[mathjax]I recently posted a bog article called QBO Model Final Stretch. The idea with that post was to give an indication that the physics and analytical math model explaining the behavior of the QBO was in decent shape. I would like to do the same thing with the ENSO model but retain the context of the QBO model.  Understanding the QBO was a boon to making progress with ENSO as it provided an excellent training ground for time-series analysis and also provided some insight into the underlying forcing functions.  In the literature, there is a clear indication that ENSO and QBO are somehow related, but the causality chain remains unclear.

The relation between ENSO and QBO is more a transitive one as opposed to direct, as the underlying mechanisms are partially common to the two, but I don’t believe that one forces the other (or vice versa) as some research proposes.

QBO forcing is resolved primarily as a Draconic or nodal lunar gravitational pull leading to an atmospheric tidal cycle. The math behind this assumes Laplace’s tidal equations, with a small-angle approximation at the equator. The key premises are that the analytically derived measure is the QBO acceleration (not velocity) and the nodal (latitudinal) forcing is strongly seasonally modulated.  The important periods are interferences of 27.212 days (the Draconic lunar month) with a strongly peaked seasonal signal.

In terms of frequencies where $$f = 365.242/27.212 y^{-1}$$, the list is

The frequency that is closest to $$1 y^{-1}$$ is $$1/f-13$$ = 0.422. This has a period of the reciprocal of this or 2.369 years. More in-depth math here.

This is sometimes referred to as “nonlinear aliasing” or “natural aliasing” as it comes about from the nonlinear product of more than 1 frequency, leading to a spread of selected harmonics.

There is a good explanation of the effect in this meteorology book : “Mesoscale Dynamics”, Yuh-Lang Lin, Cambridge University Press, 2007

That book references the AGW skeptic Roger Peilke, who wrote about the effect more recently here, “Mesoscale Meteorological Modeling”, Roger A. Pielke Sr. Elsevier, 2013

It can also lead to noisy, spiky behavior in periodograms: Assessing statistical significance of periodogram peaks

So according to textbooks on meteorology, this effect can occur. And to top that off, this behavior is described in textbooks on mesoscale phenomena, of which the QBO of stratospheric winds is a prime example occurring at the extreme upper end of the scale.

ENSO appears also to be influenced by lunar forces but strongly conflated by similar frequencies related to the Chandler wobble.  This is a conflating issue since both lunar forcing and the earth’s wobbles are both associated with periodic changes in the angular momentum of the earth, yet are difficult to separate — especially when one considers how closely the values overlap with nonlinear aliasing.  Angular momentum variations ala a wobble will definitely impact a volume with a larger mass inertia such as the ocean, while that effect is considered small for the much smaller atmospheric embodied in the QBO. Rightly so that the QBO is governed completely by the lunisolar tidal forces.

Therefore, to disambiguate the ENSO forcings, we have to see how the lunar and wobble forcings differ in their detail.

Consider the example of the nonlinear aliasing of the 1/2-Draconic tide of 13.606 days. This is the period it takes for the moon to take one excursion from the equator to a maximum latitude.  When the folded aliasing reaches its longest period — the 26th folding — then the next will actually be negative.

These are two periods corresponding to 1.185 years and -6.42 years, which are essentially close to the Chandler wobble (1.185y=433d) and to its beat frequency against the yearly wobble (6.42y).

The interesting aspect to this is that these pairs of frequencies can be equated to an exact biennial modulation of a specific frequency.

where the second term is $$2/(2/1.185-1)$$, which in this case is ~2.89 years.  For the Draconic period, the folded values are 2.369 y and -1.73 years, leading to a modulating period of $$2/(2/2.369-1)$$ or ~12.84 years.

The leading biennial term will always exist, but obscured from direct measurement because it is implicitly mixed in to the waveform.

To see this in action, consider the anomalistic period of 27.554 days. The folded values are 3.92y and -1.34 years, leading to a modulating period of $$2/(2/3.92-1)$$ or ~4.085 years.  You can see that in the cyclic representation below, where the weak interference of the 4.085 year cycle with the 2 year biennial modulation retains a clear 4 year period.   Also shown as the red-dotted line is the original 29.554 day tide modulated against a strong yearly signal. That creates the delta spikes which the paired cycles represent as the strongest Fourier factors (with the QBO extra aliased harmonics are observed).

Fig 1 : Nonlinear aliasing of the anomalistic month.

Putting all these together, the following modulating periods are associated with the following long period tides:

27.212 Draconic = 12.84 years
13.606 half-Draconic = 2.89 years
27.322 Tropical = 7.57 years
27.554 Anomalistic = 4.085 years
13.633 Mf’ = 3.43 years (weakest, cross term of Draconic and Tropical)

Also need to consider the wobble terms

Chandler wobble = 6.4 to 6.5 years
Triaxial wobble = 14 years,  according to Wang
Node modulation = 18.6 years
Half the Node modulation = 9.3 years
Chandler sideband wobble  = 1.45 years, see ref [1]

The twist for modeling a forced sloshing is to incorporate a Mathieu modulation in the wave equation. Earlier I found that a 2 year and 1 year modulation improved the fit significantly. The 1 year modulation is understandable as this can impact density differences in the water over the course of a season. But the 2 year modulation is more mysterious in that it can only occur from interactions with a 1 year cycle or other known cycle, either through a period doubling mechanism or perhaps via the aliased tidal terms above.  This modulation can also be applied to the RHS forcing — and again, often confoundingly, will imitate the biennial factor that already occurs in the tidal terms

The fit using the terms above works well using this a Mathieu formulation, with a couple of caveats.

Fig 2: ENSO model fit. The complexity of this is similar to a tidal model featuring several interfering lunisolar periods.

The big caveat is that the 14 year wobble term completely supercedes the 12.84 year modulation on the 27.212 aliased Draconic term (highlighted in bold above).  The former gives a 2.33 year Fourier component when a 14 year cycle is modulated by a biennial factor while the latter generates a 2.37 year aliased period on its own. The difference between a 2.33 year period and a 2.37 year period may not appear significant but it will generate a significant phase error against the data when extended over 100+ years.  A 2.33 year cycle also has the interesting property that it exactly aligns with a seasonal point every 7 years.

In the paper discussed in the last post, Hanson,Brier noted that both 13 year (close to 12.84) and 14 year periods appeared in the El Nino historical record. As shown below, the 14 year period is slightly stronger than the 13 year period according to an interesting statistical sampling technique they applied (2-D Buys Ballot).

This may have something to do with switching of long term cyclic states in the ENSO system. Having already identified an odd/even phase inversion of the biennial modulation between the years 1980 and 1996, a growing phase difference between the forcing cycles of 13 and 14 years may eventually trigger a shift. Hanson,Brier presented a similar argument as well.

I am curious why these future studies never occurred.

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Incidentally, we just had a Harvest Moon yesterday. Over in the Azimuth Project forum, I describe how the calculation of Easter Sunday and the Harvest Moon dates use the same aliasing math that I have used for QBO and ENSO.  That analogy proved useful to get at least one other person to understand the general idea of the seasonal aliasing argument.

References