Guest Post by Willis Eschenbach
A couple of apparently related theories have been making the rounds lately. One is by Nikolov and Zeller (N&Z), expounded here and replied to here on WUWT. The other is by Hans Jelbring, discussed at Tallblokes Talkshop. As I understand their theories, they say that the combination of gravity plus an atmosphere without greenhouse gases (GHGs) is capable of doing what the greenhouse effect does—raise the earth at least 30°C above what we might call the “theoretical Stefan-Boltzmann (S-B) temperature.”
So what is the S-B temperature, theoretical or otherwise?
A curious fact is that almost everything around us is continually radiating energy in the infrared frequencies. You, me, the trees, the ocean, clouds, ice, all the common stuff gives off infrared radiation. That’s how night-vision goggles work, they let you see in the infrared. Here’s another oddity. Ice, despite being brilliant white because it reflects slmost all visible light, absorbs infrared very well (absorptivity > 0.90). It turns out that most things absorb (and thus emit) infrared quite well, including the ocean, and plants (see Note 3 below). Because of this, the planet is often treated as a “blackbody” for IR, a perfect absorber and a perfect emitter of infrared radiation. The error introduced in that way is small for first-cut calculations.
The Stefan-Boltzmann equation specifies how much radiation is emitted at a given temperature. It states that the radiation increases much faster than the temperature. It turns out that radiation is proportional to absolute temperature to the fourth power. The equation, for those math inclined, is
Radiation = Emissivity times SBconstant times Temperature^4
where the Stefan-Boltzmann constant is a tiny number, 0.0000000567 (5.67E-8). For a blackbody, emissivity = 1.
This “fourth-power” dependence means that if you double the absolute temperature (measured in kelvins), you get sixteen (2^4) times the radiation (measured in watts per square metre, “W/m2”). We can also look at it the other way, that temperature varies as the fourth root of radiation. That means if we double the radiation, the temperature only goes up by about 20% (2^0.25)
Let me call the “theoretical S-B temperature” the temperature that an evenly heated stationary blackbody planet in outer space would have for a given level of incoming radiation in W/m2. It is “theoretical”, because a real, revolving airless planet getting heated by a sun with the same average radiation will be cooler than that theoretical S-B temperature. We might imagine that there are thousands of mini-suns in a sphere around the planet, so the surface heating is perfectly even.
Figure 1. Planet lit by multiple suns. Image Source.
On average day and night over the planetary surface, the Earth receives about 240 W/m2 of energy from the sun. The theoretical S-B temperature for this amount of radiation (if it were evenly distributed) is about -18°C, well below freezing. But instead of being frozen, the planet is at about +14°C or so. That’s about thirty degrees above the theoretical S-B temperature. So why isn’t the planet a block of ice?
Let me take a short detour on the way to answering that question in order to introduce the concept of the “elevator speech” to those unfamiliar with the idea.
The “elevator speech” is simply a distillation of an idea down to its very basics. It is how I would explain my idea to you if I only had the length of an elevator ride to explain it. As such it has two extremely important functions:
1. It forces me to clarify my own ideas on whatever I’m discussing. I can’t get into handwaving and hyperbole, I can’t be unclear about what I’m claiming, if I only have a few sentences to work with.
2. It allows me to clearly communicate those ideas to others.
In recent discussions on the subject, I have been asking for that kind of “elevator speech” distillation of Jelbring’s or Nikolov’s ideas, so that a) I can see if whoever is explaining the theory really understands what they are saying and, if so, then b) so that I can gain an understanding of the ideas of Jelbring or Nikolov to see if I am missing something important.
Let me give you an example to show what I mean. Here’s an elevator speech about the greenhouse effect:
The poorly-named “greenhouse effect” works as follows:
• The surface of the earth emits energy in the form of thermal longwave radiation.
• Some of that energy is absorbed by greenhouse gases (GHGs) in the atmosphere.
• In turn, some of that absorbed energy is radiated by the atmosphere back to the surface.
• As a result of absorbing that energy from the atmosphere, the surface is warmer than it would be in the absence of the GHGs.
OK, that’s my elevator speech about why the Earth is not a block of ice. Note that it is not just saying what is happening. It is saying how it is happening as well.
I have asked, over and over, on various threads, for people who understand either the N&Z theory or the Jelbring theory, to give me the equivalent elevator speech regarding either or both of those theories. I have gotten nothing scientific so far. Oh, there’s the usual handwaving, vague claims of things like ‘the extra heat at the surface, is just borrowed by the work due to gravity, from the higher up regions of the atmosphere‘ with no mechanism for the “borrowing”, that kind of empty statement. But nothing with any meat, nothing with any substance, nothing with any explanatory value or scientific content.
So to begin with, let me renew my call for the elevator speech on either theory. Both of them make my head hurt, I can’t really follow their vague descriptions. So … is anyone who understands either theory willing to step forward and explain it in four or five sentences?
But that’s not really why I’m writing this. I’m writing this because of the claims of the promoters of the two theories. They say that somehow a combination of gravity and a transparent, GHG-free atmosphere can conspire to push the temperature of a planet well above the theoretical S-B temperature, to a condition similar to that of the Earth.
I hold that with a transparent GHG-free atmosphere, neither the hypothetical “N&Z effect” nor the “Jelbring effect” can possibly raise the planetary temperature above the theoretical S-B temperature. But I also make a much more general claim. I hold it can be proven that there is no possible mechanism involving gravity and the atmosphere that can raise the temperature of a planet with a transparent GHG-free atmosphere above the theoretical S-B temperature.
The proof is by contradiction. This is a proof where you assume that the theorem is right, and then show that if it is right it leads to an impossible situation, so it cannot possibly be right.
So let us assume that we have the airless perfectly evenly heated blackbody planet that I spoke of above, evenly surrounded by a sphere of mini-suns. The temperature of this theoretical planet is, of course, the theoretical S-B temperature.
Now suppose we add an atmosphere to the planet, a transparent GHG-free atmosphere. If the theories of N&K and Jelbring are correct, the temperature of the planet will rise.
But when the temperature of a perfect blackbody planet rises … the surface radiation of that planet must rise as well.
And because the atmosphere is transparent, this means that the planet is radiating to space more energy than it receives. This is an obvious violation of conservation of energy, so any theories proposing such a warming must be incorrect.
Q.E.D.
Now, I’m happy for folks to comment on this proof, or to give us their elevator speech about the Jelbring or the N&Z hypothesis. I’m not happy to be abused for my supposed stupidity, nor attacked for my views, nor pilloried for claimed errors of commission and omission. People are already way too passionate about this stuff. Roger Tattersall, the author of the blog “Tallbloke’s Talkshop”, has banned Joel Shore for saying that the N&Z hypothesis violates conservation of energy. Roger’s exact words to Joel were:
… you’re not posting here unless and until you apologise to Nikolov and Zeller for spreading misinformation about conservation of energy in their theory all over the blogosphere and failing to correct it.
Now, I have done the very same thing that Joel did. I’ve said around the web that the N&Z theory violates conservation of energy. So I went to the Talkshop and asked, even implored, Roger not to do such a foolish and anti-scientific thing as banning someone for their scientific views. Since I hold the same views and I committed the same thought-crimes, it was more than theoretical to me. Roger has remained obdurate, however, so I am no longer able to post there in good conscience. Roger Tallbloke has been a gentleman throughout, as is his style, and I hated to leave. But I did what Joel did, I too said N&Z violated conservation of energy, so in solidarity and fairness I’m not posting at the Talkshop anymore.
And more to the point, even if I hadn’t done what Joel did, my practice is to never post at or even visit sites like RealClimate, Tamino’s, and now Tallbloke’s Talkshop, places that ban and censor scientific views. I don’t want to be responsible for their page views counter to go up by even one. Banning and censorship are anathema to me, and I protest them in the only way I can. I leave them behind to discuss their ideas in their now cleansed, peaceful, sanitized, and intellectually sterile echo chamber, free from those pesky contrary views … and I invite others to vote with their feet as well.
But I digress, my point is that passions are running high on this topic, so let’s see if we can keep the discussion at least relatively chill …
TO CONCLUDE: I’m interested in people who can either show that my proof is wrong, or who will give us your elevator speech about the science underlying either N&K or Jelbring’s theory. No new theories need apply, we have enough for this post. And no long complicated explanations, please. I have boiled the greenhouse effect down to four sentences. See if you can match that regarding the N&K or the Jelbring effect.
w.
NOTE 1: Here’s the thing about a planet with a transparent atmosphere. There is only one object that can radiate to space, the surface. As a result, it is constrained to emit the exact amount of radiation it absorbs. So there are no gravity/atmospheric phenomena that can change that. It cannot emit more or less than what it absorbs while staying at the same temperature, conservation of energy ensures that. This means that while the temperature can be lower than the theoretical S-B temperature, as is the case with the moon, it cannot be more than the theoretical S-B temperature. To do that it would have to radiate more than it is receiving, and that breaks the conservation of energy.
Once you have GHGs in the atmosphere, of course, some of the surface radiation can get absorbed in the atmosphere. In that case, the surface radiation is no longer constrained, and the surface is free to take up a higher temperature while the system as a whole emits the same amount of radiation to space that it absorbs.
NOTE 2: An atmosphere, even a GHG-free atmosphere, can reduce the cooling due to uneven insolation. The hottest possible average temperature for a given average level of radiation (W/m2) occurs when the heating is uniform in both time and space. If the total surface radiation remains the same (as it must with a transparent atmosphere), any variations in temperature from that uniform state will lower the average temperature. Variations include day/night temperature differences, and equator/polar differences. Since any atmosphere can reduce the size of e.g. day/night temperature swings, even a transparent GHG-free atmosphere will reduce the amount of cooling caused by the temperature swings. See here for further discussion.
But what such an atmosphere cannot do is raise the temperature beyond the theoretical maximum average temperature for that given level of incoming radiation. That’s against the law … of conservation of energy.
NOTE 3: My bible for many things climatish, including the emissivity (which is equal to the absorptivity) of common substances, is Geiger’s The Climate Near The Ground, first published sometime around the fifties when people still measured things instead of modeling them. He gives the following figures for IR emissivity at 9 to 12 microns:
Water, 0.96 Fresh snow, 0.99 Dry sand, 0.95 Wet sand, 0.96 Forest, deciduous, 0.95 Forest, conifer, 0.97 Leaves Corn, Beans, 0.94
and so on down to things like:
Mouse fur, 0.94 Glass, 0.94
You can see why the error from considering the earth as a blackbody in the IR is quite small.
I must admit, though, that I do greatly enjoy the idea of some boffin at midnight in his laboratory measuring the emissivity of common substances when he hears the snap of the mousetrap he set earlier, and he thinks, hmmm …
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“I’m a lawyer and read this website so that I can speak to issues from a different perspective than that of most of my peers. I really need that elevator speech. So, could you all stop sniping and make an attempt to attain that goal. Thanking you in advance for your cooperation – as we say in the legal profession.”
Ok.
If you had a planet which surrounded by a million [any number from a couple dozen to infinity] suns and each sun radiates 240 watts per square meter. The temperature of that planet will warm until it’s -18 C.
When you take the average amount of sunlight, that reaches earth remove 30% due to reflection
and divide the energy by 4; you also get the number 240 watts per square meter.
Summary the distance from the sun determines maximum temperature that the sun can heat any surface.
Climatic “scientist” divide the power of sunlight by 4 because the earth intercepts the sunlight as a disk, and 4 times this area is the entire surface of the earth. They are quite aware that sunlight doesn’t shine on night side of earth, but wish to consider that sunlight does warm the entire surface at one time.
This lack any sense but reason has little to do with what people want to believe,
Willis, you are correct that argon cannot have bending or stretching modes of vibration since it is a monoatomic gas. It can only have electronic transitions; the energy of which depends on the orbitals that the electrons are jumping between. However, you are overestimating the energy associated with most transitions, especially those occurring between n > 3 and n = 3. This overestimation leads you to the incorrect conclusion that argon does not absorb/emit IR radiation.
Consider the Bohr model of the hydrogen atom. There are a number of emission line series that are attributed to the excitation/relaxation of the electron. Each series is named for the person who discovered it. The lines series are observed in the UV (Lyman), the visible (Balmer), the IR (Paschen, Brackett, Pfund, and Humphreys), etc. The actual wavelength of light observed is related to how far apart the two levels that the electron is jumping between are. The farther apart, the more energy required/emitted, and hence the shorter the wavelength. Thus, the Lyman series is attributed to transitions from n > 1 to n = 1. The Balmer series is attributed to transitions from n > 2 to n = 2. Longer wavelengths are associated with transitions between the higher energy levels since these levels are even closer together. Hence, the Paschen series, which is the first series occurring in the IR spectral region, is associated with transitions from n > 3 to n = 3.
Argon is a quite a bit more complex than hydrogen. However, the simplistic Bohr model still provides a good way to begin thinking about the possible electronic transitions that will be observed. The valence (outermost) electrons in a ground state Ar atom are located in the third shell (n = 3). These electrons can be excited to the higher unoccupied orbitals (n > 3), which leads to expectation that a Paschen type absorption/emission series in the IR is probable. Indeed, this series was first reported by Paschen in the early 1900s. I would recommend that you search through this thread. Anna V had provided a link to the absorption/emission spectra for argon, which includes the IR lines that you are interested in learning more about.
Jimmi says at 1/17 11:04am:
“The word “transparent” means “does not absorb”
Right Jimmi. The transparent GHG-free atmosphere does not absorb any thermal radiation emitted from Willis’ 2nd mechanism black body surface. That thermal radiation goes right thru the transparent GHG-free atmosphere and allows Willis’ planetary black body one of the at least two physical means to cool to space and achieve equilibrium non-infinite temp.s.
In my view, Willis’ 2nd mechanism GHG-free atmosphere absorbs thru thermal conduction at his planetary black body surface interface to his atmosphere. This allows Willis’ planetary black body the second physical means to cool to the atmosphere which is matter (O, N) and this matter eventually can emit the balance of thermal radiation to space – this is the missing part which enables Willis to write a proof by contradiction that is incomplete & as consequence then violates energy conservation.
Willis leaves out of his proof by contradiction thermal conduction at his 2nd mechanism surface, ideal gas physics, and gravity so the proof is incomplete & finds a violation of energy conservation which can be resolved to no violation once these physics are employed. Nature is rescued and there could be other rescue teams.
As a consequence of the above two physical means, the same total energy from the sun is radiated back to space as in Willis’ 1st mechanism (BB only), 2nd mechanism energy is conserved, 2nd law is not violated, KE+PE = constant. This should be easy to see except for the million details & the myriad other mechanisms many others are discussing past Willis’ 2. Blogs can be inefficient. But like Waiting for Godot Act ll: “In the meantime, let us try and converse calmly, since we are incapable of keeping silent.”
I do not see where N&Z have a non-physical component assuming transparency, only see Willis 2nd mechanism incomplete proof by contradiction so far. But the play is in progress.
Deborah says:
January 17, 2012 at 7:46 pm
” I really need that elevator speech. So, could you all stop sniping and make an attempt to attain that goal.”
Done. From my perspective, which is the right one.
jjthoms says:
January 17, 2012 at 6:16 pm
“It is cooled by LW radiation. when incoming radiation = out going radiation thermal equilibrium will result. This equality only occurs at one temperature for fixed incoming radiation and surface albedo.”
So, so utterly and completely wrong. There is never a vanishing thermal gradient in the atmosphere, so long as energy cannot escape it. The temperature is of the form B/r, where B is a constant, which is a decreasing function of radius. So, heat will flow continuously up into the atmosphere until it can be released, either by high energy radiation, or “boiling” away of the atmosphere.
“There is nowhere for the heat to go – non-GHGs do not radiate and there is nothing to conduct to in space.”
Exactly. Which is why it will continue to build.
“Eventually the non-GHG will reach the same temperature as the planet surface and heating by conduction will cease.”
Nope. The surface will always be hotter, because B/r is a decreasing function of the radius. The gradient always exists, ergo conduction never stops. So there is never an equilibrium, and SB does not hold.
“The non-ghg will get no hotter than the surface.”
The non-ghg atmosphere will get hotter and hotter until there is some radiation or, if sufficient radiation does not exist to prevent the molecules from attaining escape velocity, it boils away.
GHGs short circuit the dynamic. They provide a heat sink which takes the accumulating heat energy out of the system. Without GHGs, we would not get colder. We would get hotter.
“The temperature is of the form B/r, where B is a constant, which is a decreasing function of radius.”
Actually, I originally derived B under the assumption that the time rate of change of temperature had stopped. Since the persistence of a gradient says that hasn’t, then the heat equation PDE can be solved through separation of variables and B becomes an exponential function of time. So, the actual temperature profile is going to become
T = T(0,0)*exp(lambda*t)*(r_surf/r)
T(0,0) is the temperature of the surface at time zero, t is time, lambda is the exponential rate, r_surf is the radius of the planet, and r is the radius.
This form will persist until either the atmosphere starts radiating, or it boils away.
Sorry if neglecting that little detail threw anyone off. We EEs are used to dealing with phasors and Helmholtz equations and the time varying character of the quantities is simply understood. It just didn’t occur to me to say, “oh, and you have to multiply this spatial function by the time dependent amplitude function.”
So, the actual temperature profile is going to become…”
Good thing I actually started thinking about the amplitude function, because the solution is actually not that elementary anymore, since the Laplacian of the spatial part of the temperature function is now equal to the constant lambda times the spatial temperature function. The solution is going to be of the form
T = T(0,0)*exp(lambda*t)*F(r)
where F(r) is a modified Bessel function scaled to equal one at r = r_surf. Everything still holds mutatis mutandis, The temperature function is still monotonically decreasing with altitude. I’ll work out the details later, or anyone reading this can if they like.
“Without GHGs, we would not get colder. We would get hotter.”
Agreed as regards Willis’s model with a uniform flat surface and multiple suns for even irradiation.
However a rotating uneven sphere under a single sun would have vigorous convection due to temperature differences between the day and night sides.
That should stabilise the temperature at some point from rapid surface radiation to space on the night side. The cold ground would suck conductive energy out of the fast moving air above.
Bart says: January 17, 2012 at 11:21 pm
“The non-ghg will get no hotter than the surface.”
The non-ghg atmosphere will get hotter and hotter until there is some radiation or, if sufficient radiation does not exist to prevent the molecules from attaining escape velocity, it boils away.
=======
The non-GHG is in contact with the surface heat will conduct either way – planet hotter by conduction will heat the gas: gas hotter the planet will get hotter via conduction. The gas can get neither hotter or cooler other than by conduction to the planet surface
In your mind you obviously have something heating the gas to a greater temperatuere than the surface – care to explain what this is. Where is the additional energy coming from?
If what you say is correct then we can use a heat engine to extract power from the temperature difference = free energy – and I’m sure you can see that is wrong?
Willis, sorry if I’m beating a dead horse. The adiabatic lapse-rate formula is derived from an ideal gas process — compression/expansion. The reason the earth’s lapse-rate is near this is because it is constantly convecting up & down via Hadley cells.
Thought experiment — take your steel-shell greenhouse w/an atmosphere (non-GHG or not) underneath & between shell & surface w/the heat source at the shell instead of the surface, I don’t see how convection would occur in this case, and the temp profile of the air would be nearly isothermal underneath the shell regardless of gravity, specific heat, etc. I agree that it would be difficult to completely suppress convection in a surface-warmed real case — uneven surface heating would start at least some, and some lapse rate would develop.
IMO, the lapse-rate formula assumes convection is already occuring. Fire away.
Bart:”C) heat builds up at the interface between the surface and the atmosphere until its apparent temperature suggests an exceedance of the incoming radiance based on the SB relationship”
Just curious whether over the last couple of days you were able to come up with any evidence for this conjecture.
Cheers, 🙂
JJThoms says:
January 18, 2012 at 5:06 am
“In your mind you obviously have something heating the gas to a greater temperatuere than the surface – care to explain what this is.”
No. The surface is getting hotter, too. There is no steady state equilibrium anywhere in the system until the atmosphere is able to find an outlet for the energy building up in it. So, SB does not give you any information about surface temperature. Stefan-Boltzmann is a steady state theory. When you reach steady state, when the atmosphere has either found a heat sink in the form of radiative transfer or, barring that, boiling off into space, then, and only then will SB give you the lower bound on temperature.
“If what you say is correct then we can use a heat engine to extract power from the temperature difference = free energy – and I’m sure you can see that is wrong?”
You are hung up on this idea that there is an equilibrium somewhere. There isn’t. You can extract power from a non-equilibrated system.
shawnhet says:
January 18, 2012 at 8:33 am
“Just curious whether over the last couple of days you were able to come up with any evidence for this conjecture.”
Yes, quite a bit. A) As regards apparent versus real temperature, see my exchanges with DeWitt Payne. B) As regards the SB relationship, since there are no IR emitters in the atmosphere, temperatures are going to continue building continuously. See 1st reply to JJThoms above.
beng says:
January 18, 2012 at 7:26 am
“The reason the earth’s lapse-rate is near this is because it is constantly convecting up & down via Hadley cells.”
And, that is enabled by a heat sink above in the form of so-called greenhouse gasses, which can release energy in the IR, reversing the convection. Without that heat sink, there is no adiabatic lapse rate. Instead, the environmental lapse rate is determined by the temperature profile, which is a modified Bessel function qualitatively similar to a 1/r characteristic, where r is the radius from the Earth center.
Willis, if I may be so bold to ask, can you please tell me more about the amount of suns when you state “We might imagine that there are thousands of mini-suns in a sphere around the planet, so the surface heating is perfectly even.”
With the number of them fixed, perhaps I can work out how far away they are and how brightly they shine?
Would these mini-suns be much closer than one big sun? Could they have less mass? When the atmosphere is added to the planet it will gain mass and perhaps tug all the thousands of mini-suns closer? Would this be a case of gravity induced planetary heating with no laws broken?
NoIdea
PS I am not the NoIdea who had posted before on this thread.
Bart says:
January 18, 2012 at 9:38 am
““Just curious whether over the last couple of days you were able to come up with any evidence for this conjecture.”
Yes, quite a bit. A) As regards apparent versus real temperature, see my exchanges with DeWitt Payne. B) As regards the SB relationship, since there are no IR emitters in the atmosphere, temperatures are going to continue building continuously. See 1st reply to JJThoms above.”
Let’s see if I can be a bit more focussed. You seem to be hanging your hat on the following statement from the Niclos 2004 (where you apparently get your plots from).
“For larger angles, the effect of double or multiple reflections on the sea surface produces discrepancies between measured and theoretical SSEs, and more complex models should be used to get accurate SSE values”.
You are claiming, I believe, that the measured SSEs are in fact accurate and you do not, in fact, have to account for the effects of double or multiple reflections. I, personally, maintain that the experimental setup here is not analogous to that of the movement of the Earth through space, but even if I am wrong, you are left with the conundrum that you are *estimating* the drop off of emittance as the angle approaches 90 degrees (it is not measured). You, thusly, have at least two specific things that you need to demonstrate #1. that how you estimate emittance will drop off is actually how it does drop off and #2. that multiple reflection is not a problem for measuring emittance.
Cheers, 🙂
shawnhet says:
January 18, 2012 at 11:03 am
1) So, you’re saying we have no accurate measurements? Fine with me.
2) Events have passed this whole line of argument by, and I really don’t want to dwell on it anymore – the exact emissivity of the Earth’s surface is a tangential issue. The surface of the Earth is not hotter than it would be without IR radiating gasses (IRRGs, because GHG is an even worse misnomer than commonly though). Without IRRGs , the surface would be a lot hotter.
The IRRGs are what provide the heat sink to arrest the climb in temperatures. At steady state, it is no accident that they happen to intercept enough surface radiation to establish radiative balance – that is the essence of balance. But, the steady state conditions do not establish cause, any more than the fact that the FOV of the sky being clear establishes that there is a giant holding the Earth on his shoulders.
The proof is that the temperature gradient never goes away, due to the spherical distribution of the atmosphere. As a result, without a radiative heat sink in the atmosphere, there can be no steady state.
beng says:
You are right that when there is convection, the lapse rate is going to be close to the adiabatic lapse rate but you have the cause-and-effect partly backwards: Lapse rates less steep than the adiabatic lapse rate (such as in the stratosphere) are stable and hence convection is suppressed. Lapse rates steeper than the adiabatic lapse rate are unstable to convection and hence convection occurs, transporting heat upward until the lapse rate is equal to the adiabatic lapse rate.
I wish I could provide the links to the Wikipedia articles but, of course, today that is not possible. In an atmosphere with no convection (the thought experiment world), the PDE governing the temperature is
dT/dt = alpha*del^2(T)
where alpha is the conductivity parameter and “d” is actually a partial differential operator. To solve this equation, we set T = T1(t)*T2(r), where T1 is wholly a function of t, and T2 is wholly a function of r. This leads to
(dT1/dt)/T1 = alpha*del^2(T2)/T2
where the “d’s” are now total differential operators. Since the left side is wholly a function of time, and the right wholly a function of r, they must both equal a constant, call it lambda.
Then, the solution of dT1/dt = lambda*T1 is elementary, T1 = T1(0)*exp(lambda*t). The solution of
del^2(T2) = (lambda/alpha)*T2
is a modified and adjusted zeroth order Bessel function of the second kind, which is qualitatively similar to a constant divided by radius.
Thus, as I discussed here, the full solution is
T = T(0,0)*exp(lambda*t)*F(r)
There is always a gradient downhill in the radial direction. Thus, there is always heat flow into the higher altitudes. This heat flow will continue until it stops, either by high energy radiation if the atmosphere allows, or boiling away of the atmosphere.
Bart says:
January 18, 2012 at 11:42 am
“1) So, you’re saying we have no accurate measurements? Fine with me.”
No, I’m saying that your own reference puts specific limits on the accuracy of the measurements it makes. If you are going to ignore what your own reference says, you should provide some foundation for doing so. There are plenty of other references for the emissivity of water (see from the original SOD you referenced). Unfortunately, they all disagree with you on the emissivity of water.
“2) Events have passed this whole line of argument by, and I really don’t want to dwell on it anymore – the exact emissivity of the Earth’s surface is a tangential issue. The surface of the Earth is not hotter than it would be without IR radiating gasses (IRRGs, because GHG is an even worse misnomer than commonly though). Without IRRGs , the surface would be a lot hotter.”
Well, then you should revisit your elevator speech, I suppose. These issues follow directly from what you are talking about in your item C. I will wait and see what the new version looks like 😉
Cheers, 🙂
shawnhet says:
January 18, 2012 at 12:46 pm
You are harping on items which are inconsequential at this juncture. But, OK, here is Bart’s elevator speech specifically tailored for shawnhet:
A) Radiation enters the system
B) it is absorbed by the surface, which fluoresces in the IR
C) heat builds up at the interface between the surface and the atmosphere until its apparent temperature suggests an exceedance of the incoming radiance based on the SB relationship
D) But, SB does not hold because the system is not in equilibrium. The heat that would be radiated away in steady state is, instead, being conducted to the atmosphere. SB will not hold until the system has reached an equilibrium!!!
E) the heat from the interface conducts through the atmosphere, establishing a temperature gradient as must exist in a spherically distributed atmosphere
F) the transferred heat accumulates in the atmosphere until:
1) highly energetic emissions are stimulated, which balances the energy fluxes all around in the same way GHGs would in the standard “greenhouse” theory
OR
2) the heat accumulates until the atmosphere achieves escape velocity and vanishes.
G) if greenhouse gasses which can radiate in the IR are added to the atmosphere, it will result in a net cooler atmosphere than otherwise would be the case, because there is now a lower energy outlet for heat to escape.
I wonder if I stand a whelk’s chance in a supernova of wrestling this particular bone away from shawnhet.
My apologies, shawnhet. You are right that the ES was not clear on point D. I was frustrated that you were not picking up on it, but people who see the original ES only without the subsequent discussion might be confused. Thank you for your comments.
The alpha parameter is actually called “thermal diffusivity”, and it is inversely proportional to density. So, as the altitude increases, alpha will grow larger. So, even the Bessel function solution is not precise, and will only hold approximately in the lower atmosphere and before significant thermal expansion has taken place. A precise global solution would have to take all of these factors into account.
But, there is no chance of steady state because the steady state solution is T = B/r for a constant B, and since there is a gradient, there cannot be a steady state, and that creates a contradiction. So, the conclusion remains: there is always a thermal gradient pushing heat continuously into the atmosphere, and it will not stop until either there is some kind of radiative release, or the atmosphere flees.
Bart:”D) But, SB does not hold because the system is not in equilibrium. The heat that would be radiated away in steady state is, instead, being conducted to the atmosphere. SB will not hold until the system has reached an equilibrium!!!”
I guess where I am coming from is that regardless of whether the system is in equilibrium if the energy from a surface of temperature 15C is conducting away from that surface without radiating, then the emissivity will appear to be less than if it is radiating as per SB. As such, you will still need to demonstrate some apparent emissivity that is in line with your numbers. The surface will either emit the energy it receives as radiation or it will not, but either way it will have consequences on what we measure the emissivity to be. If your hypothesis is valid, you should thusly be able to point to some evidence that (apparent) emissivity tracks with your hypothesis.
Cheers, 🙂