ESD Ideas article for review

Get a Copernicus login and comment for peer-review

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.

Complexity vs Simplicity in Geophysics

In our book Mathematical GeoEnergy, several geophysical processes are modeled — from conventional tides to ENSO. Each model fits the data applying a concise physics-derived algorithm — the key being the algorithm’s conciseness but not necessarily subjective intuitiveness.

I’ve followed Gell-Mann’s work on complexity over the years and so will try applying his qualitative effective complexity approach to characterize the simplicity of the geophysics models described in the book and on this blog.

from Deacon_Information_Complexity_Depth.pdf

Here’s a breakdown from least complex to most complex

1. Say we are doing tidal analysis by fitting a model to a historical sea-level height (SLH) tidal gauge time-series. That’s essentially an effective complexity of 1 because it just involves fitting amplitudes and phases from known lunisolar sinusoidal tidal cycles.

Figure 1: Conventional tidal analysis by fitting amplitudes and phases of known tidal periods

This image has been resized to fit in the page. Click to enlarge.

2. The same effective complexity of 1 applies for the differential length-of-day (dLOD) time-series, as it involves straightforward additive tidal cycles.

Figure 2 : Long-period tidal cycles map directly to dLOD.
The long-term modulation shown follows the 18.6 year nodal cycle

This image has been resized to fit in the page. Click to enlarge.

3. The Chandler wobble model developed in Chapter 13 has an effective complexity of 2 because it takes a single monthly tidal forcing and it multiplies it by a semi-annual nodal impulse (one for each nodal cycle pass). Just a bit more complex than #1 or #2 but the complexity already may be too great for geophysicists to accept, as the consensus instead argues for a stochastic forcing stimulating a resonance.

Figure 3 : Chandler Wobble model consisting of a lunisolar tidal forcing.
The beat frequency is annual (sun) nodal against lunar nodal cycle.

This image has been resized to fit in the page. Click to enlarge.

4. The QBO model described in Chapter 11 is also estimated at an effective complexity of 2, as it is impulse-modulated by nearly the same mechanism as for the Chandler wobble of #3. But instead of a bandpass filter for the Chandler wobble, the QBO model applies an integrating filter to create more of a square-wave-like time-series. Again, this is too complex for consensus atmospheric physics to accept.

Figure 4 : QBO model has a fundamental cycle of ~2.33 years.
This is an aliasing of lunar nodal cycle against the annual cycle

This image has been resized to fit in the page. Click to enlarge.

5. The ENSO model described in Chapter 12 is an effective complexity of 3 because it adds the nonlinear Laplace’s Tidal Equation (LTE) modulation to the square-wave-like fit of #4 (QBO), tempered by being calibrated by the tidal forcing model for #2 (dLOD). Of course this additional level of physics “complexity” is certain to be above the heads of ocean scientists and climate scientists, who are still scratching their heads over #3 and #4.

Figure 5 : Higher complexity of ENSO model due to nonlinear modulation of the LTE solution

This image has been resized to fit in the page. Click to enlarge.

The ENSO model is complex due to the non-linearity of the solution. The cyclic tidal factors can create harmonics from both the inverse cubic gravitational pull and from the LTE solution, and together with the annual impulse modulation creates an additional nasty aliasing that requires painstaking analysis to reveal.

By comparison, most GCMs of climate behaviors have effective complexities much more than this because — as Gell-Man defined it — the shortest algorithmic description would require pages and pages of text to express. To climate scientists perhaps the massive additional complexity of a GCM is preferred over the intuition required for enabling incremental complexity.

Since this post started with a Gell-Mann citation, may as well stick one here at the end:

“Battles of new ideas against conventional wisdom are common in science, aren’t they?”

“It’s very interesting how these certain negative principles get embedded in science sometimes. Most challenges to scientific orthodoxy are wrong. A lot of them are crank. But it happens from time to time that a challenge to scientific orthodoxy is actually right. And the people who make that challenge face a terrible situation. Getting heard, getting believed, getting taken seriously and so on. And I’ve lived through a lot of those, some of them with my own work, but also with other people’s very important work. Let’s take continental drift, for example. American geologists were absolutely convinced, almost all of them, that continental drift was rubbish. The reason is that the mechanisms that were put forward for it were unsatisfactory. But that’s no reason to disregard a phenomenon. Because the theories people have put forward about the phenomenon are unsatisfactory, that doesn’t mean the phenomenon doesn’t exist. But that’s what most American geologists did until finally their noses were rubbed in continental drift in 1962, ’63 and so on when they found the stripes in the mid-ocean, and so it was perfectly clear that there had to be continental drift, and it was associated then with a model that people could believe, namely plate tectonics. But the phenomenon was still there. It was there before plate tectonics. The fact that they hadn’t found the mechanism didn’t mean the phenomenon wasn’t there. Continental drift was actually real. And evidence was accumulating for it. At Caltech the physicists imported Teddy Bullard to talk about his work and Patrick Blackett to talk about his work, these had to do with paleoclimate evidence for continental drift and paleomagnetism evidence for continental drift. And as that evidence accumulated, the American geologists voted more and more strongly for the idea that continental drift didn’t exist. The more the evidence was there, the less they believed it. Finally in 1962 and 1963 they had to accept it and they accepted it along with a successful model presented by plate tectonics….”

With all that, progress is being made in earth geophysics by looking at other planets. My high-school & college classmate Dr. Alex Konopliv of NASA JPL has lead the first research team to detect the Chandler wobble on another planet (this case for Mars), see “Detection of the Chandler Wobble of Mars From Orbiting Spacecraft” Geophysical Research Letters(2020).

In the body of the article, a suggestion is made as to the source of the forcing for the Martian Chandler wobble. The Martian moon Phobos is quite small and cycles the planet in ~7 hours, so this may not have the impact that the Earth’s moon has on our Chandler Wobble.

The wobble is small, about 10 cm on average.

Since a Mars year is 687 Earth days, only the 3rd harmonic (229 days) is close to the measured wobble of 206.9 days. With the Earth, it’s quite simple how the nodal lunar cycle interferes with the annual cycle to line up exactly with the Earth’s 433 day Chandler wobble (see Figure 3 up-thread in this post) , creating that wobble as a forced response, but nothing like that on Mars, which may be a natural response wobble.

The SAO and Annual Disturbances

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.

Figure 1 : The SAO modeled with the GEM software fit to 1 hPa data along the equator
Continue reading

Why couldn’t Lindzen figure out QBO?

Background: see Chapter 11 of the book.

In research articles published ~50 years ago, Richard Lindzen made these assertions:

“For oscillations of tidal periods, the nature of the forcing is clear”

Lindzen, Richard S. “Planetary waves on beta-planes.” Mon. Wea. Rev 95.7 (1967): 441-451.


5. Lunar semidiurnal tide

One rationale for studying tides is that they are motion systems for which we know the periods perfectly, and the forcing almost as well (this is certainly the case for gravitational tides). Thus, it is relatively easy to isolate tidal phenomena in the data, to calculate tidal responses in the atmosphere, and to compare the two. Briefly, conditions for comparing theory and observation are relatively ideal. Moreover, if theory is incapable of explaining observations for such a simple system, we may plausibly be concerned with our ability to explain more complicated systems.

Lunar tides are especially well suited to such studies since it is unlikely that lunar periods could be produced by anything other than the lunar tidal potential.

Lindzen, R.S. and Hong, S.S., 1974. “Effects of mean winds and horizontal temperature gradients on solar and lunar semidiurnal tides in the atmosphere“. Journal of the atmospheric sciences31(5), pp.1421-1446.
Continue reading

Characterizing Wavetrains

In Chapter 11 and Chapter 12 of the book we characterize deterministic and stochastic variability in waves. While reviewing the presentations at last week’s EGU meeting, one study covered some of the same ground [1] and was worth a more detailed look. The distribution of stratospheric wind wave energy collected by Nastrom [2] (shown below) that we model in Chapter 11 is apparently still not completely understood.

From Mathematical Geoenergy, Chap 11
Continue reading

Stratospheric Sudden Warming

Chapter 11 of the book describes a model for the QBO of stratospheric equatorial winds. The stratified layers of the atmosphere reveal different dependencies on the external forcing depending on the altitude, see Fig 1.

Figure 1 : At high altitudes, only the sun’s annual cycle impacts the stratospheric as a semi-annual oscillation (SAO). Below that the addition of the lunar nodal cycle forces the QBO. The earth itself shows a clear wobble with the lunar cycle interacting with the annual.

Well above these layers are the mesosphere, thermosphere, and ionosphere. These are studied mainly in terms of space physics instead of climate but they do show tidal interactions with behaviors such as the equatorial electrojet [1].

The behaviors known as stratospheric sudden warmings (SSW) are perhaps a link between the lower atmospheric behaviors of equatorial QBO and/or polar vortex and the much higher atmospheric behavior comprising the electrojet. Papers such as [1,2] indicate that lunar tidal effects are showing up in the SSW and that is enhancing characteristics of the electrojet. See Fig 2.

Figure 2 : During SSW events, a strong modulation of period ~14.5 days emerges, close to the lunar fortnightly period as seen in these spectrograms. Taken from ref [2] and see quote below for more info.

“Wavelet spectra of foEs during two SSW events exhibit noticeable enhanced 14.5‐day modulation, which resembles the lunar semimonthly period. In addition, simultaneous wind measurements by meteor radar also show enhancement of 14.5‐day periodic oscillation after SSW onset.”

Tang et al [2]

So the SSW plays an important role in ionospheric variations, and the lunar tidal effects emerge as the higher atmospheric density of a SSW upwelling becomes more sensitive to lunar tidal forcing. That may be related to how the QBO also shows a dependence on lunar tidal forcing due to its higher density.


  1. Siddiqui, T. A. Relationship between lunar tidal enhancements in the equatorial electrojet and stratospheric wind anomalies during stratospheric sudden warmings. (2020). Originally presented at AGU 2018 Fall Meeting
  2. Tang, Q., Zhou, C., Liu, Y. & Chen, G. Response of Sporadic E Layer to Sudden Stratospheric Warming Events Observed at Low and Middle Latitude. Journal of Geophysical Research: Space Physics e2019JA027283 (2020).

Chandler Wobble according to Na

In Chapter 13 of the book, we have a description of the mechanism forcing the Chandler Wobble in the Earth’s rotation. As a counter to a recent GeoenergyMath post suggesting there is little consensus behind this mechanism, a recent paper by Na et al provides a foundation to understand how the lunar forcing works. 

Chandler wobble and free core nutation are two major modes of perturbation in the Earth rotation. Earth rotation status needs to be known for the coordinate conversion between celestial reference frame and terrestrial reference frame. Due mainly to the tidal torque exerted by the moon and the sun on the Earth’s equatorial bulge, the Earth undergoes precession and nutation.

Na, S.-H. et al. Chandler Wobble and Free Core Nutation: Theory and Features. Journal of Astronomy and Space Sciences 36, 11–20 (2019).
Continue reading

AO, PNA, & SAM Models

In Chapter 11, we developed a general formulation based on Laplace’s Tidal Equations (LTE) to aid in the analysis of standing wave climate models, focusing on the ENSO and QBO behaviors in the book.  As a means of cross-validating this formulation, it makes sense to test the LTE model against other climate indices. So far we have extended this to PDO, AMO, NAO, and IOD, and to complete the set, in this post we will evaluate the northern latitude indices comprised of the Arctic Oscillation/Northern Annular Mode (AO/NAM) and the Pacific North America (PNA) pattern, and the southern latitude index referred to as the Southern Annular Mode (SAM). We will first evaluate AO and PNA in comparison to its close relative NAO and then SAM …

Continue reading


In Chapter 11 of the book, we concentrated on the mechanism behind the QBO of stratospheric equatorial winds. In a related topic (but only briefly touched on in the book), there is interesting data from a presentation on the equatorial-only Semi-Annual Oscillation (SAO) of the upper stratosphere and lower mesosphere wind pattern [1]. The distinction between QBO and the SAO is that the QBO has a longer periodic cycle and exists at altitudes lower in the stratosphere than the SAO.

[1] T. Hirooka, T. Ohata, and N. Eguchi, “Modulation of the Semiannual Oscillation Induced by Sudden Stratospheric Warming Events,” in ISWA2016, Tokyo, Japan, 2016, p. 16.

— presentation slides from International Symposium on the Whole Atmosphere

What’s interesting at the core fundamental level is that the SAO is understood by consensus to be forced by a semi-annual cycle (a resonant condition happening to match 1/2 year is just too coincidental) whereas there is no consensus behind the mechanism behind the QBO period (the tidal connection is only available from Chapter 12). To make the mathematical connection, the following shows how the SAO draws from the QBO tidal model.

Continue reading

Teleconnections vs Common-mode mechanisms

The term teleconnection has long been defined as interactions between behaviors separated by geographical distances. Using Google Scholar, the first consistent use in a climate context was by De Geer in the 1920’s [1]. He astutely contrasted the term teleconnection with telecorrelation, with the implication being that the latter describes a situation where two behaviors are simply correlated through some common-mode mechanism — in the case that De Geer describes, the self-registration of the annual solar signal with respect to two geographically displaced sedimentation features.

As an alternate analogy, the hibernation of groundhogs and black bears isn’t due to some teleconnection between the two species but simply a correlation due to the onset of winter. The timing of cold weather is the common-mode mechanism that connects the two behaviors. This may seem obvious enough that the annual cycle should and often does serve as the null hypothesis for ascertaining correlations of climate data against behavioral models.

Yet, this distinction seems to have been lost over the years, as one will often find papers hypothesizing that one climate behavior is influencing another geographically distant behavior via a physical teleconnection (see e.g. [2]). This has become an increasingly trendy viewpoint since the GWPF advisor A.A. Tsonis added the term network to indicate that behaviors may contain linkages between multiple nodes, and that the seeming complexity of individual behavior is only discovered by decoding the individual teleconnections [3].

That’s acceptable as a theory, but in practice, it’s still important to consider the possible common-mode mechanisms that may be involved. In this post we will look at a possible common-mode mechanisms between the atmospheric behavior of QBO (see Chapter 11 in the book) and the oceanic behavior of ENSO (see Chapter 12). As reference [3] suggests, this may be a physical teleconnection, but the following analysis shows how a common-mode forcing may be much more likely.

Continue reading