Model of the Quasi-Biennial Oscillation

(update here)

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.


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

31 thoughts on “Model of the Quasi-Biennial Oscillation

  1. I still have the complete lunar program from Chapront-Touzé & Chapront (Willman-Bell 1991) on my computer. If it would be useful, I can dig it out and give you exact ephemerides, including lunar radius vector, at any timescale you like.


    • Since these periods aren’t *really* sine waves, having more exact knowledge of, e.g., radius vector, gives you a more exact knowledge of lunar grav potential, hence might improve your model. Ditto with exact knowledge of the Moon’s ecliptic longitude and longitude of node, which form the zero points of the tropical and anomalistic months.


      • Keith, yup. There are definitely harmonics which will change the shape away from purely sinusoidal. What will be interesting to see is if these have a significant impact.

        Since harmonics are higher in frequency, they may get damped out in the forcing response. Won’t know for certain until a test fit, but it is good to think about ahead of time.

        Ray spatially averaged out the lunar forcing but yes, the exact geospatial location of the maximum effect may also be important.


  2. I’ve still got the original QBASIC version, and I’ve since translated it into FreeBasic-compatible. Or I can send you a couple of .exe files.


  3. Paul,

    Great work!! Looking forward to the publication in Physical Review Letters!

    A small point:

    You have said: “…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.”

    You might want to refer to the following quote from my paper [2012] paper which discusses the necessity of allowing for annual aliasing when it comes to the interaction of the lunar tidal cycles with the annual solar cycle (i.e. the seasons):

    Wilson IRG. Lunar tides and the long-term variation of the peak
    latitude anomaly of the summer Sub-Tropical High Pressure Ridge
    over Eastern Australia. 2012. Open Atmos Sci J 2012; 6: 49-60.

    Page 50:

    “There are two methods that can be used to work out the possible periods for long-term atmospheric tides. The first method assumes that lunar-tidal forces act independently of the other forcing factors that produce significant long-term variations in atmospheric pressure (e.g. seasonal variations in solar heating). Under this assumption, you would expect to see long-term periodicities in the pressure records that would match periodicities of the most extreme peak lunar tides. The second method investigates what happens when this assumption breaks down.

    The most significant of the large-scale systematic variations of the atmospheric surface pressure, on an inter-annual to decadal time scale, are those caused by the seasons. These variations are predominantly driven by changes in the level of solar insolation with latitude that are produced by the effects of the Earth’s obliquity and its annual motion around the Sun. This raises the possibility that the lunar tides could act in “resonance” with (i.e. subordinate to) the atmospheric pressure changes caused by the far more dominant solar driven seasonal cycles. With this type of simple “resonance” model, it is not so much in what years do the lunar tides reach their maximum strength, but whether or not there are peaks in the strength of the lunar tides that re-occur at the same time within the annual seasonal cycle.”


    • Ian, Sorry about missing that. I will include it when the reviewers ask for a reference. I left that assertion uncited when I said “This is a well-known mechanism that occurs in many different natural phenomena.”

      The machine learning uncovered it as it picked out frequencies differing by 2pi. The problem is that machine learning does not know how to cite the known literature.


      • Thanks, I appreciate the offer. I think that your important break through will eventually be recognized once people really begin to understand what you are on about.


  4. Paul,
    I know that you do not have a high opinion of Prof. Lindzen but be careful about what you say about his work [in toto]. What he says about the magnitude of the effects of the luni-solar tides upon the atmosphere is in large part correct.

    Below ~ 3000 m in altitude (750 mb) the tides caused by the daily thermal expansion of the atmosphere are orders of magnitude strong than than the tiny atmospheric tides produced by the luni-solar tidal cycles.

    Luni-solar tides only become visible above about 3000 m in altitude once you have got above about 40 % of the total atmosphere and so it is only in the upper troposphere and stratosphere were they begin to have an effect. [Hence, the obvious effects upon the QBO induced stratospheric winds].

    Lindzen may have been technically correct to point out at that the dominance of the thermal tides (driven by the Sun) in the lower troposphere dominates the effects of the luni-solar tides but he did not fully appreciate that effects the luni-solar tides in the upper troposphere and stratosphere could be amplified when they are in resonance with the seasons (i.e. annually aliased).


    • The bottom-line is that Lindzen failed miserably as a scientist in not being able to pick this obvious behavior out of the data. He is considered the authority on QBO and the upper atmosphere, after all.

      By golly, Lindzen had over 50 years of looking at the data to discover it and could not ! Cripes, I had been looking at the data for only a short time before it was obvious that a correlation existed.

      He also fails miserably as a concerned scientist over the plight of the environment. He is clearly paid off by big oil and gives those unctuous presentations where he acts like the smartest guy in the room, while sneering at his fellow climate scientists.

      I have no doubt that if Lindzen wasn’t the “authority” on QBO, then this behavior would have been found long ago. But because of his authoritarian, egotistical style (remember, he’s from MIT! ) others likely cowered from working the issue with the diligence it deserved.

      Really, there is no other way to explain it, because the lunar-related behavior is that obvious in retrospect.

      If I am wrong about this, remember that you get what you paid for, and I did this work on my own time and expense.



      • Science is a rather cold mistress. You present your best argument and if people look at the details and find that it fits the data/observations then it will become more widely accepted. That’s why it is such a cathartic process submitting a paper for peer review. It helps crystallize your thoughts and helps others see the detail and content of your hypothesis.

        I do not know about you but I always find most of the referee reports are painful to read because they expose some of the weakness and flaws in my cherished arguments. It becomes even more painful if I have invested my blood and sweat in compiling the manuscript.


      • I don’t think that it will be painful to read reviewer’s comments (if it even gets that far) because there is not much meat to criticize. The ball is in whoever’s court to explain away the significant coincidence of a zero-degree-of-freedom-complexity almost perfectly aligned fit. Any information criteria metric such as AIC or BIC applied to this fit would be though the roof in comparison to an equivalent GCM-based fit, just because the latter would have loads of adjustable parameters that would reduce its score in comparison.

        I intentionally kept the paper short for that reason — to make the reader focus on the salient points. No use trying to oversell the argument at this stage, because that is where you will get criticized the most.


  5. Here is my connection between the QBO and the lunar tidal cycles.

    (8/20.2937) + (8/18.6000) + (4/8.8505) = 3/(2.3506)

    20.2937 tropical yrs = time for new moon at closest perigee to re-occur.

    18.6000 tropical yrs = time for the lunar line-of-nodes to precess around the Earth wrt. the stars.

    8.8505 tropical yrs = time for the lunar line-of-apse to precess around the Earth wrt. the stars.

    2.3506 tropical years = 28.21 months ~= the average length of the QBO.

    Of course, the equation could also be written as:

    (4/10.1469) + (4/9.3000) + (4/8.8505) = 3/(2.3506)

    This requires that the Chandler wobble be 430 days if the mean QBO period is precisely twice the Chandler wobble i.e. (2 x 430 days)/(365.242189 days) = 2.354 tropical yrs = 28.248 months.

    It is generally believed that the Chandler wobble is closer to 433 days so you would expect a mean QBO period equal to:

    (2 x 433 days) /(365.242189 days) = 2.371 tropical yrs = 28.452 months.

    Malkin and Miller (2009) get a period for the Chandler wobble of 1.185 Julian yrs = 432.8 +/- 0.3 days

    Click to access 0908.3732v1.pdf

    Hence, the lunar tidal synchronization is close to, but not precisely at, the QBO = 2 x Chandler wobble oscillation period. However, drifts of the lunar cycles around the nominal periods cited above would cause some overlap with time.


    • Ian, Here is how I see it in cyclic terms. The significant lunar month periods after aliasing are ~ 2.37 years and 2.715 years. The first synchronizes with the Metonic cycle of 19 years after 8 periods, and the second takes 7 periods.

      8*2.37=18.96 years and 7*2.715=19.005 years.

      and the beat period between 2.37 and 2.715 is 18.65 years


  6. You have indicated that the significant lunar month periods after aliasing are ~ 2.37 years and 2.715 years. You have also noted that the first synchronizes with the Metonic cycle of 19 years after 8 periods, and the second takes 7 periods.

    8*2.37=18.96 years and 7*2.715=19.005 years.

    A preliminary analysis on my part shows that if you sample the lunar monthly cycles on an annual basis (i.e. once every tropical years) you get the following periods:

    Draconic month (27.21222 days) aliases to 2.3501 tropical years
    anomalistic month (27.55455 days) aliases to 3.9984 tropical years
    Synodic month (29.5305889 days) aliases to 2.7108 tropical years
    Tropical month (27.32158 days) aliases to 2.7108 tropical years.

    1. “The aliasing period of the anomalistic month is the well know 4.00 year seasonal (anomalistic) tidal cycle. The underlying reason for the four year cycle is the fact that for short term periods of a few years, the specific FMC period i.e. the average of 14 Synodic months (= 413.428244 days (J2000)) and 15 Anomalistic months (= 413.318248days (J2000)), which is 413.373246 days (J2000) or 1.131778 tropical years, should be used rather than the long-term mean FMC period i.e. 411.784430 days (J2000). Hence, 3.5 specific FMC’s = 3.96122463 tropical years ~ 49 Synodic months. This falls short of exactly four tropical years by 14.16 days which is very close to half a synodic month. Thus, if we start with a new moon at perigee on a given day of the calendar year, we end up with a full moon at or close to perigee (3.5 specific FMC’s + 0.5 Synodic months) later = 4.001651 tropical years or 4 years and 0.602895 days (J2000).”

    Reference: Wilson, I.R.G. Are the Strongest Lunar Perigean Spring Tides Commensurate with the Transit Cycle of Venus?, Pattern Recogn. Phys., 2, 75-93.

    2. The Tropical and Synodic months both have the same alias to 2.7108 years and so this period would almost certainly be related to the well known 19.0 tropical year Metonic cycle since this is basis for religious calendars to align the phase of the Moon with the seasonal calendar.

    “Higher than normal spring tides occur once every semi-synodic month (Msf), whenever the Sun, Earth and Moon are co-aligned at either New or Full Moon. It turns out that 12.5 synodic months are 3.890171 days longer than one tropical year (N.B. from this point forward, the word “year” will mean one tropical or seasonal year = 365.2421897 days (J2000) (McCarthy and Seidelmann [11], unless indicated). Hence, if a spring tide occurs on a given day of the year,
    3.796 tropical years will pass before another spring tide occurs on the same day of the year. This occurs because: (0.5 synodic months)/(12.5 synodic months – tropical year) = (14.7652944 days/3.890171 days) = 3.796 years.

    In addition, it can be shown that multiples of half of the lunar synodic cycle (Msf) are almost exactly equal to whole multiples of a year, for 4.0 years, 4.0 + 4.0 = 8.0 years, 4.0 + 4.0 + 3.0 = 11.0 years, 4.0 + 4.0 + 3.0 + 4.0 = 15.0 years, and
    4.0 + 4.0 + 3.0 + 4.0 + 4.0 = 19.0 years. Hence, spring tides that occur on roughly the same day of the year follow a 4:4:3:4:4 year spacing pattern (with an average spacing of (4 + 4 + 3 + 4 + 4)/5 = 3.8 years), with the pattern repeating itself after a period of almost exactly 19 years. The 19.0 year period is known as the Metonic cycle. This cycle results from the fact that 235 Synodic months = 6939.688381 days = 19.000238 Tropical years.”

    Reference: Wilson, I.R.G., Lunar Tides and the Long-Term Variation of the Peak Latitude Anomaly of the Summer Sub-Tropical High Pressure Ridge over Eastern Australia The Open Atmospheric Science Journal, 2012, 6, 49-60.

    3. The aliasing period of the Draconic of 2.350 tropical year does not appear to perfectly match the 2.370 year period. However, it is close enough that normal deviations of the lunar variables from their nominal values would cause an overlap of these two numbers [N.B. 2 x 432.8 days/365.242189 days = 2.370 tropical years.] Hence, I believe that 2.370 feature is related to the Chandler wobble with the lunar tidal cycles coming into supplementing them at random periods.


    • Ian,
      Aliasing calculations can be made very precise, and so I am using as known a value for the tropical year as I have been able to find: 365.24219 days (or calendar year 365.2425 days)

      I think I was able to reverse engineer your calculations and it appears that you use these values for the year
      draconic 365.338
      anomalistic 365.0998
      synodic 365.2607
      tropical 365.2593

      Perhaps you are estimating the aliasing from some charts, instead of calculating directly? They should all be self-consistent in using the same value for a year.

      These numbers really do make a difference because even slight variations will make a model of QBO go out of sync with the data after enough cycles.


      • My results for the aliases are only crude estimations since the mean month lunar cycles are just that, means. The mean synodic period of the Moon is 29.3505889 days for J2000. but in real life it can vary by up to 6.5 hours = 0.27 days from the mean in an y given year.

        This means that any aliasing process must be done upon real lunar data rather than the nominal means.


      • Yup, we have discussed that variation over at the Azimuth Forum, and it likely has some impact to the overall fit. But you have to remember that the mean value is all that is important to establish a long-term coherence of the model with the QBO time-series. I am fitting a model to the series from 1953 until 2014, and if I get the mean wrong, it will get out of phase after a number of cycles.

        On the other hand, if I include the variation as a slight frequency modulation, it may improve the shape of the individual periods, but by how much?

        I am sure the question also comes up during the harmonic analysis of ocean tides. There they use the mean values to first-order, and all those other terms contribute to the second-order fit.


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