The new research by Professor Michael Mann in his peer-reviewed article called “Absence of internal multidecadal and interdecadal oscillations in climate model simulations” asks whether the decadal >10 year (and perhaps faster cycles) in the AMO and PDO behave as an internal property of the ocean or whether the cycles are externally forced. The quandary is that Mann does not deny that the ENSO behavior ** has** a strong internal oscillation, so what if anything makes ENSO special?

Like the AMO, the characteristic of PDO that distinguishes it from ENSO is in the longer decadal variation that it exhibits. In** Fig 1** below, we show a model fit to PDO that relies on essentially the same input tidal forcing that was applied to the ENSO model. What is most impressive about the fit is how naturally the interdecadal cyclic variation emerges.

The comparison between the ENSO and PDO tidal forcing is shown in** Fig 2** below. As with AMO, the clue to where the interdecadal cycle arises from is in the slight modulated curvature in the profile.

What is causing the curvature is the interference between two closely aliased primary tidal forcings. These are the fortnightly tropical cycle of 13.66 days and the monthly anomalistic cycle of 27.55 days as described here in LOD characterization and see** Fig A2** at the end of the AMO post.

The two primary forcings acting mutually are also observed in tidal tables, in what is known as the long-period 9-day tide labelled “Mt” . The period of this tide is ~9.133 days and so when applied to an annual impulse we get 365.242/9.133 = 39.99146. This number is very close to an integral value but not quite, so it will only reach a constructive interference against an annual impulse every 1/(40-39.99146) ~ 120 years.

Since the LTE model is nonlinear, the 120 year underlying cycle can readily transform into a 60 year harmonic. So — just as for the AMO — the ~60 year cycle emerges in the PDE time series with the appropriate LTE modulation as in **Fig 3** below.

Although this analysis is slightly different than what we presented at the 2018 AGU meeting, the same bottomline stands, in that the tidal forcing is nearly equivalent for the ENSO, PDO, and AMO climate indices as show in **Fig 4** below. This also applies to the IOD index, so that any long-term predictability may arise solely due to the ability to precisely refine this forcing along with the LTE modulation for the appropriate oceanic dipole.

The reality is that with the combination of AGW and the additive impact of multiple concurrently peaking oceanic dipoles, extremes in temperature can occur that may be larger than any time in modern times. Consider the possibility in **Fig 4 **of the IOD, ENSO, and SAM peaking simultaneously and the impact that has in Australia. And that is at least partly what is happening now.

TIW wavetrains are also observed in the equatorial Atlantic so would be considered alongside the AMO there as the high wavenumber and low wavenumber pairing.

According to the Laplace’s Tidal Equation (LTE) model that we applied to ocean thermocline dynamics, TIW wavetrains originate from the higher-order solutions to the LTE differential equation. As with any waveguide containing standing-wave behaviors, the spatio-temporal solution allows for multiple wavelength/frequency combinations — in the case of LTE, it’s solved resulting in a linear dispersion relation, *k ~ f*. So for the Pacific ENSO + TIW pairing, the ratio is ~15:1, with the TIW wavelength ~15 times shorter than the ENSO standing wave (see **Fig 1**) and the TIW LTE amplitude modulation also 15 times the ENSO LTE modulation. This faster modulation is shown in the inset in the upper panel in **Fig 2** below. The specific combination of slower and faster modulation provides the proper mix of harmonics to recreate the spikiness in the ENSO time-series.

As with any application of harmonics, the most important aspect is in modifying the shape (e.g. triangle, square, sawtooth) of the waveform and not in the overall period. Adding the higher-order modulation via the 15x TIW wavenumber is shown in **Fig 3 **below, with a clear enhancement of the sharpness in the overall fit by comparing the upper plot to the lower plot.

A recent paper titled “A simple theory for the modulation of tropical instability waves by ENSO and the annual cycle” [1] suggests a similar close relationship between ENSO and TIW. They refer to it as a simple model in that specific harmonics related to the ENSO cycle and annual cycle comprise the TIW wavetrain, which they empirically establish from an isolated TIW time-series. In contrast, in our model, the relation is determined by the nature of the LTE modulation. **Fig 4** below shows our unfitted TIW model (extracted as the 15x factor) alongside the model from the Boucherel paper.

This is unfitted in the sense that the principal higher wavenumber solution derives only from a best fit of the LTE model to the NINO34/SOI time-series data — essentially the higher harmonics contribution of **Fig 3**. This has the byproduct of fitting the observed TIW as in **Fig 4**, thus creating the faster surface temperature cycles.

What is not called out by the Boucherel paper is that the TIW may actually amplify ENSO, whereas they show it having an opposite polarity relationship. That is dependent on where they are measuring the amplitude of the standing wave as shown in **Fig 5** below. To investigate this further, it will be useful to have access to the complete set of data and reproduce their regression procedure to isolate the TIW component.

[1] Boucharel, J. and Jin, F.F., 2020. A simple theory for the modulation of tropical instability waves by ENSO and the annual cycle. *Tellus A: Dynamic Meteorology and Oceanography*, *72*(1), pp.1-14.

The new research on AMO by Professor Michael Mann appears to be meant to be somewhat provocative, which is OK as it spurred some discussion on Twitter. His peer-reviewed article is called “Absence of internal multidecadal and interdecadal oscillations in climate model simulations” and its takeaway is right in the title. Essentially, Mann *et al* are asking whether the ~60 year oscillation (and perhaps faster cycles) in the AMO behave as an internal property of the Atlantic ocean or whether the cycles are externally forced. ** Fig A1 **in the Appendix provides some background on the AMO time-series data.

The characteristic of AMO that distinguishes it from ENSO is the multidecadal variation of ~60 years that it exhibits. In** Fig 1** below, we show a model fit to AMO that relies on essentially the same input tidal forcing that was applied to the ENSO model. What is most impressive about the fit is how naturally the ~60 year cyclic variation emerges.

The comparison between the ENSO and AMO tidal forcing is shown in** Fig 2** below. The clue to where the 60 year cycle arises from is in the slight modulated curvature in the profile.

What is causing the curvature is the interference between two closely aliased primary tidal forcings. These are the fortnightly tropical cycle of 13.66 days and the monthly anomalistic cycle of 27.55 days as described here in LOD characterization and see** Fig A2** at the end of this post. These two, when interacting against a yearly annual impulse, produce a clear ~120 year repeat pattern as shown in **Fig 3**.

The two primary forcings acting mutually are also observed in tidal tables, in what is known as the long-period 9-day tide labelled “Mt” . From **Fig 4** below the period of this tide is precisely 9.132933 days and so when applied to an annual impulse we get 365.242/ 9.132933 = 39.99175. This will reach a constructive interference against an annual impulse every 1/(40-39.9917)=121.3 years.

This value is important to consider for understanding how a ~60 year cycle comes about from the model. Since the LTE model is nonlinear, the 120 year underlying cycle can readily transform into a 60 year harmonic. Or these longer periods may not be as obvious, as with the ENSO model. So it just so happens that the 60 year cycle emerges in the AMO time series with the appropriate LTE modulation as in **Fig 5** below (as it does with PDO).

The 60 year modulation also appears as an intra-spectral cross-correlation as described in the previous post .

Furthermore, scientists at NASA JPL and the Paris Observatory have long known about a 60 year link between LOD and climate variation see **Fig 6** below,.

The idea is that this multi-decadal period is integrated over time, creating a mutual interaction between the long-period forcing of the lunar + solar tides and the sloshing response of the ocean basins. The LOD or Universal Time (UT1) measure becomes a correlating measure of these forcing constituents. So the longer the period of potential constructive interference, such as with the 60 year near aliasing of the *Mt* constituent against a yearly impulse, the more that an inertial response can accumulate [4]. It may in fact be that the entirety of the LOD variations are due to the lunar + solar forcing and this is the unification between LOD and the climate dipole standing-wave behavior.

From [4]

- Cleverly, J.
*et al.*The importance of interacting climate modes on Australia’s contribution to global carbon cycle extremes.*Sci Rep***6**, 1–10 (2016). - Desai, S. D. & Wahr, J. M. Empirical ocean tide models estimated from TOPEX/POSEIDON altimetry.
*Journal of Geophysical Research: Oceans***100**, 25205–25228 (1995). - Marcus, S. L. Does an Intrinsic Source Generate a Shared Low-Frequency Signature in Earth’s Climate and Rotation Rate?
*Earth Interact.***20**, 1–14 (2015). - Dickman, S. R. Dynamic Ocean-Tide Effects On Earth’s Rotation.
*Geophysical Journal International***, 112**, 448–470 (1993).

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 12, 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.

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