The context is looking for autocorrelations in the frequency domain of a time-series. Although not as common as performing autocorrelations in the time domain, it is equally as powerful.

The earlier idea was to look for harmonics in the periodicity of the ENSO signal, and the chart described in the post showed clear annual and higher harmonics in the time series. This was via a straightforward sliding autocorrelation in the power spectra.

As an additional technique, we can look for symmetric sidebands of the annual fundamental and harmonics frequencies by folding the spectra over about the annual frequency and performing a direct correlation calculation.

This correlation is painfully obvious and is well beyond statistically significant in demonstrating that an annual impulse signal is modulating another much more complex forcing signal (likely of tidal origin). This is actually a well-known process known as a double-sideband suppressed carrier modulation, used most commonly in facilitating broadcast transmissions. As shown in the equations below, the modulation acts to completely suppress the carrier (i.e. annual) frequency.

Read the previous post for more detail on the approach.

Ordinarily, the demodulation is straightforward via a standard mixing approach, as the carrier signal is a much higher frequency than the informational signal, but since annual and long-period tides are of roughly similar periods, the demodulation will only complicate the spectrum. This is not a big deal as we need to fit the peaks via the LTE formulation in any case.

This is a new and novel finding and not to be found anywhere in the ENSO research literature. Why it hasn’t been uncovered yet is a bit of a mystery, but the fact that the annual signal is completely suppressed may be a hint. It may be that we need to understand why the dog didn’t bark.

Gregory (Scotland Yard detective): “Is there any other point to which you would wish to draw my attention?”

— “

Holmes: “To the curious incident of the dog in the night-time.”

Gregory: “The dog did nothing in the night-time.”

Holmes: “That was the curious incident.”The Adventure of Silver Blaze” by Sir Arthur Conan Doyle

If that is not the case, and this has been published elsewhere, will update this post.

]]>A recent article uploaded to arXiv [1] gives an alternate treatment to the one we described. This follows Beck’s original approach more than our simplified formulation but each is an important contribution to understanding and applying the math of wind variability. The introduction to their article is valuable in providing a rationale for doing the analysis.

“Mitigating climate change demands a transition towards renewable electricity generation, with wind power being a particularly promising technology. Long periods either of high or of low wind therefore essentially define the necessary amount of storage to balance the power system. While the general statistics of wind velocities have been studied extensively, persistence (waiting) time statistics of wind is far from well understood. Here, we investigate the statistics of both high- and low-wind persistence. We find heavy tails and explain them as a superposition of different wind conditions, requiring q-exponential distributions instead of exponential distributions. Persistent wind conditions are not necessarily caused by stationary atmospheric circulation patterns nor by recurring individual weather types but may emerge as a combination of multiple weather types and circulation patterns. Understanding wind persistence statistically and synoptically, may help to ensure a reliable and economically feasible future energy system, which uses a high share of wind generation. “

[1]Weber, J.et al.“Wind Power Persistence is Governed by Superstatistics”.arXiv preprint arXiv:1810.06391(2019).

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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 Sciences36, 11–20 (2019).

The tidal torque of the earth and sun are *sufficient*, and the non-spherical bulge is *essential *to start any kind of precession — as a perfectly-uniform rotating spherical object can’t be influenced by an external torque, explained by symmetry arguments alone.

Na *et al* estimate the Chandler Wobble cycle from the empirical time-series to be 432.2 days, compared to our estimate of 432.4 days based on the lunar nodal cycle synchronized to a semi-annual cycle.

They continue with a further rationale and formulate a precession matrix that one can apply.

For precession, one may assume the lunar and solar masses as circularly distributed around the Earth like donuts. In fact, the moon and the sun give periodic torques as oscillatory perturbations and lead Earth nutation. By analogy of harmonic oscillator to periodic forces, amplitude of long period nutation is larger than short period one.

Na, S.-H.et al.Chandler Wobble and Free Core Nutation: Theory and Features.Journal of Astronomy and Space Sciences36, 11–20 (2019).

So the foundation is in place but they have yet to make an association of the ~432 day cycle to the known simultaneous nodal crossing period.

Another finding that they make is the apparent measured decrease in the amplitude of the Chandler wobble over time.

This may be related to the Earth’s spin axis drift over recent years, perhaps revealing a change in the moment of inertia — caused by redistribution of mass such as in glaciers, glacial rebound, and/or mantle convection. Less likely, but perhaps coincidental, could this decrease be related to the QBO anomaly of 2016, which also is synchronized to the nodal crossing cycle?

In any case, it will be interesting if the cycle period remains constant with further reduction in wobble amplitude.

]]>The Pacific North America oscillation stretches across the continent, as described in Wikipedia: *“The positive phase of the PNA pattern features above-average barometric pressure heights in the vicinity of Hawaii and over the inter-mountain region of North America, and below-average heights located south of the Aleutian Islands and over the southeastern United States. The PNA pattern is associated with strong fluctuations in the strength and location of the East Asian jet stream.”* Both the PNA and the Arctic Oscillation can be easily fit from a perturbation of the NAO model, which can be deduced from the known similarity between the AO and NAO — (“*The North Atlantic oscillation (NAO) is a close relative of the AO*“).

**Fig 1** shows the common tidal forcing for each of these models, with the LTE modulation in the lower panel. The tidal forcing has a strong semi-annual factor, as does the QBO (see Chapter 11).

The LTE modulation differs subtly between the three, as the multipliers are slightly different for NAO and AO and within ~15% for PNA. They are in sync at the yellow arrows shown in the lower panel of **Fig 1**. The LTE modulation is dependent on the fundamental spatial wavenumber defining the dipole, which should be different for each of the regions.

**Fig 2** shows the fits for each of the time-series, starting from the NAO described in an earlier post:

You can see how the NAO and AO are vaguely similar and the the PNA is similar but flipped in polarity. It is known that the QBO has a connection to the polar vortex, so the semi-annual commonality between QBO and AO makes some sense [1].

The only major index left is the Southern Annular Mode (SAM) index associated with the Antarctic Oscillation (data source). As a side note and based of the complexity of these waveforms, this should have taken a long time to adequately fit a model if starting from scratch. Yet, since the tidal forcing is nearly identical for each of these indices (see **Fig 3** below), the computation took relatively little time to converge to a good fit.

The LTE modulation of SAM was close to that of the complementary AO/NAM, as it retains the same phase over a greater range of forcing levels (indicated by the yellow arrow in **Fig 4**):

Like the others, the fit for SAM is also very good (Fourier spectrum comparison in lower panel of **Fig 5**)

As a bottom-line, these climate indices are likely not related as teleconnections (which is the current consensus idea), but more likely by a common-mode forcing . The set is synchronized by the common lunisolar tidal forces operating across the earth and individually distinguished by the standing wave constraints of each region. Moreover, it’s highly unlikely that the quality of these model fits is due to overfitting as there are very few DOF available given the common-mode forcing constraint shared by each model.

In general, what we characterize as the LTE multiplier may require a new vocabulary to describe the resultant behavior. Since there is nothing in the research literature that approximates this solution, there is no lingo or common understanding to draw from. The LTE modulation factor is perhaps something akin to a Reynolds number (Re) or a Richardson number (Ri) defined in fluid dynamics, which makes it a single scalar that describes the breaking or folding of the waves (like a turbulence factor but not chaotic) and relates to the primary wavenumber of the standing wave dipole.

The trend of the LTE value is closer to zero if the climate index is measured close to the equator (QBO is the lowest) and it tends to increase as the index moves away from the equator. The ordering is approximately this:

QBO < ENSO < (AMO ~ IOD) < PDO < ( NAO ~ AO ~ SAM ~ PNA)

The wavenumber of QBO approaches zero because the standing wave encircles the equator and cycles in unison. Correspondingly the wavenumber values may be required to increase away from the equator — where the dimensionality naturally shrinks closer to the poles — but it also may be due to the specific waveguide bounding box of the index. For example, the equatorial Pacific is the widest dimension of the oceanic indices and thus ENSO has the lowest primary wavenumber next to QBO.

The PDO has a significant LTE sin() modulation that is the same as ENSO, but also has a strong factor with a wavenumber that is 5 times as rapid. In contrast the AMO wavenumber modulation is 3 times as fast as ENSO (with a much weaker modulation that’s the same as ENSO). See **Fig 6** below.

These also have tidal forcings that are similar (see **Fig 7** below) but distinct from that of the upper latitude group of NAO, PNA, AO, and SAM

What’s interesting about the common tidal forcing of (AO, NAO, PNA, SAM) is that there is a distinct visible period in the time-series which is the lunar *tropical* month (27.321582 days) aliased against the annual signal. This can be determined from counting the major periods in **Fig 3**.

1/(365.242/(27.321582)-13) = 2.72 years

For the QBO, the forcing and response are very close to each other (due to the low LTE factor) and the tidal forcing is the lunar *draconic* month (27.2122 days) aliased against the annual signal. This gives the measured QBO periodicity of:

1/(365.242/(27.2122)-13) = 2.37 years

There are many papers suggesting that there is a connection between QBO and polar behavior, see [1], but it is not always apparent from the data. The wavenumber=0 symmetry of the QBO precludes any tropical (synodic) dependence so the cycle is draconic while the the other indices require a tropical dependence, as they are geospatially specific. The two distinct cycles will go in and out of sync gradually with an 18.6 year cycle.

In conclusion, the commonality of these 9 indices in terms of a common tidal forcing and distinct LTE modulations provides a convincing cross-validation of the LTE formulation described in our book Mathematical Geoenergy.

[1] One such paper from earlier this year claiming a QBO to AO teleconnection: Observed and Simulated Teleconnections Between the Stratospheric Quasi‐Biennial Oscillation and Northern Hemisphere Winter Atmospheric Circulation

]]>In Chapter 11, we concentrated on the Pacific ocean dipole referred to as ENSO (El Nino/Southern Oscillation). A dipole that shares some of the characteristics of ENSO is the neighboring Indian Ocean Dipole and its gradient measure the Dipole Mode Index.

The IOD is important because it is correlated with India subcontinent monsoons. It also shows a correlation to ENSO, which is quite apparent by comparing specific peak positions, with a correlation coefficient of 0.2. This post will describe the differences found via perturbing the ENSO model …

As a starting rationale for explaining why the correlation isn’t higher, there is likely another standing wave solution that complements the major standing wave that stretches across the equatorial Pacific. The latter contributes the majority of ENSO but only a portion of IOD, so the mystery standing wave is what generates the busier cyclic behavior of IOD.

As with the other oceanic indices, the IOD model generates a similar tidal forcing to ENSO, with R^2>0.95.

What differs from the ENSO model is the Laplace’s Tidal Equation (LTE) modulation — the IOD consists of a LTE background similar to ENSO, but also a faster modulation that is 3 to 4 that of the background. This can be seen in **Fig 2** below.

The LTE modulation is applied to the tidal forcing during the model fitting process. The fit over the entire time span is good, with the Fourier spectrum in the lower panel of **Fig 3 **below.

Each of ENSO, PDO, AMO [1], and now IOD have a nearly identical set of fundamental forced tidal cycles but distinct standing mode modulations. The QBO is the only completely atmospheric behavior, and it has a distinct tidal forcing (mainly draconic as opposed to tropical for the oceanic indices). The NAO model has the semiannual forcing of QBO but the same tidal forcing as ENSO, PDO, AMO, and IOD.

[1] “Ephemeris calibration of Laplace’s tidal equation model for ENSO“, 2018 AGU Fall Meeting. *Note: PDO, AMO, and NAO models were evaluated and compared to ENSO but this was too late for inclusion in the book*

The models for ENSO, PDO, and AMO share a common lunisolar forcing pattern, which is essentially an annual impulse which modulates the fortnightly and monthly tidal cycles.. This is then integrally lagged to create the ragged square-wave time-series shown in **Fig 1** below

To create the distinct responses for each index, a specific level of Laplace’s Tidal Equation (LTE) modulation (as derived in Chap 11) is applied to the (close to) common tidal forcing. These are shown in** Fig 2 **below and are associated with a distinct spatio-temporal standing wave pattern.

The difference with the NAO solution is that the annual modulation is suppressed and requires not as sharp an annual impulse, resulting in a semiannual sinusoidal modulation along with the monthly values bleeding through to the lag integrator. This sub-annual variation is essentially what generates a faster cycling. Yet, the underlying raw forcing (shown in **Fig 3** below with correlation coefficient > 0.99) is identical for the two, reducing the number of degrees of freedom varied during the fit..

The LTE modulation for NAO is strong, approximately that used for the PDO model (in **Fig 2** above). Perhaps this is expected as both NAO and PDO are each northern/higher latitude behaviors.

With the LTE modulation providing the primary fitting parameter, the NAO results are better correlated to that published for the AGU presentation. A short cross-validation training interval in **Fig 5** is over-fitted but still shows agreement outside of that band.

**Summary**: It appears that the model for NAO is not as sensitive to sharp annual impulses as the other behaviors (ENSO, PDO, AMO) require for modeling. So instead of an impulse, it appears to more directly correspond to monthly tidal variations with a sinusoidal semi-annual (like for QBO) modulation aiding in the fit. Considering the commonality of the tidal forcing signals within the collection of oceanic indices, the quality of the results indicates that either the LTE approach is meaningful *or* that we have stumbled on some other pattern that connects the behaviors together.

PubPeer provides a good way to debunk poorly researched work as shown in the recent comments pertaining to the Zharkova paper published in Nature’s Scientific Reports journal.

An issue with the comment policy at Amazon is that one can easily evaluate the contents of a book via the “Look Inside” feature or through the Table of Contents. Often there is enough evidence to provide a critical book review just through this feature — in a sense, a statistical sampling of the contents — yet Amazon requires a full purchase before a review is possible. Even if one can check the book out at a university library this is not allowable. Therefore it favors profiting by the potential fraudster because they will get royalties in spite of damaging reviews by critics that are willing to sink money into a purchase.

In the good old days at Amazon, one could actually warn people about pseudo-scientific research. This is exemplified by Curry’s Bose-Einstein statistics debacle, where unfortunately political cronies and acolytes of Curry’s have since purchased her book and have used the comments to do damage control. No further negative comments are possible since smart people have not bought her book and therefore can no longer comment.

PubPeer does away with this Catch-22 situation.

]]>[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.

According to the Hirooka paper, the SAO flips by 180 degrees between the stratosphere (the SSAO) and the mesosphere (the MSAO). You can see this in the upper panel in **Figure 1** below where the intense westerlies (in RED) occur during the beginning and middle of each year for the MSAO, and they occur between these times (Spring and Fall) for the SSAO. The direction times are complementary for the easterlies in BLUE. At altitudes between the MSAO and SSSAO, the strength of the SAO is significantly reduced as you can see in the lower panel showing the spectral lines.

This may be explained by the Laplace’s Tidal Equation analytic solution that we have been applying to the ENSO and QBO models. The analytic solution expressed in a parametric form is :

sin(A sin(4πt+ϕ)+θ(z)))

If the LTE model phase (θ) varies in altitude (z) due to differing characteristics of the atmospheric density, the sense of the sinusoidal modulation will flip. This is for a value of *A* that is large enough to cause a strong modulation. For phases halfway between where the sign flips, the modulation bifurcates the semi-annual oscillation such that the 1/2-year period disappears and is replaced by (in-theory) a 1/4-year or 90-day cycle. This can be seen in the **Figure 1** power-spectra .

Below in **Figure 2** is the theoretical LTE model plot alongside the Hirooka et al plot. The contour colors don’t quite match up, because of the limitations of the plotting tool — the data is represented as a true density plot whereas the model includes contour line artefacts.

As an additional observation from** Figure 1**: If one considers that an annual oscillation occurs both above and below the altitudes of the QBO, it would be odd if the stratospheric layer for QBO* wasn’t* similarly forced by a tidal oscillation.

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.

]]>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.

From the outset, the forcing to the modeled behaviors of ENSO and QBO were split into just a few categories. There is (1) the declination of the lunar cycle, (2) the ellipticity of the lunar cycle, and (3) the earth’s orbit around the sun while it is rotating itself, as potential null hypothesis forcings. Each of these categories has similar complexity with cross-terms similar to those found with conventional tidal analysis.

As it turns out, the *declination forcing* for ENSO, when isolated on its own and retaining the 2^{nd}-order detail in the Draconic month variation, provides an almost exact match as a QBO forcing, as shown in **Figure 1** below.

It makes sense that the QBO model only uses a single forcing factor considering how regular the cycle appears. Yet the regularity is obscured on a closer look by a variation in the 2.38 year average period. The common-mode mechanism is that the model from ENSO assumes a specific variation in the Draconic month [4], see **Figure 2** for the alignment of the QBO forcing applied — they are essentially the same.

Without the Draconic variation, the correlation coefficient of the QBO data with a model applying a purely sinusoidal forcing is only ~0.5, instead of reaching 0.75.

In contrast, the Draconic forcing alone is not enough to provide the variation required to model ENSO, and so the orthogonal Anomalistic and Synodic factors provide the additional detailed forcing. The common-mode Draconic forcing is shown in green in **Figure 4.**

The utility of this common-mode mechanism is that it may enable an additional constraint to fitting ENSO and reducing the number of degrees of freedom.

[1] G. De Geer, “Teleconnections contra so-called telecorrelations,” *Geologiska Föreningen i Stockholm Förhandlingar*, vol. 57, no. 2, pp. 341–346, 1935.

[2] D. I. V. Domeisen, C. I. Garfinkel, and A. H. Butler, “The Teleconnection of El Niño Southern Oscillation to the Stratosphere,” *Reviews of Geophysics*, vol. 57, no. 1, pp. 5–47, Mar. 2019.

[3] A. A. Tsonis and K. L. Swanson, “Topology and predictability of El Nino and La Nina networks,” *Physical Review Letters*, vol. 100, no. 22, p. 228502, 2008.

[4] P. R. Pukite, D. Coyne, and D. Challou, “Ephemeris calibration of Laplace’s tidal equation model for ENSO,” presented at the AGU Fall Meeting Abstracts, 2018.

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