOIM and the Paul Trap
  3. The Chandler Wobble and the SOIM

The first post introduced the Mathieu equation and established a premise for mathematically modeling the historical SOI time-series of ENSO, the Southern Oscillation part of the El Nino/Southern Oscillation phenomenon.  The second post was an initial evaluation of a multivariate fit, evaluated by exploring the parameter space.  The third post was a bit of a breakthrough, which focused on a specific periodic process -- the Chandler Wobble (CW) -- which appeared to have a strong causal connection to the underlying SOI model.

This short post effectively substantiates the Chandler Wobble connection and provides nearly as strong support that other tidal beat periodicities force the modulation as well.

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