Trying to understand QBO may lead to madness, if the plights of Richard Lindzen (Macbeth) and Timothy Dunkerton (Hamlet) are any indication. It was first Lindzen — the primary theorist behind QBO — in his quest for scientific notoriety that led to lofty pretentiousness and eventually bad blood with his colleagues. Now it’s the Lindzen-acolyte Dunketon’s turn, avenging his “uncle” with troubling behavior

Regarding the gravity waves concentrically emanating from the Tonga explosion

“It’s really unique. We have never seen anything like this in the data before,” says Lars Hoffmann, an atmospheric scientist at the Jülich Supercomputing Centre in Germany.

“That’s what’s really puzzling us,” says Corwin Wright, an atmospheric physicist at the University of Bath, UK. “It must have something to do with the physics of what’s going on, but we don’t know what yet.”

The discovery was prompted by a tweet sent to Wright on 15 January from Scott Osprey, a climate scientist at the University of Oxford, UK, who asked: “Wow, I wonder how big the atmospheric gravity waves are from this eruption?!” Osprey says that the eruption might have been unique in causing these waves because it happened very quickly relative to other eruptions. “This event seems to have been over in minutes, but it was explosive and it’s that impulse that is likely to kick off some strong gravity waves,” he says. The eruption might have lasted moments, but the impacts could be long-lasting. Gravity waves can interfere with a cyclical reversal of wind direction in the tropics, Osprey says, and this could affect weather patterns as far away as Europe. “We’ll be looking very carefully at how that evolves,” he says.

This (“cyclical reversal of wind direction in the tropics”) is referring to the QBO, and we will see if it has an impact in the coming months. Hint: the QBO from the last post is essentially modeling gravity waves arising from the tidal forcing as driving the cycle. Also, watch the LOD.

Perhaps the lacking is in applying this simple scientific law: for every action there is a reaction. Always start from that, and also consider: an object that is in motion, tends to stay in motion. Is the lack of observed Coriolis effects to first-order part of why the scientists are mystified? Given the variation of this force with latitude, the concentric rings perhaps were expected to be distorted according to spherical harmonics.

The forcing spectrum like this, with the aliased draconic (27.212d) factor circled:

For QBO, we remove all the lunar factors except for the draconic, as this is the only declination factor with the same spherical group symmetry as the semi-annual solar declination.

And after modifying the annual (ENSO spring-barrier) impulse into a semi-annual impulse with equal and opposite excursions, the resultant model matches well (to first order) the QBO time series.

Although the alignment isn’t perfect, there are indications in the structure that the fit has a deeper significance. For example, note how many of the shoulders in the structure align, as highlighted below in yellow

The peaks and valleys do wander about a bit and might be a result of the sensitivity to the semi-annual impulse and the fact that this is only a monthly resolution. The chart below is a detailed fit of the QBO using data with a much finer daily resolution. As you can see, slight changes in the seasonal timing of the semi-annual pulse are needed to individually align the 70 and 30 hBar QBO time-series data.

The underlying forcing of the ENSO model shows both an 18-year Saros cycle (which is an eclipse alignment cycle of all the tidal periods), along with a 6-year anomalistic/draconic interference cycle. This modulation of the main anomalistic cycle appears in both the underlying daily and monthly profile, shown below before applying an annual impulse. The 6-year is clearly evident as it aligns with the x-axis grid 1880, 1886, 1892, 1898, etc.

The 6-year cycle in the LOD is not aligned as strictly as the tidal model and it tends to wander, but it seems a more plausible and parsimonious explanation of the modulation than for example in this paper (where the 6-year LOD cycle is “similarly detected in the variations of C22 and S22, the degree-2 order-2 Stokes coefficients of the Earth’s gravitational field”).

Cross-validation confidence improves as the number of mutually agreeing alignments increase. Given the fact that controlled experiments are impossible to perform, this category of analyses is the best way to validate the geophysical models.

Revisiting earlier modeling of the North Atlantic Oscillation (NAO) and Arctic Oscillation (AO) indices with the benefit of updated analysis approaches such as negative entropy. These two indices in particular are intimidating because to the untrained eye they appear to be more noise than anything deterministically periodic. Whereas ENSO has periods that range from 3 to 7 years, both NAO and AO show rapid cycling often on a faster-than-annual pace. The trial ansatz in this case is to adopt a semi-annual forcing pattern and synchronize that to long-period lunar factors, fitted to a Laplace’s Tidal Equation (LTE) model.

Start with candidate forcing time-series as shown below, with a mix of semi-annual and annual impulses modulating the primarily synodic/tropical lunar factor. The two diverge slightly at earlier dates (starting at 1880) but the NAO and AO instrumental data only begins at the year 1950, so the two are tightly correlated over the range of interest.

The intensity spectrum is shown below for the semi-annual zone, noting the aliased tropical factors at 27.32 and 13.66 days standing out.

The NAO and AO pattern is not really that different, and once a strong LTE modulation is found for one index, it also works for the other. As shown below, the lowest modulation is sharply delineated, yet more rapid than that for ENSO, indicating a high-wavenumber standing wave mode in the upper latitudes.

The model fit for NAO (data source) is excellent as shown below. The training interval only extended to 2016, so the dotted lines provide an extrapolated fit to the most recent NAO data.

Same for the AO (data source), the fit is also excellent as shown below. There is virtually no difference in the lowest LTE modulation frequency between NAO and AO, but the higher/more rapid LTE modulations need to be tuned for each unique index. In both cases, the extrapolations beyond the year 2016 are very encouraging (though not perfect) cross-validating predictions. The LTE modulation is so strong that it is also structurally sensitive to the exact forcing.

Both NAO and AO time-series appear very busy and noisy, yet there is very likely a strong underlying order due to the fundamental 27.32/13.66 day tropical forcing modulating the semi-annual impulse, with the 18.6/9.3 year and 8.85/4.42 year providing the expected longer-range lunar variability. This is also consistent with the critical semi-annual impulses that impact the QBO and Chandler wobble periodicity, with the caveat that group symmetry of the global QBO and Chandler wobble forcings require those to be draconic/nodal factors and not the geographically isolated sidereal/tropical factor required of the North Atlantic.

It really is a highly-resolved model potentially useful at a finer resolution than monthly and that will only improve over time.

(as a sidenote, this is much better attempt at matching a lunar forcing to AO and jet-stream dynamics than the approach Clive Best tried a few years ago. He gave it a shot but without knowledge of the non-linear character of the LTE modulation required he wasn’t able to achieve a high correlation, achieving at best a 2.4% Spearman correlation coefficient for AO in his Figure 4 — whereas the models in this GeoenergyMath post extend beyond 80% for the interval 1950 to 2016! )

In Chapter 12, we described the model of QBO generated by modulating the draconic (or nodal) lunar forcing with a hemispherical annual impulse that reinforces that effect. This generates the following predicted frequency response peaks:

The 2nd, 3rd, and 4th peaks listed (at 2.423, 1.423, and 0.423) are readily observed in the power spectra of the QBO time-series. When the spectra are averaged over each of the time series, the precisely matched peaks emerge more cleanly above the red noise envelope — see the bottom panel in the figure below (click to expand).

The inset shows what these harmonics provide — essentially the jagged stairstep structure of the semi-annual impulse lag integrated against the draconic modulation.

It is important to note that these harmonics are not the traditional harmonics of a high-Q resonance behavior, where the higher orders are integral multiples of the fundamental frequency — in this case at 0.423 cycles/year. Instead, these are clear substantiation of a forcing response that maintains the frequency spectrum of an input stimulus, thus excluding the possibility that the QBO behavior is a natural resonance phenomena. At best, there may be a 2nd-order response that may selectively amplify parts of the frequency spectrum.

The simple idea is that tidal forces play a bigger role in geophysical behaviors than previously thought, and thus helping to explain phenomena that have frustrated scientists for decades.

The idea is simple but the non-linear math (see figure above for ENSO) requires cracking to discover the underlying patterns.

The rationale for the ESD Ideas section in the EGU Earth System Dynamics journal is to get discussion going on innovative and novel ideas. So even though this model is worked out comprehensively in Mathematical Geoenergy, it hasn’t gotten much publicity.

In Chapter 11 of the book Mathematical GeoEnergy, we model the QBO of equatorial stratospheric winds, but only touch on the related cycle at even higher altitudes, the semi-annual oscillation (SAO). The figure at the top of a recent post geometrically explains the difference between SAO and QBO — the basic idea is that the SAO follows the solar tide and not the lunar tide because of a lower atmospheric density at higher altitudes. Thus, the heat-based solar tide overrides the gravitational lunar+solar tide and the resulting oscillation is primarily a harmonic of the annual cycle.

Our book Mathematical Geoenergy presents a number of novel approaches that each deserve a research paper on their own. Here is the list, ordered roughly by importance (IMHO):

Laplace’s Tidal Equation Analytic Solution. (Ch 11, 12) A solution of a Navier-Stokes variant along the equator. Laplace’s Tidal Equations are a simplified version of Navier-Stokes and the equatorial topology allows an exact closed-form analytic solution. This could classify for the Clay Institute Millenium Prize if the practical implications are considered, but it’s a lower-dimensional solution than a complete 3-D Navier-Stokes formulation requires.

Model of El Nino/Southern Oscillation (ENSO). (Ch 12) A tidally forced model of the equatorial Pacific’s thermocline sloshing (the ENSO dipole) which assumes a strong annual interaction. Not surprisingly this uses the Laplace’s Tidal Equation solution described above, otherwise the tidal pattern connection would have been discovered long ago.

Model of Quasi-Biennial Oscillation (QBO). (Ch 11) A model of the equatorial stratospheric winds which cycle by reversing direction ~28 months. This incorporates the idea of amplified cycling of the sun and moon nodal declination pattern on the atmosphere’s tidal response.

Origin of the Chandler Wobble. (Ch 13) An explanation for the ~433 day cycle of the Earth’s Chandler wobble. Finding this is a fairly obvious consequence of modeling the QBO.

The Oil Shock Model. (Ch 5) A data flow model of oil extraction and production which allows for perturbations. We are seeing this in action with the recession caused by oil supply perturbations due to the Corona Virus pandemic.

The Dispersive Discovery Model. (Ch 4) A probabilistic model of resource discovery which accounts for technological advancement and a finite search volume.

Ornstein-Uhlenbeck Diffusion Model (Ch 6) Applying Ornstein-Uhlenbeck diffusion to describe the decline and asymptotic limiting flow from volumes such as occur in fracked shale oil reservoirs.

The Reservoir Size Dispersive Aggregation Model. (Ch 4) A first-principles model that explains and describes the size distribution of oil reservoirs and fields around the world.

Origin of Tropical Instability Waves (TIW). (Ch 12) As the ENSO model was developed, a higher harmonic component was found which matches TIW

Characterization of Battery Charging and Discharging. (Ch 18) Simplified expressions for modeling Li-ion battery charging and discharging profiles by applying dispersion on the diffusion equation, which reflects the disorder within the ion matrix.

Anomalous Behavior in Dispersive Transport explained. (Ch 18) Photovoltaic (PV) material made from disordered and amorphous semiconductor material shows poor photoresponse characteristics. Solution to simple entropic dispersion relations or the more general Fokker-Planck leads to good agreement with the data over orders of magnitude in current and response times.

Framework for understanding Breakthrough Curves and Solute Transport in Porous Materials. (Ch 20) The same disordered Fokker-Planck construction explains the dispersive transport of solute in groundwater or liquids flowing in porous materials.

Wind Energy Analysis. (Ch 11) Universality of wind energy probability distribution by applying maximum entropy to the mean energy observed. Data from Canada and Germany. Found a universal BesselK distribution which improves on the conventional Rayleigh distribution.

Terrain Slope Distribution Analysis. (Ch 16) Explanation and derivation of the topographic slope distribution across the USA. This uses mean energy and maximum entropy principle.

Thermal Entropic Dispersion Analysis. (Ch 14) Solving the Fokker-Planck equation or Fourier’s Law for thermal diffusion in a disordered environment. A subtle effect but the result is a simplified expression not involving complex errf transcendental functions. Useful in ocean heat content (OHC) studies.

The Maximum Entropy Principle and the Entropic Dispersion Framework. (Ch 10) The generalized math framework applied to many models of disorder, natural or man-made. Explains the origin of the entroplet.

Solving the Reserve Growth “enigma”. (Ch 6) An application of dispersive discovery on a localized level which models the hyperbolic reserve growth characteristics observed.

Shocklets. (Ch 7) A kernel approach to characterizing production from individual oil fields.

Reserve Growth, Creaming Curve, and Size Distribution Linearization. (Ch 6) An obvious linearization of this family of curves, related to Hubbert Linearization but more useful since it stems from first principles.

The Hubbert Peak Logistic Curve explained. (Ch 7) The Logistic curve is trivially explained by dispersive discovery with exponential technology advancement.

Laplace Transform Analysis of Dispersive Discovery. (Ch 7) Dispersion curves are solved by looking up the Laplace transform of the spatial uncertainty profile.

Gompertz Decline Model. (Ch 7) Exponentially increasing extraction rates lead to steep production decline.

The Dynamics of Atmospheric CO2 buildup and Extrapolation. (Ch 9) Convolving a fat-tailed CO2 residence time impulse response function with a fossil-fuel emissions stimulus. This shows the long latency of CO2 buildup very straightforwardly.

Reliability Analysis and Understanding the “Bathtub Curve”. (Ch 19) Using a dispersion in failure rates to generate the characteristic bathtub curves of failure occurrences in parts and components.

The Overshoot Point (TOP) and the Oil Production Plateau. (Ch 8) How increases in extraction rate can maintain production levels.

Lake Size Distribution. (Ch 15) Analogous to explaining reservoir size distribution, uses similar arguments to derive the distribution of freshwater lake sizes. This provides a good feel for how often super-giant reservoirs and Great Lakes occur (by comparison).

The Quandary of Infinite Reserves due to Fat-Tail Statistics. (Ch 9) Demonstrated that even infinite reserves can lead to limited resource production in the face of maximum extraction constraints.

Oil Recovery Factor Model. (Ch 6) A model of oil recovery which takes into account reservoir size.

Network Transit Time Statistics. (Ch 21) Dispersion in TCP/IP transport rates leads to the measured fat-tails in round-trip time statistics on loaded networks.

Particle and Crystal Growth Statistics. (Ch 20) Detailed model of ice crystal size distribution in high-altitude cirrus clouds.

Rainfall Amount Dispersion. (Ch 15) Explanation of rainfall variation based on dispersion in rate of cloud build-up along with dispersion in critical size.

Earthquake Magnitude Distribution. (Ch 13) Distribution of earthquake magnitudes based on dispersion of energy buildup and critical threshold.

IceBox Earth Setpoint Calculation. (Ch 17) Simple model for determining the earth’s setpoint temperature extremes — current and low-CO2 icebox earth.

Global Temperature Multiple Linear Regression Model (Ch 17) The global surface temperature records show variability that is largely due to the GHG rise along with fluctuating changes due to ocean dipoles such as ENSO (via the SOI measure and also AAM) and sporadic volcanic eruptions impacting the atmospheric aerosol concentrations.

GPS Acquisition Time Analysis. (Ch 21) Engineering analysis of GPS cold-start acquisition times. Using Maximum Entropy in EMI clutter statistics.

1/f NoiseModel (Ch 21) Deriving a random noise spectrum from maximum entropy statistics.

Stochastic Aquatic Waves (Ch 12) Maximum Entropy Analysis of wave height distribution of surface gravity waves.

The Stochastic Model of Popcorn Popping. (Appx C) The novel explanation of why popcorn popping follows the same bell-shaped curve of the Hubbert Peak in oil production. Can use this to model epidemics, etc.

Dispersion Analysis of Human Transportation Statistics. (Appx C) Alternate take on the empirical distribution of travel times between geographical points. This uses a maximum entropy approximation to the mean speed and mean distance across all the data points.

The modeled QBO cycle is directly related to the nodal (draconian) lunar cycle physically aliased against the annual cycle. The empirical cycle period is best estimated by tracking the peak acceleration of the QBO velocity time-series, as this acceleration (1st derivative of the velocity) shows a sharp peak. This value should asymptotically approach a 2.368 year period over the long term. Since the recent data from the main QBO repository provides an additional acceleration peak from the past month, now is as good a time as any to analyze the cumulative data.

The new data-point provides a longer period which compensated for some recent shorter periods, such that the cumulative mean lies right on the asymptotic line. The jitter observed is explainable in terms of the model, as acceleration peaks are more prone to align close to an annual impulse. But the accumulated mean period is still aligned to the draconic aliasing with this annual impulse. As more data points come in over the coming decades, the mean should vary less and less from the asymptotic value.

The fit to QBO using all the data save for the last available data point is shown below. Extrapolating beyond the greenarrow, we should see an uptick according to the red waveform.

Adding the recent data-point and the bluewaveform does follow the model.

There was a flurry of recent discussion on the QBO anomaly of 2016 (shown as a split peak above), which implied that perhaps the QBO would be permanently disrupted from it’s long-standing pattern. Instead, it may be a more plausible explanation that the QBO pattern was not simply wandering from it’s assumed perfectly cyclic path but instead is following a predictable but jittery track that is a combination of the (physically-aliased) annual impulse-synchronized Draconic cycle together with a sensitivity to variations in the draconic cycle itself. The latter calibration is shown below, based on NASA ephermeris.

This is the QBO spectral decomposition, showing signal strength centered on the fundamental aliased Draconic value, both for the data and the set by the model.

The main scientist, Prof. Richard Lindzen, behind the consensus QBO model has been recently introduced here as being “considered the most distinguished living climate scientist on the planet”. In his presentation criticizing AGW science [1], Lindzen claimed that the climate oscillates due to a steady uniform force, much like a violin oscillates when the steady force of a bow is drawn across its strings. An analogy perhaps better suited to reality is that the violin is being played like a drum. Resonance is more of a decoration to the beat itself.
Keith ?

[1] Professor Richard Lindzen slammed conventional global warming thinking warming as ‘nonsense’ in a lecture for the Global Warming Policy Foundation on Monday. ‘An implausible conjecture backed by false evidence and repeated incessantly … is used to promote the overturn of industrial civilization,’ he said in London. — GWPF

The Arctic Oscillation (AO) dipole has behavior that is correlated to the North Atlantic Oscillation (NAO) dipole. We can see this in two ways. First, and most straight-forwardly, the correlation coefficient between the AO and NAO time-series is above 0.6.

Secondly, we can use the model of the NAO from the last post and refit the parameters to the AO data (data also here), but spanning an orthogonal interval. Then we can compare the constituent lunisolar factors for NAO and AO for correlation, and further discover that this also doubles as an effective cross-validation for the underlying LTE model (as the intervals are orthogonal).

Top panel is a model fit for AO between 1900-1950, and below that is a model fit for NAO between 1950-present. The lower pane is the correlation for a common interval (left) and for the constituent lunisolar factors for the orthogonal interval (right)

Only the anomalistic factor shows an imperfect correlation, and that remains quite high.