In Chapter 12 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.

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Peak Oil Barrel

The first part of our book Mathematical GeoEnergy deals with the mathematics behind the depletion of fossil fuels, and specifically crude oil. One of the co-authors, Dennis, helps maintain and moderate the Peak Oil Barrel blog. Recently, Dennis posted a blog entry on Oil Shock model scenarios, which is based partly on the mathematics described in Chapter 5 (and elsewhere in the book, as the shock model is a fundamental aspect of modeling oil depletion).

There’s lots of commentary on the POB blog, including climate science topics on the Non-Petroleum comment threads, so worthwhile to have it bookmarked.

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.

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Detailed Forcing of QBO

In Chapter 11 of the book, we present the geophysical recipe for the forcing of the QBO of equatorial stratospheric winds. As explained, the fundamental forcing is supplied by the lunar draconic cycle and impulse modulated by a semi-annual (equatorial) nodal crossing of the sun. It’s clear that the QBO cycle has asymptotically approached a value of 2.368 years, which is explained by its near perfect equivalence to the physically aliased draconic period. Moreover, there is also strong evidence that the modulation/fluctuation of the QBO period from cycle to cycle is due to the regular variation in the lunar inclination, thus impacting the precise timing and shape of the draconic sinusoid. That modulation is described in this post.

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Applying Wavelet Scalograms

Dennis suggested:

“I was thinking about ENSO model and the impulse function used to drive it.  Could it be the wind shift from the QBO that is related to that impulse function.

My recollection was that it was a biennial pulse, which timing wise might fit with QBO. “


They are somehow related but more than likely through a common-mode mechanism. Consider that QBO has elements of a semi-annual impulse, as the sun crosses the equator twice per year.  The ENSO model has an impulse of once per year, with more recent evidence that it may not have to be biennial (i.e. alternating sign in consecutive years) as we described it in the book.

I had an evaluation Mathematica license for a few weeks so ran several wavelet scalograms on the data and models. Figure 1 below is a comparison of ENSO to the model

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Synchronization, critical points, and relaxation oscillators

In Chapter 18 of the book, we discuss the behavior around critical points in the context of reliability, both at the small-scale in terms of component breakdown, and in the large-scale in the context of earthquake triggering which was introduced in Chapter 13. The connection is that things break at all scales, with the common mechanism of a varying rate of progression to the critical point:


As indicated in the figure caption, the failure rate is generally probabilistic but with known external forcings, there is the potential for a better deterministic prediction of the breakdown point, which is reviewed below:

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Forced Natural Responses to LTE Solution

In Chapter 12 of the book, we describe in detail the solution to Laplace’s Tidal Equations (LTE), which were introduced in Chapter 11.  Like the solution to the linear wave equation, where there are even (cosine) and odd (sine) natural responses, there are also  even and odd responses for nonlinear wave equations such as the Mathieu equation, where the natural response solutions are identified as MathieuC and MathieuS.  So we find that in general the mix of even and odd solutions for any modeled problem is governed by the initial conditions of the behavior along with any continuing forcing. We will describe how that applies to the LTE system next:

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Mathematical GeoEnergy, update

The book is now available from these sellers: 1119434297[1]

Amazon: Mathematical Geoenergy: Discovery, Depletion and Renewal

Barnes&Noble: Mathematical Geoenergy: Discovery, Depletion and Renewal

Publisher – Wiley/AGU: Mathematical Geoenergy: Discovery, Depletion and Renewal


Outline and Chapter overviews

Errata Page 

Presentations at the American Geophysical Union (AGU) meeting in Washington D.C.