Paul L. Vaughan, M.Sc.
Without a good handle on its simple geometry, a seemingly complex time series can appear as a changeling yielding to the pressures of mysterious statistical manipulation.
For example, a fundamentally important seminal observation reported by Le Mouël, Blanter, Shnirman, & Courtillot (2010) revealed the quasistationary 11 year solar cycle in the rate of change of length of day (LOD’), but newcomers taking a preliminary look at daily resolution LOD’ are more likely to fixate on the 18.6 year lunisolar envelope.
Multiscale variance summaries highlight obvious envelopes:
Zooming in, a semi-annual envelope is also evident:
(WIDE GRAPH ABOVE –Click to view elongate graph^1 & then click again to magnify.)
(WIDE GRAPH ABOVE –Click to view elongate graph^2 & then click again to magnify.)
A parsimonious weekly-to-monthly timescale model of daily LOD’, explaining ~93% of the variance (r = 0.965), can be constructed using the following information (with model terms in bold italics):
| Year | Period (days) | Half-Period (days) | Defined by… |
| Tropical | 365.24219 | 182.621095 | equinoxes |
| Lunar Month | Period (days) | Half-Period (days) | Defined by… |
| Tropical | 27.321582 | 13.660791 | equator/equinoxes |
| Nodal or Draconic | 27.212221 | 13.6061105 | ecliptic |
| Anomalistic | 27.55455 | 13.777275 | apogee/perigee |
| Synodic | 29.530589 | 14.7652945 | new/full moon |
(27.321582)*(27.212221) / (27.321582 – 27.212221)
= 6798.410105 days = 18.61343046 years
(6798.410105)*(13.6061105) / (6798.410105 – 13.6061105)
= 13.63339592 days
(27.55455)*(13.660791) / (27.55455 + 13.660791)
= 9.132933018 days
Noteworthy envelopes apparent in the variance structure of LOD’ relate to:
1) lunar nodal cycle (LNC) = 18.6 years
2) lunar apse cycle (LAC) = 8.85 years
3) terrestrial year (1 year)
4) harmonics (e.g. 0.5 years & 4.42 years)
| Beat Period | (years) | Tropical | Nodal | Anomalistic | Synodic |
| 27.321582 | 27.212221 | 27.55455 | 29.530589 | ||
| Tropical | 27.321582 | – | 18.6134 | 8.8475 | 1.0000 |
| Nodal | 27.212221 | 18.6134 | – | 5.9970 | 0.9490 |
| Anomalistic | 27.55455 | 8.8475 | 5.9970 | – | 1.1274 |
| Synodic | 29.530589 | 1.0000 | 0.9490 | 1.1274 | – |
| Beat Period | (years) | Tropical/2 | Nodal/2 | Anomalistic/2 | Synodic/2 |
| 13.660791 | 13.6061105 | 13.777275 | 14.7652945 | ||
| Tropical/2 | 13.660791 | – | 9.3067 | 4.4238 | 0.5000 |
| Nodal/2 | 13.6061105 | 9.3067 | – | 2.9985 | 0.4745 |
| Anomalistic/2 | 13.777275 | 4.4238 | 2.9985 | – | 0.5637 |
| Synodic/2 | 14.7652945 | 0.5000 | 0.4745 | 0.5637 | – |
Beat Period = (A*B) / ( |A-B| )
| | indicates absolute value
The model:
| Relative | Cumulative | ||||
| Term | Period (days) | Amplitude | r^2 | r | Contribution |
| 1 | 13.660791 | 1 | 0.713 | 0.844 | | polarity | |
| 2 | 13.63339592 | 0.41 | 0.824 | 0.908 | LNC |
| 3 | 9.132950896 | 0.30 | 0.881 | 0.939 | LAC alternation |
| 4 | 27.55455 | 0.26 | 0.926 | 0.962 | LAC alternation |
| 5 | 14.7652945 | 0.08 | 0.931 | 0.965 | semi-annual |
(WIDE GRAPH ABOVE – Click to view elongate graph^3 & then click again to magnify.)
eLOD’ = estimated LOD’
The above tables & figures, while certainly nothing new to science, have been summarized here for the benefit of those striving to efficiently develop the foundations necessary to appreciate and build upon the recent seminal work of Le Mouël, Blanter, Shnirman, & Courtillot (2010). From their conclusions:
“The solid Earth behaves as a natural spatial integrator and time filter, which makes it possible to study the evolution of the amplitude of the semi-annual variation in zonal winds over a fifty-year time span. We evidence strong modulation of the amplitude of this lod spectral line by the Schwabe cycle (Figure 1a). This shows that the Sun can (directly or undirectly) influence tropospheric zonal mean-winds over decadal to multi-decadal time scales. Zonal mean-winds constitute an important element of global atmospheric circulation. If the solar cycle can influence zonal mean-winds, then it may affect other features of global climate as well […]”
[Typos: 1) “evidence” should read “observe”. 2) “undirectly” should read “indirectly”.]
Caution
Exclusive &/or excessive focus on the first moment (the mean) should not be at the expense of attention to higher moments (such as the variance), as the following graph should emphasize:
SOI = Southern Oscillation Index (an index of El Nino / La Nina)
[ ] indicates boxcar averaging [applied here to highlight interannual variability]
When studying the preceding graph, it is important to understand that the blue line is the normalized interannual average of the black line. (Take a minute to think about this carefully.)
To reinforce this point, here is another graph of the normalized mean at the semi-annual to annual timescale:
The occurrence of such patterns in the mean despite the maintenance of stationary variance limits suggests a need to carefully consider which equators (geographic, celestial, magnetic, meteorological, etc.) are relevant to the phenomena under study. (See for example Leroux (1993).)
Multimoment multiscale spatiotemporal integration reveals nonrandom harmonic pattern-summary discontinuities, exposing the comedy tragically advocated by deceitful &/or naive theoreticians who are in part constrained by a dominant culture that clings seemingly religiously to maladaptive traditions such as unjustifiable assumptions of randomness, independence, uniformity, linearity, etc. that are routinely misapplied (for example to conveniently render abstract conceptions mathematically tractable).
Bear in mind that for some phenomena, such as ice-jacking freeze/thaw cycles, the properties of the variance play a critically fundamental role in dynamics.
Conclusion
With awareness of key wavelengths and a solid conceptual understanding of the effect of integration across harmonics, we arrive at something truly simple: Earth, Sun, Moon.
Both of the ~11 year waves summarize the semi-annual wave, which summarizes biweekly & monthly LOD’ variations bounded by lunisolar limits.
While the magenta wave is isolated via complex wavelet methods, the sky-blue wave is accessible to any member of the general public with an understanding of this article, 5 minutes to spare, & a spreadsheet.
Acknowledgement
Tim Channon generously shared LOD’ models developed using his synthesizer software. Access to Tim’s models facilitated expeditious cross-checking of lunisolar theory, mainstream literature, & data.
Suggestion
I encourage responsible readers to download & archive daily LOD data. Scientifically-engaged citizens can keep a vigilant watch on potentially-arising future data vandalism.
Data
LOD
International Earth Rotation Service (IERS)
http://www.iers.org/IERS/EN/DataProducts/EarthOrientationData/eop.html
Related Reading
Li, G.-O.; & Zong, H.-F. (2007). 27.3-day and 13.6-day atmospheric tide. Science in China Series D – Earth Sciences 50(9), 1380-1395.
http://www.scichina.com:8080/sciDe/fileup/PDF/07yd1380.pdf
Sidorenkov, N.S. (2007). Long-term changes in the variance of the earth orientation parameters and of the excitation functions.
http://syrte.obspm.fr/journees2005/s3_07_Sidorenkov.pdf
Sidorenkov, N.S. (2005). Physics of the Earth’s rotation instabilities. Astronomical and Astrophysical Transactions 24(5), 425-439.
http://images.astronet.ru/pubd/2008/09/28/0001230882/425-439.pdf
Gross, R.S. (2007). Earth rotation variations – long period. In: Herring, T.A. (ed.), Treatise on Geophysics vol. 11 (Physical Geodesy), Elsevier, Amsterdam, in press, 2007.
http://geodesy.eng.ohio-state.edu/course/refpapers/Gross_Geodesy_LpER07.pdf
http://geodesy.geology.ohio-state.edu/course/refpapers/Gross_Geodesy_LpER07.pdf
Schwing, F.B.; Jiang, J.; & Mendelssohn, R. (2003). Coherency of multi-scale abrupt changes between the NAO, NPI, and PDO. Geophysical Research Letters 30(7), 1406. doi:10.1029/2002GL016535.
Maraun, D.; & Kurths, J. (2005). Epochs of phase coherence between El Nino-Southern Oscillation and Indian monsoon. Geophysical Research Letters 32, L15709. doi10.1029-2005GL023225.
http://www.cru.uea.ac.uk/~douglas/papers/maraun05a.pdf
Leroux, M. (1993). The Mobile Polar High: a new concept explaining present mechanisms of meridional air-mass and energy exchanges and global propagation of palaeoclimatic changes. Global and Planetary Change 7, 69-93.
http://ddata.over-blog.com/xxxyyy/2/32/25/79/Leroux-Global-and-Planetary-Change-1993.pdf
Trenberth, K.E.; Stepaniak, D.P.; & Smith, L. (2005). Interannual variability of patterns of atmospheric mass distribution. Journal of Climate 18, 2812-2825.
http://www.cgd.ucar.edu/cas/Trenberth/trenberth.papers/massEteleconnJC.pdf
Abarca del Rio, R.; Gambis, D.; & Salstein, D.A. (2000). Interannual signals in length of day and atmospheric angular momentum. Annals Geophysicae 18, 347-364.
http://hal-insu.archives-ouvertes.fr/docs/00/32/91/24/PDF/angeo-18-347-2000.pdf
Abarca del Rio, R.; Gambis, D.; Salstein, D.; Nelson, P.; & Dai, A. (2003). Solar activity and earth rotation variability. Journal of Geodynamics 36, 423-443.
http://www.cgd.ucar.edu/cas/adai/papers/Abarca_delRio_etal_JGeodyn03.pdf
Le Mouël, J.-L.; Blanter, E.; Shnirman, M.; & Courtillot, V. (2010). Solar forcing of the semi-annual variation of length-of-day. Geophysical Research Letters 37, L15307. doi:10.1029/2010GL043185.
Vaughan, P.L. (2010). Semi-annual solar-terrestrial power.
Technical Aside
For those interested in exploring LOD’ variance patterns that are not necessarily evident at first glance, another noteworthy envelope is the following:
(13.777275)*(13.63339592) / (13.777275 – 13.63339592)
= 1305.478517 days = 3.574281812 years
This polar-equatorial eclipse cycle is evident in the sequence of diagrams here:
http://eclipse.gsfc.nasa.gov/5MCLE/5MCLE-Figs-10.pdf (1733-2151)
From:
Espenak, F.; & Meeus, J. (2009). Five millennium canon of solar eclipses: -1999 to +3000 (2000 BCE to 3000 CE). NASA Technical Publication TP-2009-214172.
http://eclipse.gsfc.nasa.gov/SEpubs/5MCLE.html
h/t to WUWT commenter “lgl” for initially drawing attention to this pattern some time ago.
Earlier & Future Articles
I wrote the following articles before (a) acquiring access to Le Mouël, Blanter, Shnirman, & Courtillot (2010), (b) coming across Leroux (1993), and (c) re-reading Sidorenkov (2005) with consequently improved awareness:
1) http://wattsupwiththat.com/2010/08/18/solar-terrestrial-coincidence/
2) http://wattsupwiththat.com/2010/09/04/the-north-pacific-solar-cycle-change/
3) http://wattsupwiththat.com/2010/09/11/solar-cycle-length-its-rate-of-change-the-northern-hemisphere/
Related articles could have been written on All India Rainfall Index & other variables, but the audiences’ handle on the solar, lunisolar, & spatiotemporal nature of interannual variations was revealed to be inadequate in comments here:
4) http://wattsupwiththat.com/2010/10/11/atlantic-hurricanes-the-sun/
[Some audience members may benefit from careful consideration of issues raised by Tomas Milanovic at Dr. Judith Curry’s blog Climate Etc.]
Le Mouël, Blanter, Shnirman, & Courtillot’s (2010) game changing observation rendered earlier results much less mysterious:
For capable individuals striving to render these & related findings disgestible by a mainstream audience, I strongly recommend:
A) gleaning the primary point made by Schwing, Jiang, & Mendelssohn (2003) about the effect of windowing parameters on apparent phase, which can be reversed by spatial patterns, not just temporal evolution.
B) heeding the advice of Maraun & Kurths (2005) about “periods of coupling which are invisible to linear methods.”
Future posts in this series (if it continues) may draw attention to:
a) nonrandom relations between interannual terrestrial oscillations and interannual [not to be confused with decadal] rates of change of solar variables.
b) the guaranteed potential for naive investigators to be irrecoverably derailed by Simpson’s Paradox due to stubborn &/or blind adherence to seriously misguided conventional mainstream statistical inference paradigms & malpractices that rigidly & dogmatically insist on falsely assuming independence when none exists.
c) the [counterintuitive &/or paradoxical for some] influence of grain & extent – & aggregation criteria more generally – on summaries of spatiotemporal pattern.
“Grain” & “extent“?…
Grain is another term for spatiotemporal resolution. Important: Extent is a term which concisely encompasses the properties of spatiotemporal summary windows. The vast majority of mainstream researchers are either absolutely ignorant or insufficiently cognizant of the effect of extent on integrals across spatiotemporal harmonics (including the nonstationary variety). The consequences are serious: blindness and rejection of valid findings on nonsensical grounds.
Best Regards to All.
Paul,
Frankly, I do not know if this is relevant, but after staring at hundreds of land (and some ocean and some satellite where applicable) temperature graphs, many from around Australia, there are 4 persistent peaks that show hot years about 28 years apart. These are in years 1915, 1943, 1970 and 1998, most +/- 1 year. There are cold years 14 years +/-1 after these, in 1929, 1956, 1985, ?2012. At a given site, not all of these need be present, but commonly at least 5 are. I have the impression of an alternating global factor related to the Sun’s energy input, moderated at any particular site by local events such as cloudiness for a part of the year. Am I reading too much into noisy data?
Leif Svalgaard says, “Nonsense, the paradox comes from combining unequal group sizes, and no real scientists [but many economists] do such a silly thing.”
You clearly have not thought carefully about the ways in which Simpson’s Paradox can arise. While it’s good that fred berple is thinking about these things, the page to which he linked isn’t dealing with a continuous variable with a spatial dimension, nor does it address another form of the paradox which arises when assumptions of independence are systematically flawed. Your comments also suggest that you pay little attention to the regional leverage of spatial amplitude variability (e.g. heat capacity weighting) on global means and that you do not understand what is meant by grain & extent (nor does fred berple – no offense intended fred – just alerting readers to be careful when reading exchanges above that misrepresent the meaning of grain).
Regarding P. Solar’s comments about smoothing & Leif Svalgaard’s comments about degrees of freedom:
Have either of you noticed that LOD’ is clearly not a random time series? Suggestion: Do a color-contoured time-integrated autocorrelation plot for some insight. Clearer awareness of the effect of integration across harmonics is needed (rather than reliance on rote memorization of procedures advocated by some for dealing with less well-structured time series).
Re: izen & P. Solar
I have provided a link to the data. The units are indicated. Take the first difference.
Quantities like correlation are not affected by normalization. Data visualization objectives used here (simply using the whole graph instead of leaving most of it blank & white) may be at odds with your analysis priorities. Please share whatever analysis you have to contribute.
P. Solar charged, “vitriolic criticism”
On the contrary, I have volunteered an objective assessment.
Paul Vaughan says:
April 12, 2011 at 6:29 am
You clearly have not thought carefully about the ways in which Simpson’s Paradox can arise.
Of course, I have. It is not that complicated. Any variation between the groups not compensated for by careful weighting will get you into trouble. A classical example actually occurred at a company where I once worked. The CEO had gotten one of those new-fangled PCs with Lotus 1-2-3 [early spreadsheet program] and used that to report on the profitability of the company. We had three divisions. for the Quarter in question the fiscal results were [profit in col.3]:
Div1: 10M -0.5M = -5%
Div2: 10M -1.0M = -10%
Div3: 1M +0.5M = +50%
Total: 12% profit, when it actually was a 1.6% loss.
Hint: calculate the average percentage profits as (-5-10+50)/3
you do not understand what is meant by grain & extent
Of course I do, how about: grain and extent are the upper and lower limits of data resolution. You can’t find patterns finer than the grain or coarser than the extent. But it is not relevant to the time series comparisons made in the Le Mouel paper. And, BTW, it is not helpful to stoop to the spatiotemporal cult’s jargon to explain something that simple.
P. Solar, what is at issue is not what some abstractly-idealized time-filter does, but rather what the Earth does.
Re: Agile Aspect
I have not performed any inverse transforms. I have provided a link to the data. If you run wavelet analyses, be sure to vary the extent (rather than just picking one extent as per conventional mainstream misguidance).
Re: Richard Holle
“…Workin’ on a mystery,
Goin’ wherever it leads…” – Tom Petty
Leif
Fred is trying to convince you that TSI is a measure of the rate of change of T. Unfortunately, he may have succeeded.
Fortunately he knows what heat capacity means.
lgl says:
April 12, 2011 at 7:29 am
Fortunately he knows what heat capacity means.
Show us what the evidence for that is.
don penman says:
April 12, 2011 at 12:16 am
wikipedia.org/wiki/Day
Hope that link works it say that the length of day noon to noon over the year is about + or- 7.9 seconds.
You and I may be off topic but the LOD you refer to is solar, not sidereal. It has to do with how long the earth faces the sun, and depends on which part of the orbit the earth is in. A Foucalt Pendulum, if it were accurate enough, would measure a sidereal day, not a solar day, since its movement is governed by an inertial frame of reference. The LOD used by JPL and the IERS is sidereal, not influenced by the annual revolution, and is measured with great accuracey, to within hundredths of milliseconds. It varies from fortnight to fortnight by 2ms, and from winter to summer by 2ms. The seven seconds you speak of is another matter entirely, referring to the length of visible day and night.
True, sidereal LOD is converted to a mean solar day equal to 86,400 seconds, but it remains an inertial frame of reference, and when astronomically measured (by star transit, or whatever) is initially sidereal.
Leif
Are you kidding? ΔT=q/C , where q is heat in joule an C is heat capacity. TSI is in J/s so to know what ΔT is you need to know for how long TSI is applied.
“TSI is a measure of the rate of change of T”, and not a measure of T.
lgl says:
April 12, 2011 at 9:10 am
Are you kidding? ΔT=q/C , where q is heat in joule an C is heat capacity.
“TSI is a measure of the rate of change of T”, and not a measure of T.
No, as T=100K is TSI=5.67 W/m2, T=1000K is TSI=56,700 W/m2. What comes in must go out. At the Sun’s temperature TSI is 62,970,000 W/m2. Since we are 215 solar radii away, at Earth that reduces to 62,970,000/215^2 = 1362 W/m2. Reduce that by a factor of four, subtract what is reflected, add the greenhouse effect from having an atmosphere you calculate T to be 289K. TSI is very much an measure of T.
TSI is in J/s so to know what ΔT is you need to know for how long TSI is applied.
TSI is applied all the time and has been for billions of years. That is ~200 J for some 4 billion years, for a ‘q’ of 100,000,000,000,000,000 J. Quite a ΔT you’ll get from that…
What struck me is that mainstream science is likely to reject the 11 solar cycle affecting the LOD.
What is not accounted for in climate and solar science is that the solar cycle is a cycle. It can create resonance and thus amplify its effects, well in excess of the calculated forcing.
Thus, one must be extremely careful in assuming a cyclical forcing is too small to cause the observed effect. A cyclical forcing can create effects significantly larger than might be naively assumed.
From Wikipedia:
“In physics, resonance is the tendency of a system to oscillate with larger amplitude at some frequencies than at others. These are known as the system’s resonant frequencies. At these frequencies, even small periodic driving forces can produce large amplitude oscillations, because the system stores vibrational energy.”
“Resonance occurs widely in nature”
http://en.wikipedia.org/wiki/Resonance
ferd berple says:
April 12, 2011 at 9:47 am
What is not accounted for in climate and solar science is that the solar cycle is a cycle. It can create resonance and thus amplify its effects, well in excess of the calculated forcing.
what you are missing [and I have pointed that to you before, but you are a slow learner, apparently] is that for resonance to occur, the forcing must occur at a natural frequency of the system, so the climate must already have an 11-yr cycle [due to other things] for the solar cycle to be creating resonance.
I always have a hard time navigating the IERS site, but here’s an easy reference for the tinkering novice–the latest LOD (up to 3900 days) with or without tidal variations (best to remove them for annual variations–not for stat analysis):
http://hpiers.obspm.fr/eop-pc/index.php?index=realtime&lang=en
Leif
What comes in must go out
No, that’s what heat capacity is about. Some of what comes in can remain in the ocean for centuries. On a very long time scale on avarage what comes in must go out, but not on a decadal scale.
@-Paul Vaughan says:
April 12, 2011 at 6:52 am
“I have provided a link to the data. The units are indicated. Take the first difference.”
Is there some reason you declined to simply state – ‘the graphs represent changes of a few milliseconds’ instead of this indirect hinting ?
So the periodic variation in the sidereal day length (not solar) is several hundred times LESS than the time a single frame appears in a movie.
Given the infinitesimally small energy variation this represents it must be swamped by several orders of magnitude by… well just about every other variable that affects the amount of energy the Earth receives. There is no possible physical process I can envisage that could causally connect such minute fluctuations with GCR flux or SOI changes.
Any correlation must be just that – a spurious correlation.
The quick and dirty way to check for that is to move one of the curves an arbitrary number of peaks laterally in either direction and see if the correlation changes significantly…. just tried it with the LOD/SOI graph, the match looks almost as good if you move the red curve forward three years and somewhat better if you move it back four…!
lgl says:
April 12, 2011 at 10:59 am
No, that’s what heat capacity is about. Some of what comes in can remain in the ocean for centuries. On a very long time scale on avarage what comes in must go out, but not on a decadal scale.
Explain that to Paul and Le Mouel who claim
1: no lags
2) direct correlation with cosmic rays and LOD’ [assumed to be climate related – although Paul never commits to anything]. http://www.leif.org/research/Courtillot-GRL-Cosmic-Rays.png or the discussion here: http://wattsupwiththat.com/2011/03/17/tisdale-update-on-ocean-heat-content/
Some, a very small part, will stay deep for centuries, but most clearly will not.
izen says:
April 12, 2011 at 11:05 am
Any correlation must be just that – a spurious correlation.
that is what you get when the data is tortured by people disclaiming any physics.
Leif
No need to complicate this. ΔT=q/C says T is determined by J. TSI is J/s and has to be integrated to become J, very simple.
The GCR-LOD link is not a A causing B, it’s a case of C causing both A and B. The planets accelerate both the Sun and the Earth, changing the rotation of both. Probably the same with sun-climate link.
lgl says:
April 12, 2011 at 11:54 am
No need to complicate this. ΔT=q/C says T is determined by J. TSI is J/s and has to be integrated to become J, very simple.
Too simple. As it is assumed that there is no heat losses. The oceans radiate to space all the time. Study Tisdale’s discussion to learn more: http://wattsupwiththat.com/2011/03/17/tisdale-update-on-ocean-heat-content/
lgl says:
April 12, 2011 at 11:54 am
No need to complicate this. ΔT=q/C says T is determined by J. TSI is J/s and has to be integrated to become J, very simple.
so, integrated over a solar cycle, the temperature increases by 164,000C. This makes sense to you? If so, you should join Al Gore and his 2 million degree temperature of the Earth’s interior.