# Tidal Locking and ENSO

As I have been formulating a model for ENSO, I always try to relate it to a purely physical basis. The premise I have had from the beginning is that some external factor is driving the forcing of the equatorial Pacific thermocline. This forcing stimulus essentially causes a sloshing in the ocean volume due to small changes in the angular momentum of the rotating earth. I keep thinking that the origin is lunar as the success of the QBO model in relating lunisolar forcing to the oscillatory behavior of the QBO winds is enough motivation to keep on a lunar path.

Yet, I am finding that the detailed mechanism for ENSO differs from that of QBO. An interesting correlation I found is in the tidal-locking of the Earth to the moon. I think this is a subset of the more general case of spin-orbit resonance, where the rotation rate of a satellite is an integral ratio of the main body. In the case of the moon and the earth, it explains why the same moon face is always directed at the earth -- as they spin at the same rate during their mutual orbit, thus compensating via a kind of counter-rotation as shown in the left figure below.

Fig 1: Tidal Locking (left) results in the spin of the moon

# Biennial Mode of SST and ENSO

This recent 2016 paper [1] by Kim is supporting consensus to my model of a modulated biennial forcing to ENSO. I had read some of Kim's earlier papers [2] where he introduced the idea of cyclostationary behavior.

The insight that they and I share is that the strictly biennial oscillation is modulated by longer frequencies such that +/- sideband frequencies are created around the 2-year period.

$sin(\pi t) \cdot sin(\omega_m t) = \frac{1}{2} ( cos(\pi t - \omega_m t) - cos(\pi t + \omega_m t) )$
$.$

This is not aliasing but essentially a non-aliased frequency modulation of the base cycle. The insight is clarified by Kim with respect to Meehl's [3] tropospheric biennial oscillation (TBO).

From [1]

That biennial mode locks the base frequency in place to a seasonal cycle, with the modulation creating what looks like a more chaotic pattern. That's why ENSO has been so stubborn to analysis, in that the number of Fourier frequencies doubles with each modulating term. Yet in reality it's likely half as complicated as most scientists have been lead to believe.

# Seasonal Aliasing of Tidal Forcing in Mean Sea Level Height

I applied the QBO aliased lunar tidal model to another measure that seems pretty obvious -- long term monthly time series data of sea-level height (SLH), in this particular case tidal gauge readings in Sydney harbor (I wrote about correlating Sydney data to ENSO before -- post 1 and post 2).

The key here is that I used the second-derivative of the tidal data for the multiple regression fit:

Fig 1:  QBO factors over the training interval

This is remarkable as it applies the as-is QBO factors to Sydney training data from 1940 to 1970. That is, the parameters are solely derived from the critical aliased lunisolar periods used to optimize the QBO fit.

# 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:

# Inferring forced response from QBO wave equation

If the wave equation is to be effective in modeling a forced behavior such as QBO then certain invariant properties would need to be observed.

$f''(t) + \omega_0^2 f(t) = F(t)$

So if we split this into two parts

$f''(t) \approx k_1 F(t)$

and

$\omega_0^2 f(t) \approx k_2 F(t)$

I have been applying multiple linear regression to fit the wave equation via either of the above two forms

So if we combine these two and see how well the Fourier series coefficients align -- keeping care to compensate for the $\omega^2$ scaling factor after taking the second derivative of a sinusoid -- the linear result should be:

$\frac{f''(t)}{f(t)} = \omega_0^2 k_1/k_2$

and indeed ! (see Figure 1 below)

Fig 1: Correlation between direct f(t) and second-derivative f''(t) coefficient strengths. It should be linear with an intercept at the origin. The strongest term in the upper right is the aliased Draconic tidal period.  The rest are other lunar periods, solar periods, and aliased lunar harmonics.

# Forced versus Natural Response, not a secret

This is a concept that is beaten into the head of electrical engineers.

Engineers tend to assume the forced response first, as being pessimists they always assume the worst case.  For example: Why is that annoying 60 Hz hum coming out of that circuit?  hmmm .. probably because the AC power somehow leaked into the input and the hum resulted from the forced response of the internal transfer function?  Issue solved.

Physicists tend to look for a natural response first. Being optimists, a research physicist first and foremost wants to make a discovery. A natural response would tell us something interesting about the behavior ... and perhaps a new physical law!  Look and See! The eigenvalues!   Often to them, the forced response is a nuisance that needs to be removed to observe the natural response.

But what happens when the forced response is actually the interesting physical behavior?

A prototypical example of a forced response is the phenomenon of cyclic ocean tides. The periods of tides, though complex in breadth, are merely a reflection of the orbital rates of the moon and sun.

Pictures of (left) high tide (right) low tide, linked from Science Buddies

Obviously the ocean has a natural response to a forcing stimulus, obeying some variation of the wave equation, but the overriding output directly links to the observed orbits. Even the second-order effects such as the 9-day tides are simply beats arising from nonlinear interactions of the fortnightly and monthly long period tides.  Everything from the semi-diurnal periods to the long-period monthly periods are forced responses to the gravitational tug of the lunisolar orbits.  Any resonant frequencies in the ocean's natural response are over-ridden by the forced response.

And so it likely goes for the quasi-biennial oscillations (QBO) of the equatorial stratospheric winds, as these most definitely are a forced response to the inputs ... in spite of what the current literature infers.

# QBO Model Validation

The strongly periodic Quasi-Biennial Oscillation (QBO) of stratospheric winds is externally forced by a cyclic mechanism. This is obvious in spite of what the current scientific literature says about the nature of QBO;  the general consensus being that the QBO emerges as a complex natural resonant response of other atmospheric factors [1]. Yet, an experienced experimental scientist should not consider that as the only plausible premise. It is in fact exceedingly rare for a phenomena of that global a scale to be the result of a natural resonance, and for the QBO to be a resonance is clearly an untenable hypothesis based on historical precedent. There simply aren't any climate phenomena that behave similarly and are not reliant on lunar or solar periods.

What causes the oscillation then?

# 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.

# If the glove don't fit ...

To finish off that sentence, "If the glove don't fit, you must acquit". One of the most common complaints that I've received whenever I describe fitting models to data, is that "correlation does not equal causation". I especially get this at Daily Kos with respect to any comment I make to science posts. One knee jerk follows me around and reminds me of this bit of wisdom like clockwork.  I can't really argue the assertion out of hand because it's indeed true that there are plenty of coincidental correlations that don't imply anything significant in terms of causation.

Fig 1:  Correlation of the QBO 30 hPa time-series data.  Does this excellent correlation of data to a model of aliased tidal periods lead to a causation conclusion?  Or is it just a coincidence of the fit chosen?

# Possible Luni-Solar Tidal Mechanism for the Chandler Wobble

Thanks to KO, I have started subscribing to Robert Grumbine's blog. The topic currently is on what provides the pacemaker for the Chandler wobble.

My own opinion, which I hinted at before (and Keith Pickering also pointed out), is the same general mechanism as for the model of the QBO -- but twice as fast. This idea essentially relates that the modulation of the yearly solar orbit (365.242 days) with the draconic (or nodal) lunar month of 27.21222 days sets up the perfect cyclic forcing for the 433 +/- 1.1 days Chandler wobble period.

Think in terms of the maximum declination of (1) the moon with respect to the equator along with (2) the maximum declination of the sun with respect to the equator. For (1) this happens once every ½ of the 27.2122 day nodal cycle or 13.60611 days and for (2) this happens twice a year (once for the southern hemisphere summer and once for the northern hemisphere summer).

Calculating this out, the closest aliased value is $2\pi (365.242/13.606) - 52\pi$ = 5.303 rads/year, equivalent to a period of 432.77 days. That is close to the generally accepted value of 433 days for the wobble [1]. One can also produce this value graphically by sampling a sinusoid of period 13.60611 days every half a year (see below). That's enough to reveal the solar-lunar declination synchronization pretty clearly. In 120 years, I count a little over 101 complete cycles, which is close to the 433 day period.

Sampling of the Draconic sinusoidal signal (period 13.6 days) with a sampling of twice per year.

The Earth has a large inertia compared to the moon, so it is essentially picking up differential changes in gravitational mass forcing, which is enough to get the wobble in motion.

So that's my simple explanation for what drives the Chandler wobble, yet this differs from Grumbine's idea, and from many other theories, many of which refer to it as a free nutation stemming from a resonance [2]. I am closer to Grumbine, who thinks it is planetary-solar while I think it is luni-solar. Without getting into the plausibility of the gravitational dynamics, it's just too simple and parsimonious to pass up without posting a comment. It is also in line with my general thinking that very few natural processes follow a natural resonance, and that external forcing should always be the initial hypothesis. That works for the QBO in particular; in that case, the primary forcing period is the full Draconic cycle, likely due to the stronger asymmetry in lithosphere response for the two hemispheres.

Model of the second derivative of the QBO, featuring factors related only to lunar and solar gravitational forcing cycles.

And besides the QBO, this potential mechanism likely holds more clues to the behavior behind the ENSO model.  As I commented at Grumbine's blog:

@whut said...The Chandler Wobble, QBO, LOD variation, ENSO, Angular Atmospheric Momentum, and oceanic and atmospheric tides all have varying degrees of connection, likely ultimately tied to luni-solar origins.

Keep plugging away at what you are doing because there are likely common origins to much of the behavior. If you can figure it all out, it will be useful in establishing the natural variability in climate, and thereafter the GCMs can then use this information in their models.

## Refs

[1] 435 days also happens to be a commonly cited number (see here), but I get the smaller 433 days when I look at the data myself . And Nastula and Gross recently came up with a value of 430.9 days from newly available satellite measurements.
[2] "On the maintenance of the Chandler wobble" Alejandro Jenkins http://arxiv.org/abs/1506.02810