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 research category is topological considerations of Laplace’s Tidal Equations (LTE = a shallow-water simplification of Navier-Stokes) applied to the equatorial thermocline — the following citations provides an evolutionary understanding that I have developed via presentations and publications over the last 6 years (working backwards)

Given that I have worked on this topic persistently over this span of time, I have gained considerable insight into how straightforward it has become to generate relatively decent fits to climate dipoles such as ENSO. Paradoxically, this is both good and bad. It’s good because the model’s recipe is algorithmically simply described in terms of plausibility and parsimony. That’s largely because it’s a straightforward non-linear extension of a conventional tidal analysis model. However that non-linearity opens up the possibility for many similar model fits that are equally good, yet difficult to discriminate between. So it’s bad in the sense that I can come to an impasse in selecting the “best” model.

This is oversimplifying a bit but the framing issue is if you knew the answer was 72, but have a hard time determining whether the question being posed was one of 2×36, 3×24, 4×18, 6×12, 8×9, 9×8, 12×6, 18×4, 24×3, or 36×2. Each gives the right answer, but potentially not the right mechanism. This is a fundamental problem with non-linear analysis.

A conventional tidal analysis by itself is just a few fundamental tidal factors (exactly 4) but made devastatingly accurate by the introduction of 2nd-order harmonics and cross-harmonics. All these harmonics are generated by non-linear effects but the frequency spectrum is so clean and distinct for a sea-level-height (SLH) time-series that the equivalent solution to k × F = 72 is essentially a scaling identification problem where the k is the scale factor for the corresponding cyclic tidal factors F.

Yet, by applying the non-linear LTE solution to the problem of modeling ENSO, we quickly discover that the algorithm is a wickedly effective harmonics and cross-harmonics generator. Any number of combinations of harmonics can develop an adequate fit depending on the variable LTE modulation applied. So it could be a small LTE modulation mixed with a wide assortment of tidal factors (the 2×36 case) or it could be a large LTE modulation mixed with a minimum of tidal factors (the 18×4 case). Or it could be something in between (e.g. the 8×9 case). This is all a result of the sine-of-a-sine non-linearity of the LTE formulation, related to the Mach-Zehnder modulation used in optical cryptography applications. The latter bit is the hint that things may not be unambiguously decoded given the fact that M-Z has been discovered to be nature’s own built-in encryption device.

However, there remains lots of light at the end of this tunnel, as I have also discovered that the tidal factor spread is likely largely governed by a single lunar tidal constituent, the 27.55 day anomalistic Mm cycle interfering with an annual impulse. That’s essentially 2 of the 4 tidal factors, with the other 2 lunar factors providing a 2nd-order correction. For the longest time I had been focused on the 13.66 day tropical Mf cycle as that also lead to a decent fit over the years, specifically since the first beat harmonic of the Mf cycle with the annual impulse is 3.8 years while the Mm cycle is 3.9 years. These two terms are close enough that they only go out-of-phase after ~130 years, which is the extent of the ENSO time-series. Only when you try to simplify a model fit by iterating over the space of factor combinations will you discover the difference between 3.8 and 3.9.

In terms of geophysics, the Mf factor is a tractional tidal forcing operating parallel to the ocean’s surface influenced by the moon’s latitudinal declination, while the Mm factor is a largely perpendicular gravitational forcing influenced by the perigean cycle of the Moon-to-Earth distance. The latter may be the “right mechanism” as each can give close to the “right answer”.

So the gist of the fitting observations is that far fewer harmonic factors are required for a decent Mm-based model than for a Mf-based model. This is slightly at the expense of a stronger LTE modulation, but the parsimony of an Mm-based model can’t be beat, as I will show via the following analysis charts…

This is a good model fit based on a slightly modified Mm-based factorization, with a sample-and-hold placed on a strong annual impulse

The comparison of the modified Mm tidal factorization to the pure Mm is below (the reason the 27.55 day periodicity doesn’t appear is because of the monthly aliasing used in plotting).

The slight pattern on top of pure signal is due to a 6-year beat of the Mm period with the 27.212 day lunar draconic pattern indicating the time between equatorial nodal crossings of the moon. This is the strongest factor of the ascension cycle described in the solar and lunar ephemeris published recently by Sung-Ho Na. As highlighted above by numbered cycles, ~20 occur in the span of 120 years.

Below, an expanded look showing how slight a correction is applied

The integrated forcing after the annual impulse is shown below. The sample-and-hold integration exaggerates low-frequency differences so the distinction between the pure Mm forcing and Mm+harmonics is more apparent. The 6-year periodicity is obscured by longer term variations.

The log-scaled power spectra of the integrated tidal forcing is shown below. Note the overwhelmingly strong peak near the 0.25/year frequency (3.9 year cycle). The rest of the peaks are readily matched to periodicities in the Na ephemerides [1] according to their strength.

The LTE modulation is quite strong for this factorization. As shown below, the forcing levels need to sinusoidally fold several times over to match the observed ENSO behavior. See the recent post Inverting non-autonomous functions for a recipe to aid in iterating for the underlying LTE modulation.

The parsimony of this model can’t be emphasized enough. It’s equivalent to the agreement of a conventional tidal forcing analysis to a SLH time-series in that only a single lunar tidal factor accounts for a majority of the modulation. Only the challenge of finding the correct LTE modulation stands in the way of producing an unambiguously correct model for the underlying ENSO behavioral dynamics.

References

[1] Sung-Ho Na, Chapter 19 – Prediction of Earth tide, Editor(s): Pijush Samui, Barnali Dixon, Dieu Tien Bui, Basics of Computational Geophysics, Elsevier, 2021, Pages 351-372, ISBN 9780128205136, https://doi.org/10.1016/B978-0-12-820513-6.00022-9. (note: the ephemerides for the Earth-Moon-Sun system matches closely the online NASA JPL ephemerides generator available at https://ssd.jpl.nasa.gov/horizons, but this paper is more useful in that it algorithmically states the contributions of the various tidal factors in the source code supplied. Source code also available at https://github.com/pukpr/GeoEnergyMath/tree/master/src)

Following up from this post, there is a recent sequence of articles in an AGU journal on Water Resources Research under the heading: “Debates: Does Information Theory Provide a New Paradigm for Earth Science?”

By anticipating all these ideas, you can find plenty of examples and derivations (with many centered on the ideas of Maximum Entropy) in our book Mathematical Geoenergy.

Here is an excerpt from the “Emerging concepts” entry, which indirectly addresses negative entropy:

That is, there are likely many earth system behaviors that are highly ordered, but the complexity and non-linearity of their mechanisms makes them appear stochastic or chaotic (high positive entropy) yet the reality is that they are just a complicated deterministic model (negative entropy). We just aren’t looking hard enough to discover the underlying patterns on most of this stuff.

Addendum: After this presentation was submitted, a ground-breaking paper by a group at the University of Paris came on-line. Their paper, “On the Shoulders of Laplace” covers much the same ground as the EGU presentation linked above.

F. Lopes, J.L. Le Mouël, V. Courtillot, D. Gibert, On the shoulders of Laplace, Physics of the Earth and Planetary Interiors, 2021, 106693, ISSN 0031-9201, https://doi.org/10.1016/j.pepi.2021.106693.

Excerpts from the paper “On the shoulders of Laplace”

Moreover Lopes et al claim that this celestial gravitational forcing carries over to controlling cyclic climate indices, following Laplace’s mathematical formulation (now known as Laplace’s Tidal Equations) for describing oceanic tides.

This view also aligns with the way we model climate indices such as ENSO and QBO via a solution to Laplace’s Tidal Equations, as described in the linked EGU presentation above.

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 12 of the book, we provide an empirical gravitational forcing term that can be applied to the Laplace’s Tidal Equation (LTE) solution for modeling ENSO. The inverse squared law is modified to a cubic law to take into account the differential pull from opposite sides of the earth.

The two main terms are the monthly anomalistic (Mm) cycle and the fortnightly tropical/draconic pair (Mf, Mf’ w/ a 18.6 year nodal modulation). Due to the inverse cube gravitational pull found in the denominator of F(t), faster harmonic periods are also created — with the 9-day (Mt) created from the monthly/fortnightly cross-term and the weekly (Mq) from the fortnightly crossed against itself. It’s amazing how few terms are needed to create a canonical fit to a tidally-forced ENSO model.

The recipe for the model is shown in the chart below (click to magnify), following sequentially steps (A) through (G) :

The tidal forcing is constrained by the known effects of the lunisolar gravitational torque on the earth’s length-of-day (LOD) variations. An essentially identical set of monthly, fortnightly, 9-day, and weekly terms are required for both a solid-body LOD model fit and a fluid-volume ENSO model fit.

If we apply the same tidal terms as forcing for matching dLOD data, we can use the fit below as a perturbed ENSO tidal forcing. Not a lot of difference here — the weekly harmonics are higher in magnitude.

So the only real unknown in this process is guessing the LTE modulation of steps (F) and (G). That’s what differentiates the inertial response of a spinning solid such as the earth’s core and mantle from the response of a rotating liquid volume such as the equatorial Pacific ocean. The former is essentially linear, but the latter is non-linear, making it an infinitely harder problem to solve — as there are infinitely many non-linear transformations one can choose to apply. The only reason that I stumbled across this particular LTE modulation is that it comes directly from a clever solution of Laplace’s tidal equations.

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 challenge of validating the models of climate oscillations such as ENSO and QBO, rests primarily in our inability to perform controlled experiments. Because of this shortcoming, we can either do (1) predictions of future behavior and validate via the wait-and-see process, or (2) creatively apply techniques such as cross-validation on currently available data. The first is a non-starter because it’s obviously pointless to wait decades for validation results to confirm a model, when it’s entirely possible to do something today via the second approach.

There are a variety of ways to perform model cross-validation on measured data.

In its original and conventional formulation, cross-validation works by checking one interval of time-series against another, typically by training on one interval and then validating on an orthogonal interval.

Another way to cross-validate is to compare two sets of time-series data collected on behaviors that are potentially related. For example, in the case of ocean tidal data that can be collected and compared across spatially separated geographic regions, the sea-level-height (SLH) time-series data will not necessarily be correlated, but the underlying lunar and solar forcing factors will be closely aligned give or take a phase factor. This is intuitively understandable since the two locations share a common-mode signal forcing due to the gravitational pull of the moon and sun, with the differences in response due to the geographic location and local spatial topology and boundary conditions. For tides, this is a consensus understanding and tidal prediction algorithms have stood the test of time.

In the previous post, cross-validation on distinct data sets was evaluated assuming common-mode lunisolar forcing. One cross-validation was done between the ENSO time-series and the AMO time-series. Another cross-validation was performed for ENSO against PDO. The underlying common-mode lunisolar forcings were highly correlated as shown in the featured figure. The LTE spatial wave-number weightings were the primary discriminator for the model fit. This model is described in detail in the book Mathematical GeoEnergy to be published at the end of the year by Wiley.

Another common-mode cross-validation possible is between ENSO and QBO, but in this case it is primarily in the Draconic nodal lunar factor — the cyclic forcing that appears to govern the regular oscillations of QBO. Below is the Draconic constituent comparison for QBO and the ENSO.

The QBO and ENSO models only show a common-mode correlated response with respect to the Draconic forcing. The Draconic forcing drives the quasi-periodicity of the QBO cycles, as can be seen in the lower right panel, with a small training window.

This cross-correlation technique can be extended to what appears to be an extremely erratic measure, the North Atlantic Oscillation (NAO).

Like the SOI measure for ENSO, the NAO is originally derived from a pressure dipole measured at two separate locations — but in this case north of the equator. From the high-frequency of the oscillations, a good assumption is that the spatial wavenumber factors are much higher than is required to fit ENSO. And that was the case as evidenced by the figure below.

ENSO vs NAO cross-validation

Both SOI and NAO are noisy time-series with the NAO appearing very noisy, yet the lunisolar constituent forcings are highly synchronized as shown by correlations in the lower pane. In particular, summing the Anomalistic and Solar constituent factors together improves the correlation markedly, which is because each of those has influence on the other via the lunar-solar mutual gravitational attraction. The iterative fitting process adjusts each of the factors independently, yet the net result compensates the counteracting amplitudes so the net common-mode factor is essentially the same for ENSO and NAO (see lower-right correlation labelled Anomalistic+Solar).

Since the NAO has high-frequency components, we can also perform a conventional cross-validation across orthogonal intervals. The validation interval below is for the years between 1960 and 1990, and even though the training intervals were aggressively over-fit, the correlation between the model and data is still visible in those 30 years.

NAO model fit with validation spanning 1960 to 1990

Over the course of time spent modeling ENSO, the effort that went into fitting to NAO was a fraction of the original time. This is largely due to the fact that the temporal lunisolar forcing only needed to be tweaked to match other climate indices, and the iteration over the topological spatial factors quickly converges.

Many more cross-validation techniques are available for NAO, since there are different flavors of NAO indices available corresponding to different Atlantic locations, and spanning back to the 1800’s.