In formal mathematical terms of geometry/topology/homotopy/homology, let’s try proving that a wavenumber=0 cycle of east/west direction inside an equatorial toroidal-shaped waveguide, can only be forced by the Z-component of a (x,y,z) vector where x,y lies in the equatorial plane.

To address this question, let’s dissect the components involved and prove within the constraints of geometry, topology, homotopy, and homology, focusing on valid mathematical principles.

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That’s simply a proof of a mathematical construction. Proofs are possible in math since the premise and rules are absolutely known. However, in real-world physics we can’t absolutely prove anything because we do not know if the premise holds and whether the math rules actually apply. Yet, in the context of the construction supplied above, we can suggest a hypothesis that would apply to a real-life torus in the form of the QBO of equatorial stratospheric winds.¹ This takes the shape of a toroidal waveguide with longitudinal winds that travel strictly east or west, thus a wavenumber=0 mode. That set of characteristics is based on observations of the QBO’s behavior since 1952.

So, the question arises as to whether the QBO has been cast in these terms in the research literature. It’s entirely possible that no one has ever thought through the constraints of QBO topology and what is allowed or not in terms of behavior. For example, the forcing in vector (x,y,z) terms with respect to lunar tidal forces would be — save for phase shifts and perigean amplitude scaling² — (x,y,z) = (sin(w_Tt)cos(w_Dt), cos(w_Tt)cos(w_Dt), sin(w_Dt)) in the long-period case. Here, w_T is the tropical or sidereal angular frequency corresponding to 27.3216 days, and w_D is the draconic (sometimes draconitic, also known as a nodal month or nodical month³ ) angular frequency corresponding to 27.2122 days. Note that these aren’t the diurnal frequencies taking account the Earth’s daily rotation, but the path taken at a snapshot taken once per day tracing out the traversal of the Moon in terms of longitude and latitude. (The exact diurnal periods can be calculated from these values). And of course, we can’t forget that the seasonal forcing due to the Sun — sin(w_St) — is superficially homologous to the Draconic cycle, as both are declinations in Z.

Now we can start to put 2 and 2 together, mapping the lunar orbit to the mathematical QBO model and determining the resultant critically constrained terms. This will allow us to do a literature search and find out whether other past research have in fact made any claims or considerations to the applicability of these terms to a model of QBO.

First, consider that according to the proof of the purely mathematical toroidal waveguide model, only the Draconic cycle (and seasonal solar cycle) can contribute to a wavenumber=0 behavior (as observed in the real QBO). However, the Tropical/Synodic cycle can’t since that only exists in the equatorial x,y plane.

Second, consider that the Draconic cycle could interact in an aliased or modulo fashion with the seasonal cycle, when one considers that they may be constructively reinforcing at a specific time of the year, thus introducing satellite sub-bands to the spectrum. This would mean that the following aliased frequency is relevant:

w_D / w_S mod 1 = 0.422 per year

In fact, that frequency corresponds to 2.37 years which is the nominal period for QBO and what has been used to fit to the data, as the last blog post has demonstrated.

So that connects the mathematical proof of the abstract toroidal waveguide allowing only wavenumber=0 cycles with an empirical model of QBO forced by Draconic and seasonal cycles, with the result parsimoniously fitting that observed in the real QBO (and the SAO if only the seasonal cycle is considered).

If this is not a novel concept, a literature search of the terms [Draconic, “QBO”, aliased, sub-bands, …] should return hits. Via Google Scholar, prompting with only the terms “QBO” and “Draconic” returns 4 hits to my model and only 1 to a paper that likely only fortuitously matches, highlighted in violet below.

As a cross-check search on “27.212” and “QBO” reveals only the following, which is actually an editorial comment of mine made during a 2020 EGU meeting presentation (comments no longer linked to, but I had made passing reference to them on the blog).

Unfortunately, Google Scholar does not cite the main source Mathematical Geoenergy. Have to search Google Books to find that reference:

All this is perhaps understandable in that the Draconic period is not a cliche “go to” tidal cycle to cite when looking for causal factors. This is rationalized because the Tropical/Sidereal factor explains much more in terms of geophysical behaviors in the lithosphere. In contrast to the stratosphere, the lithosphere provides longitudinal torque points for a tidal force to apply longitudinal-specific traction to. For example, the elevated continents and Himalayas are considered as the drag that provides the angular momentum response for changes in the Earth’s Length of Day (LOD) due to the tropical Mf lunar tidal factor (i.e. 2w_T and the Mf’ w_T+w_D). Same consideration obviously applies with localized ocean tides — it’s primarily the Mf (and Mm perigean w_A) that supplants the diurnal/semidurnal cycles. Yet, the stratosphere is sufficiently decoupled from this tractive effect that the Tropical lunar factor is at best a secondary factor — indeed, I do add the other tidal factors (including the Mm anomalistic 27.554 day cycle) to justifiably improve the fit to higher wavenumber characteristics of the QBO behavior, as QBO is observationally and likely not a pure wavenumber=0 behavior.

It may be informative to try to understand how the role of tides in the QBO behavior may have evolved over time in the research community. It likely starts and stops with the research of the atmospheric physicist Richard Lindzen, sometimes referred to as the godfather of QBO modeling. Apparently, he made early claims to the inapplicability of any tidal factors, inferring that he had considered causal effects but was not able to discern any correlations between the measured periods. These were comments made in 1967 and 1974.

And that’s where the trail essentially ends in investigating any causal link between QBO and tides. Lindzen (born 1940) was likely the last arbiter and adjudicator of applicability in the research timeline. The tribal knowledge has become the conventional wisdom that “tides have nothing to do with QBO”. That’s too bad, but as with all science, it advances one funeral at a time.⁴ The next generation of researchers, or perhaps the machine learning community, will revisit this when Lindzen’s authoritarian legacy ends. They, especially ML, may not accept the preconceived notions and tribal knowledge as to what is supposed to work or not, see for example this topological research or this example applying persistent homology based on topological data analysis.

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Footnotes

¹ If the physical application was a circular or toroidal coil oriented in an x-y plane as used in a metal detector device, the real-world analogy could be validated via a controlled experiment. The model would apply the laws of electro-magnetism (Maxwell’s equations, Faraday’s law, Lenz’s law), while the experimental test would involve moving a metal piece along the z-axis and noting a reversal in coil current when that metal piece was reversed in motion. No change in current would be induced if the metal piece was moved laterally by a delta amount in the x-y plane. An experiment would confirm this. Unfortunately, controlled experiments are not possible with the Earth, and only the rough equivalence between E-M fields and gravitational fields would serve as a sanity check analogy or as a thought experiment.

² The perigean amplitude involves the anomalistic month of 27.554 days which modulates the distance R from the Moon to the center of the Earth over time, affecting all vector components via R*(x,y,z). This fluctuation in R, modeled as R(t) = R_mean ​+ΔR sin(w_A​t), where ΔR is the variation in the Moon’s distance from the Earth due to its elliptical orbit. Of course, the gravitational forcing strength is inverse to R. The vector v=R*(x,y,z) then obeys the trig rule |v|² = R² * (x²+y²+z²) = R².

³ Don’t confuse the nodal month with the 18.6 year nodal cycle here. The 18.6 year nodal cycle comes about from the beat frequency interference between the nodal month and tropical month via 1/(1/27.2122 – 1/27.3216) = 18.6 * 365.242 days. So that care must be taken in precisely stating the lunar factor ascribed to — whether it is a nodal crossing cycle (Draconic) or a nodal extremum cycle at a specific longitude (18.6 year cycle) — otherwise ambiguity may arise.

⁴ The adage that “science advances one funeral at a time” is often attributed to Max Planck. This statement reflects Planck’s observation on the resistance within the scientific community to revolutionary ideas and paradigms. He is quoted as saying, “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.”