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.
Leif
And over 4 billion years it is even worse than we thought: 6.3*10^30 J
No, the 6.3*10^23 J will be lost the next 400 years when the forcing is -0.1 W/m2, but I agree 400 years is probably long enough for most of the ocean to reach equilibrium so my number is to high.
Schwartz has a nice analysis of the energy balance
Good, then you realize TSI is not a measure of T “C*dT/dt=dH/dt=Q-E”
and “Time constant τ varies linearly with heat capacity”
The longer cycles will mix more of the ocean, thus larger heat capacity and larger time constant. This paper confirms my main points.
lgl says:
April 14, 2011 at 11:15 am
Good, then you realize TSI is not a measure of T “C*dT/dt=dH/dt=Q-E”
and “Time constant τ varies linearly with heat capacity”
The longer cycles will mix more of the ocean, thus larger heat capacity and larger time constant. This paper confirms my main points.
I don’t think so. The forcing is dTSI. dT=q/C means that your q is really a dq as I don’t think the total energy input from the Sun is only 0.1 W/m2, more like 1360 W/m2. So we have dT ~ dq ~ dTSI, which is what I have been saying all along [dT/T= 1/4 dTSI/TSI, remember]. You mistake is to forget that the q is actually a dq.
In any event the time constant seems to be ~10 years, not 1/4 of [2*400] years. Do me a favor and repeat your plot using all 2000 years.
ferd berple says:
April 12, 2011 at 1:32 pm
ENSO for example averages about 2x the frequency of the solar cycle. The longer ocean cycles also appear to be close integer multiples of the solar cycle, similar to the resonance we see in orbital cycles.
—————————————————-;
Which is the Hale cycle or the rotation of the center of mass of Sun.
A G Foster says:
April 12, 2011 at 10:55 am
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)
—————————————————————————————————-;
Great – I’ll take a look. Thanks for posting the URL.
Make that 4900 days. Have fun.
Backtracking here, I believe the effect of precession on core mantle coupling would be many orders of magnitude greater than the miniscule variation in LOD. Like I said, I’m in over my head.
Leif
The example of 0.1 W/m2 was net forcing, not TSI.
So we have dT ~ dq
In the real world we have C*dT ~ dq
In any event the time constant seems to be ~10 years
The Schwartz paper says: “The time constant of Earth’s climate system is 5 ± 1 years
OR 16 ± 3 years” My guess is when they do not detrend they get the contribution from the longer cycle and ends up with 16 years. But back to 1880 still isn’t long enough to get the right picture. Secondly, time constant is not the same as time lag. I agree we will not see a 200 years lag on a 800 years cycle. The heat capacity is big but not that big.
There is no point plotting 1000 years of inaccurate data. The troughs are often caused by volcanoes and the timing of those pre-1000 is very uncertain so impossible to make the right adjustment.
lgl says:
April 14, 2011 at 2:26 pm
So we have dT ~ dq
In the real world we have C*dT ~ dq
But C does not vary much so we are back to dT ~ dq, and thus dT ~ dTSI.
There is no point plotting 1000 years of inaccurate data. The troughs are often caused by volcanoes and the timing of those pre-1000 is very uncertain so impossible to make the right adjustment.
The trough in 1810 was very likely caused by volcanoes [perhaps also around 1700]. The timing is pretty good. 14C and 10Be agree pretty well [and they have different timing issues]. See slide 20 of http://www.leif.org/research/Does%20The%20Sun%20Vary%20Enough.pdf
Of course, if your definition of bad data is that data that don’t fit are bad, then you may have a point, otherwise not. Please don’t chicken out, but make that plot [you can already see the result on slide 20] so you can see for yourself.
“””””” A G Foster says:
April 14, 2011 at 2:16 pm
Backtracking here, I believe the effect of precession on core mantle coupling would be many orders of magnitude greater than the miniscule variation in LOD. Like I said, I’m in over my head. “””””
Well I had one of those Tiger Woods moments; and ended up busting my number 3 sand stick over my knees, after thinking about your Antarctic melt water situation.
I guess I should have seen that having the ice slip off the beach in Antarctica, and melt at sea level was peanuts compared to moving most of it up past Australia, to where it can really rotate.
So I’m going to just sit over here in the corner, with the dunce hat on, and spend a while whittling me a new sand stick; well hell that other one had got a little blunt anyway.
Sometimes it helps to draw a little 1000 word picture, and then you can see what matters, and what doesn’t.
But note that I did say that it was an elastic bounce that promptly followed the exit of the ice water; and I realize the plastic readjustment would take a few days; maybe more.
And Phillip Mulholland’s dissertation was very instructive too. Well I come here to learn.
George E. Smith April 14, 2011 at 2:50 pm
George,
Your observation on the significance of the instantaneous elastic response had never occurred to me, so answering your question forced me to learn too.
Win win I say.
Leif Svalgaard wrote, “by simple examples you feed the students up front.”
Exactly. (Just need more time.)
Paul Vaughan says:
April 14, 2011 at 8:03 pm
Leif Svalgaard wrote, “by simple examples you feed the students up front.”
Exactly. (Just need more time.)
Time is something you ‘make’ according to your priorities. Often people spend more time explaining that they don’t have enough time than it takes to do what they say they don’t have time for…
Leif
But C does not vary much so we are back to dT ~ dq, and thus dT ~ dTSI
No we are not, we are back to C*dT/dt=dH/dt=Q-E and it does not say d(Q-E) so you are wrong. It says a constant positive forcing will increase the heat content and temperature. A short cycle will mostly mix the top 300 m where C is 6.5 (Schwartz), a long cycle will mix all the ocean with a C of 14 (or 17). In the first case temp will rise faster (short lag) than in the latter (long lag). This is getting boring. Why don’t we switch to F=ma and discuss how many ds and ~s are hidden in there.
I ment the timing (guess dating is the right word) of volcanoes, not solar. Look at the link I gave you. And newer temperature data is better that older data. That’s just a fact and not my ‘definition’.
lgl says:
April 15, 2011 at 3:05 am
“But C does not vary much so we are back to dT ~ dq, and thus dT ~ dTSI”
No we are not, we are back to C*dT/dt=dH/dt=Q-E and it does not say d(Q-E)
Q is the energy supplied over a certain time, hence is actually dQ.
I meant the timing (guess dating is the right word) of volcanoes, not solar. Look at the link I gave you. And newer temperature data is better that older data.
Timing of volcanoes is actually much better than timing of solar changes. Both written records and physical evidence show that, e.g. Vesuvius :
“Mount Vesuvius has erupted many times. The famous eruption in 79 AD was preceded by numerous others in prehistory, including at least three significantly larger ones, the best known being the Avellino eruption around 1800 BC which engulfed several Bronze Age settlements. Since 79 AD, the volcano has also erupted repeatedly, in 172, 203, 222, possibly 303, 379, 472, 512, 536, 685, 787, around 860, around 900, 968, 991, 999, 1006, 1037, 1049, around 1073, 1139, 1150, and there may have been eruptions in 1270, 1347, and 1500.[14] The volcano erupted again in 1631, six times in the 18th century, eight times in the 19th century (notably in 1872), and in 1906, 1929, and 1944. There has been no eruption since 1944, and none of the post-79 eruptions were as large or destructive.”
“…but newcomers taking a preliminary look at daily resolution LOD’ are more likely to…”
See it as curious, and swiftly move on to something meaningful, as LOD variations do not force or drive anything.
Tim Channon wrote, “[…] I assume no-one is interested in an elephant.”
–
“Even on a cloudy day,
I keep my eyes fixed on the sun.” – Cage the Elephant.
Leif
Q is J/s. It isn’t actually something else just because you like that better. It equals dH/dt. What’s that actually then? The second derivative of heat content?
Look at the accuracy here http://www.volcano.si.edu/world/largeeruptions.cfm the large eruptions around year 1000 for instance. +/-75, +/-100 and so on.
I think that the observed length of day is more real than the lod measured by radio telescopes using distant quasars,the fact that the Earths orbit is elliptical and that the Earth is inclined to the rotation gives us real changes here on Earth(just because something is more accurate does not mean more relevant).Why the precise rotation of the Earth varies I could speculate though not in the complicated statistical way used here.I think that statistical speculation is replacing real science in climate science ,because it is impossible to to experimentally test any theory apparently, then everything is judged on how clever your computer models are.
lgl says:
April 15, 2011 at 8:29 am
Q is J/s.
So is TSI, but in both cases, the dT is for that expected for a further increase [or decrease] from the level T is already at, due to Q or TSI in the past, so the relevant Q or TSI is the delta over those.
Look at the accuracy here http://www.volcano.si.edu/world/largeeruptions.cfm the large eruptions around year 1000 for instance. +/-75, +/-100 and so on.
Some of the biggest have pretty good dating [e.g. Vesuvius, and the Japanese ones] , and the temperature depends not only on volcanoes. Loehle’s reconstruction [see his discussion] has better timing than 100 years, and in any case should still show the larger Grand Minina/Maxima signals if they exist. You reluctance to do the analysis is not good. You have spent more time trying to avoid it than it would take to do it.
Leif
“So is TSI” exactly, and J/s will tell how fast temperature is increasing. What you are saying is you can find the temperature of the water in a kettle just by looking at the wattage input. You are right only after equilibrium is reached and in the case of the ocean that is centuries, for a small imbalance. I don’t have more time for this nonsense, all this plotting you know 🙂
Leif Svalgaard says:
April 14, 2011 at 8:13 am
Schwartz has a nice analysis of the energy balance:
http://folk.uio.no/clausn/APPC/Stephen_Schwartz.pdf
—————————————————;
Now that my taxes are out of the way, I had little time to look at the above PDF but it appears this thread has died.
The author starts with a physical model where the heat source is on the wrong side of the water/air interface – which implies the internal heat of the earth is driving the Earth’s climate to the extent of causing the oceans to boil at constant temperature.
The model is useless when looking at the Earth’s climate.
Then on page 16, after the author has progressed from making a cup of tea to the Earth’s climate, states the following equations
C*dT_s/dt=Q-E
C*dT_s/dt=gamma*J-espilson*sigma*T^4_s
where
Q is absorbed solar energy (Joules)
E is the emitted longwave flux (Watts/m^2)
However, since the difference term in the first equation is dimensionally incorrect, it’s nonsense.
Re-writing the second equation in an attempt to try and make some sense of it as
C*dT_average=gamma*J-espilson*sigma*T^4_average
since the Stefan-Boltzmann (the integral of the Planck radiation law) predicts an average temperature for a black body (one doesn’t get to choose the temperature.)
Note, it’s not possible to apply the Stefan-Boltzmann to any portion of the atmosphere since gas molecules don’t emit approximately continuous radiation, i.e., they are *not* black body radiators. Their radiation is discrete and extremely narrow band. And since atmospheric gases play a vital role in the Earth’s climate system, ignoring them be confusing masterbation with sex.
But I digress.
The average temperature calculated using the Stefan-Boltzmann equation for the Earth-atomsphere system is roughly -10C at 5km. Assuming J was measured at the top of the troposphere, in order to compare the Stefan-Boltzmann prediction to experimental measurements – which again is the only interpretation of the author’s equations which make any sense – one needs to add a thermodynamic piece to the above equation based on the equations of state (since the top of the troposphere is roughly 10 km.)
Also, note, one can equate the differences in power to anything they like but if C is set to C=1 in the above equation, then the temperature scale needs to be re-scaled to the “Swartz temperature scale”.
Agile Aspect says:
April 19, 2011 at 2:14 pm
C*dT_s/dt=Q-E
C*dT_s/dt=gamma*J-espilson*sigma*T^4_s
where
Q is absorbed solar energy (Joules)
E is the emitted longwave flux (Watts/m^2)
However, since the difference term in the first equation is dimensionally incorrect, it’s nonsense.
Nonsense is spouted all the time on this blog. E.g.
lgl says:
April 15, 2011 at 8:29 am
“Q is J/s.”
Note, it’s not possible to apply the Stefan-Boltzmann to any portion of the atmosphere since gas molecules don’t emit approximately continuous radiation, i.e., they are *not* black body radiators.
A single molecule is not, but an aggregate can be. E.g. the Sun is a gas and does emit ‘approximately continuous radiation’. The condition of the medium has to be specified correctly and precisely to use the various ‘laws’. And it is not the atmosphere that radiates into space, but the opaque surface.
Agile Aspect
However, since the difference term in the first equation is dimensionally incorrect, it’s nonsense.
It’s only dimensionally incorrect after you made it incorrect. Q=gamma*J, where gamma is planetary albedo and J is 1/4 TSI, which is W/m2. Also Q=dH/dt, which of course is in J/s, also known as W. And then you have to choose calculation /m2 or the globe in total.
Leif
And it is not the atmosphere that radiates into space, but the opaque surface
It’s both.