I just found this relatively obscure article in the [Volume 6 No. 1, March 2001 Newsletter of the Climate Variability and Predictability Programme (CLIVAR)](http://nora.nerc.ac.uk/119292/1/ex19.pdf)

*"On the Role of a quasiperiodic Forcing in the interannual and interdecadal Climate Variations"*

D.M. Sonechkin, N.N. Ivachtchenko, Hydrometeorological Research Center of Russia

> "Mention in this context the so-called Chandler wobble in the Earth’s pole motion. The mean period of this

wobble is about 14 months (1.2 year), and the ratio of its frequency to the frequency of the annual period seems to be similar to the “worst” irrational number Y=0.8393... derived from the above root of the cubic equation (Sidorenkov and Sonechkin, 1999). It is well known that the Chandler wobble excites a pole tide in the atmosphere. For certain, this pole tide forces the equatorial gravity and Kelvin waves that are known in the Lindzen–Holton theory to be the direct drivers of the QBO of the lower-stratospheric equatorial zonal winds. Although the magnitude of this pole tide is very small a nonlinear mechanism of its force enhancing like the so-called parametric resonance may be supposed. Some similar enhanced effects of the pole tide may also be supposed for the atmospheric variations within extratropics.

> Thus the atmosphere turns out to be forced by a quasi-periodic manner. Therefore, the power spectra of the atmospheric variations must reveal some, possible subtle, peaks at both annual and Chandlerian frequencies and their combinational harmonics. In particular, a sequence of some peaks must exist at the difference frequency Z=1-Y=0.1607... (the oscillation of the about 6.5 years period), Z^2 (the oscillation of the about 40 year period) etc. as a consequence of the above-mentioned self-similarity of the power spectra of the strange nonchaotic attractor. This sequence can be also shifted along the frequency axis as a whole, so that all of the underlain periods are doubled, tripled, or quadrupled. For example, it is well known that the spectrum of the equatorial QBO reveals the main peak at the frequency of the doubled Chandlerian period (of the about 28 months, i.e. 2.4 years). The doubled period corresponding to Z (of the about 13-14 years) is also observed (Vlasovaet al. 1987). Unfortunately, the length of the equatorial lower-stratospheric zonal wind record is too short to admit an accurate estimation of the energy of the wind oscillations at the frequency Z^2."

I am applying precisely this forcing configuration to the ENSO model. In the residual to the model fit, I was seeing a small but very distinct spectral component at a 3.65 year period. Now that makes sense because it is the difference frequency, $CW-QBO$.

I am beginning to think that perhaps ENSO is an analagous manifestation of QBO, but instead of the resonance operating in the atmosphere, it is manifesting as an oceanic variation.

The citations to the Sonechkin article are sparse, but there is this:

G. Fuhrmann, “Non-Smooth Saddle-Node Bifurcations of Forced Monotone Interval Maps I: Existence of an SNA,” arXiv preprint [arXiv:1307.0347](http://arxiv.org/pdf/1307.0347.pdf), 2013.

> "Quasi-periodic forcing and SNAs play an important role in a large class of models for real life systems: The Harper map is a mathematically well-understood dynamical system related to a certain kind of quasi-periodic Schrödinger equations (see below); there is numerical evidence for the existence of SNAs in the physiologically relevant Izhikevich Neuron Model [17]; [22-*Soneckin*] motivates that not just in order to get a complete description of the tides -as the result of the gravitational interaction between the Earth, the Moon, and the Sun- but even to predict interdecadal atmospheric variations, strange non-chaotic attractors have to be considered. "

Hill or Mathieu equations show up in Earth-Moon-Sun orbital perturbation theory.

---

The following paper essentially calls out the deterministic nature of these oscillations -- perhaps no different in concept to first-order tidal theory, but hampered by the non-linearities in the hydrodynamics.

S. Hameed, R. G. Currie, and H. Lagrone, “Signals in atmospheric pressure variations from 2 to ca. 70 months: Part I, simulations by two coupled ocean—atmosphere GCMs,” International journal of climatology, vol. 15, no. 8, pp. 853–871, 1995.

> "The two sets of parents and 33 tones vary in period from 2 to ca. 70 months and demonstrate that the spectrum of climate on these time-scales is ‘signal-like’ rather than ‘noise-like’ as traditionally believed"

I do not think I see any noise at all with the latest ENSO model fit.

*"On the Role of a quasiperiodic Forcing in the interannual and interdecadal Climate Variations"*

D.M. Sonechkin, N.N. Ivachtchenko, Hydrometeorological Research Center of Russia

> "Mention in this context the so-called Chandler wobble in the Earth’s pole motion. The mean period of this

wobble is about 14 months (1.2 year), and the ratio of its frequency to the frequency of the annual period seems to be similar to the “worst” irrational number Y=0.8393... derived from the above root of the cubic equation (Sidorenkov and Sonechkin, 1999). It is well known that the Chandler wobble excites a pole tide in the atmosphere. For certain, this pole tide forces the equatorial gravity and Kelvin waves that are known in the Lindzen–Holton theory to be the direct drivers of the QBO of the lower-stratospheric equatorial zonal winds. Although the magnitude of this pole tide is very small a nonlinear mechanism of its force enhancing like the so-called parametric resonance may be supposed. Some similar enhanced effects of the pole tide may also be supposed for the atmospheric variations within extratropics.

> Thus the atmosphere turns out to be forced by a quasi-periodic manner. Therefore, the power spectra of the atmospheric variations must reveal some, possible subtle, peaks at both annual and Chandlerian frequencies and their combinational harmonics. In particular, a sequence of some peaks must exist at the difference frequency Z=1-Y=0.1607... (the oscillation of the about 6.5 years period), Z^2 (the oscillation of the about 40 year period) etc. as a consequence of the above-mentioned self-similarity of the power spectra of the strange nonchaotic attractor. This sequence can be also shifted along the frequency axis as a whole, so that all of the underlain periods are doubled, tripled, or quadrupled. For example, it is well known that the spectrum of the equatorial QBO reveals the main peak at the frequency of the doubled Chandlerian period (of the about 28 months, i.e. 2.4 years). The doubled period corresponding to Z (of the about 13-14 years) is also observed (Vlasovaet al. 1987). Unfortunately, the length of the equatorial lower-stratospheric zonal wind record is too short to admit an accurate estimation of the energy of the wind oscillations at the frequency Z^2."

I am applying precisely this forcing configuration to the ENSO model. In the residual to the model fit, I was seeing a small but very distinct spectral component at a 3.65 year period. Now that makes sense because it is the difference frequency, $CW-QBO$.

I am beginning to think that perhaps ENSO is an analagous manifestation of QBO, but instead of the resonance operating in the atmosphere, it is manifesting as an oceanic variation.

The citations to the Sonechkin article are sparse, but there is this:

G. Fuhrmann, “Non-Smooth Saddle-Node Bifurcations of Forced Monotone Interval Maps I: Existence of an SNA,” arXiv preprint [arXiv:1307.0347](http://arxiv.org/pdf/1307.0347.pdf), 2013.

> "Quasi-periodic forcing and SNAs play an important role in a large class of models for real life systems: The Harper map is a mathematically well-understood dynamical system related to a certain kind of quasi-periodic Schrödinger equations (see below); there is numerical evidence for the existence of SNAs in the physiologically relevant Izhikevich Neuron Model [17]; [22-*Soneckin*] motivates that not just in order to get a complete description of the tides -as the result of the gravitational interaction between the Earth, the Moon, and the Sun- but even to predict interdecadal atmospheric variations, strange non-chaotic attractors have to be considered. "

Hill or Mathieu equations show up in Earth-Moon-Sun orbital perturbation theory.

---

The following paper essentially calls out the deterministic nature of these oscillations -- perhaps no different in concept to first-order tidal theory, but hampered by the non-linearities in the hydrodynamics.

S. Hameed, R. G. Currie, and H. Lagrone, “Signals in atmospheric pressure variations from 2 to ca. 70 months: Part I, simulations by two coupled ocean—atmosphere GCMs,” International journal of climatology, vol. 15, no. 8, pp. 853–871, 1995.

> "The two sets of parents and 33 tones vary in period from 2 to ca. 70 months and demonstrate that the spectrum of climate on these time-scales is ‘signal-like’ rather than ‘noise-like’ as traditionally believed"

I do not think I see any noise at all with the latest ENSO model fit.