Earth's baseline black-body model – "a damn hard problem"

The Earth only has an absorbing area equal to a two dimensional disk, rather than the surface of a sphere.

By Robert G. Brown, Duke University (elevated from a WUWT comment)

I spent what little of last night that I semi-slept in a learning-dream state chewing over Caballero’s book and radiative transfer, and came to two insights. First, the baseline black-body model (that leads to T_b = 255K) is physically terrible, as a baseline. It treats the planet in question as a nonrotating superconductor of heat with no heat capacity. The reason it is terrible is that it is absolutely incorrect to ascribe 33K as even an estimate for the “greenhouse warming” relative to this baseline, as it is a completely nonphysical baseline; the 33K relative to it is both meaningless and mixes both heating and cooling effects that have absolutely nothing to do with the greenhouse effect. More on that later.

I also understand the greenhouse effect itself much better. I may write this up in my own words, since I don’t like some of Caballero’s notation and think that the presentation can be simplified and made more illustrative. I’m also thinking of using it to make a “build-a-model” kit, sort of like the “build-a-bear” stores in the malls.

Start with a nonrotating superconducting sphere, zero albedo, unit emissivity, perfect blackbody radiation from each point on the sphere. What’s the mean temperature?

Now make the non-rotating sphere perfectly non-conducting, so that every part of the surface has to be in radiative balance. What’s the average temperature now? This is a better model for the moon than the former, surely, although still not good enough. Let’s improve it.

Now make the surface have some thermalized heat capacity — make it heat superconducting, but only in the vertical direction and presume a mass shell of some thickness that has some reasonable specific heat. This changes nothing from the previous result, until we make the sphere rotate. Oooo, yet another average (surface) temperature, this time the spherical average of a distribution that depends on latitude, with the highest temperatures dayside near the equator sometime after “noon” (lagged because now it takes time to raise the temperature of each block as the insolation exceeds blackbody loss, and time for it to cool as the blackbody loss exceeds radiation, and the surface is never at a constant temperature anywhere but at the poles (no axial tilt, of course). This is probably a very decent model for the moon, once one adds back in an albedo (effectively scaling down the fraction of the incoming power that has to be thermally balanced).

One can for each of these changes actually compute the exact parametric temperature distribution as a function of spherical angle and radius, and (by integrating) compute the change in e.g. the average temperature from the superconducting perfect black body assumption. Going from superconducting planet to local detailed balance but otherwise perfectly insulating planet (nonrotating) simply drops the nightside temperature for exactly 1/2 the sphere to your choice of 3K or (easier to idealize) 0K after a very long time. This is bounded from below, independent of solar irradiance or albedo (or for that matter, emissivity). The dayside temperature, on the other hand, has a polar distribution with a pole facing the sun, and varies nonlinearly with irradiance, albedo, and (if you choose to vary it) emissivity.

That pesky T^4 makes everything complicated! I hesitate to even try to assign the sign of the change in average temperature going from the first model to the second! Every time I think that I have a good heuristic argument for saying that it should be lower, a little voice tells me — T^4 — better do the damn integral because the temperature at the separator has to go smoothly to zero from the dayside and there’s a lot of low-irradiance (and hence low temperature) area out there where the sun is at five o’clock, even for zero albedo and unit emissivity! The only easy part is to obtain the spherical average we can just take the dayside average and divide by two…

I’m not even happy with the sign for the rotating sphere, as this depends on the interplay between the time required to heat the thermal ballast given the difference between insolation and outgoing radiation and the rate of rotation. Rotate at infinite speed and you are back at the superconducting sphere. Rotate at zero speed and you’re at the static nonconducting sphere. Rotate in between and — damn — now by varying only the magnitude of the thermal ballast (which determines the thermalization time) you can arrange for even a rapidly rotating sphere to behave like the static nonconducting sphere and a slowly rotating sphere to behave like a superconducting sphere (zero heat capacity and very large heat capacity, respectively). Worse, you’ve changed the geometry of the axial poles (presumed to lie untilted w.r.t. the ecliptic still). Where before the entire day-night terminator was smoothly approaching T = 0 from the day side, now this is true only at the poles! The integral of the polar area (for a given polar angle d\theta) is much smaller than the integral of the equatorial angle, and on top of that one now has a smeared out set of steady state temperatures that are all functions of azimuthal angle \phi and polar angle \theta, one that changes nonlinearly as you crank any of: Insolation, albedo, emissivity, \omega (angular velocity of rotation) and heat capacity of the surface.

And we haven’t even got an atmosphere yet. Or water. But at least up to this point, one can solve for the temperature distribution T(\theta,\phi,\alpha,S,\epsilon,c) exactly, I think.

Furthermore, one can actually model something like water pretty well in this way. In fact, if we imagine covering the planet not with air but with a layer of water with a blackbody on the bottom and a thin layer of perfectly transparent saran wrap on top to prevent pesky old evaporation, the water becomes a contribution to the thermal ballast. It takes a lot longer to raise or lower the temperature of a layer of water a meter deep (given an imbalance between incoming radiation) than it does to raise or lower the temperature of maybe the top centimeter or two of rock or dirt or sand. A lot longer.

Once one has a good feel for this, one could decorate the model with oceans and land bodies (but still prohibit lateral energy transfer and assume immediate vertical equilibration). One could let the water have the right albedo and freeze when it hits the right temperature. Then things get tough.

You have to add an atmosphere. Damn. You also have to let the ocean itself convect, and have density, and variable depth. And all of this on a rotating sphere where things (air masses) moving up deflect antispinward (relative to the surface), things moving down deflect spinward, things moving north deflect spinward (they’re going to fast) in the northern hemisphere, things moving south deflect antispinward, as a function of angle and speed and rotational velocity. Friggin’ coriolis force, deflects naval artillery and so on. And now we’re going to differentially heat the damn thing so that turbulence occurs everywhere on all available length scales, where we don’t even have some simple symmetry to the differential heating any more because we might as well have let a five year old throw paint at the sphere to mark out where the land masses are versus the oceans, and or better yet given him some Tonka trucks and let him play in the spherical sandbox until he had a nice irregular surface and then filled the surface with water until it was 70% submerged or something.

Ow, my aching head. And note well — we still haven’t turned on a Greenhouse Effect! And I now have nothing like a heuristic for radiant emission cooling even in the ideal case, because it is quite literally distilled, fractionated by temperature and height even without CO_2 per se present at all. Clouds. Air with a nontrivial short wavelength scattering cross-section. Energy transfer galore.

And then, before we mess with CO_2, we have to take quantum mechanics and the incident spectrum into account, and start to look at the hitherto ignored details of the ground, air, and water. The air needs a lapse rate, which will vary with humidity and albedo and ground temperature and… The molecules in the air recoil when the scatter incoming photons, and if a collision with another air molecule occurs in the right time interval they will mutually absorb some or all of the energy instead of elastically scattering it, heating the air. It can also absorb one wavelength and emit a cascade of photons at a different wavelength (depending on its spectrum).

Finally, one has to add in the GHGs, notably CO_2 (water is already there). They have the effect increasing the outgoing radiance from the (higher temperature) surface in some bands, and transferring some of it to CO_2 where it is trapped until it diffuses to the top of the CO_2 column, where it is emitted at a cooler temperature. The total power going out is thus split up, with that pesky blackbody spectrum modulated so that different frequencies have different effective temperatures, in a way that is locally modulated by — nearly everything. The lapse rate. Moisture content. Clouds. Bulk transport of heat up or down via convection. Bulk transport of heat up or down via caged radiation in parts of the spectrum. And don’t forget sideways! Everything is now circulating, wind and surface evaporation are coupled, the equilibration time for the ocean has stretched from “commensurate with the rotational period” for shallow seas to a thousand years or more so that the ocean is never at equilibrium, it is always tugging surface temperatures one way or the other with substantial thermal ballast, heat deposited not today but over the last week, month, year, decade, century, millennium.

Yessir, a damn hard problem. Anybody who calls this settled science is out of their ever-loving mind. Note well that I still haven’t included solar magnetism or any serious modulation of solar irradiance, or even the axial tilt of the earth, which once again completely changes everything, because now the timescales at the poles become annual, and the north pole and south pole are not at all alike! Consider the enormous difference in their thermal ballast and oceanic heat transport and atmospheric heat transport!

A hard problem. But perhaps I’ll try to tackle it, if I have time, at least through the first few steps outlined above. At the very least I’d like to have a better idea of the direction of some of the first few build-a-bear steps on the average temperature (while the term “average temperature” has some meaning, that is before making the system chaotic).



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seems to me that the 33 degree greenhouse effect is wildly overstated …

Great article Dr Brown!
I guess it depends what you are setting out to prove as to whether all the effort is worthwhile. perhaps Hans Jelbring made a smart move with his model atmosphere and isometrically heated planet surface in his 2003 paper:
At least if he is right you won’t have to worry about including radiative effects of GHG’s

Randall Harris

It is a very tough problem. I hope you can get some sleep tonight.

G. Karst

Yes, it truly is a Gorgon’s knot. We are all waiting for an Alexander, to appear, with a very sharp sword. GK

If the IPCC had spent $1b trying to do this before they started telling everyone what the solution was, we may have not only got somewhere by now but also had some respect for them. As it is, I get the feeling those telling us how to solve the problem don’t even know what the question is — or even where to start looking for the question.

Victor Barney

BERESHITH 8:22…as long as the earth remains, seedtime and harvest, and cold and heat, and winter and summer, and day and night shall not cease.” AND, it doesn’t matter if you believe it or not because it’s going to occur anyhow… because that’s exactly what’s written!


Are you serious? You know there are a few books which might help you understand how the system works.


Would it be impossile to build a physical model?

Mike M

But but.. the ‘settled science’ is so settled that we’re already spending over $2.5 billion per year to ‘combat climate change’. Ain’t no physics in the universe gonna slow down a gravy train with that much inertia.

When I see those magic words “lapse rate”, I hope that what you are talking about makes sense.


And then you haven’t even mentioned that the rotation of the earth around the sun isn’t a perfect circle, which influences the amount of radiation from the sun on the earth.

michael hart

The problem is [well, one of the many problems], that I fully expect climate modellers to say “Yes, of course we know all this. We take it fully into account. Now, run along, while we tell politicians how to run the world.”
But where can us lesser mortals examine the algorithms and computer code, not to mention the assumptions about which bits can be safely ignored?


You need to write a CSharp program and combine it with a ball made in WPF. WPF (Windows presentation Foundation) is perfect for this. We use it in my business every day.
If we could get some money from some rich guy, and I could take a 2 years leave from my day job, I would love to join you.
We could patch up the globe with small triangles with texture. The globe can be made in any 3-D package and exported to xaml. You can then import it into a WPF project and then write a CSharp program that starts turning the globe….and start calculating stuff…..ahh…a fantastic project…. Click “play” and it comes alive….
But for crying out loud; Dont make a “Report for Policy-Makers” when we are finished!!!!!


I’d be tempted to try the Earth as a simple disc (ie, what the sun ‘sees’), flipping over once per day, to give a global average. Then let temperature rise by 1 degree for each doubling of CO2. Then use average albedo, clouds increasing with temperature (more water vapour), hence more albedo and less temperature. Fudge something in for leads and lags if you must.
I’m interested to see if more CO2 eventually ends up with a faster cooling Earth, and therefore a route to the next ice age.
My view is that if the actual situation is too complicated to model, then move to a simpler picture and play around with it.

Looking back at the historical record for clues regarding the character or behavior of these many factors might helpt he effort, at least to decide whether a given addition is going in the right direction.
I like to point out that maybe kids are right to not like vegetables. After all, everybody who ate green peas during the Civil War died

Arno Arrak

You are on the right track. Have you looked at what Ferenc Miskolczi has done? You should because he has some very important conclusions about the greenhouse effect.

Joules Verne

I’m surprised at your surprise. Anyone who’s been paying attention to this blog knows that the 255K baseline is an ideal grey sphere which has no mass and superconducts. The only difference between gray and black body is albedo. The only difference between sphere and uniformly lit plane is angle of incidence adjustment.
We can at least use some good old experimental science to get us to what an airless world made out of the same rocks as the earth at the same distance from the sun does as far as average surface temperature. At mid-latitudes on the moon that measured number is 250K.
The moon’s slower spin and thermal conductivity of rocks combine to lower its temperature some more from a gray body and the earth’s faster spin would then serve to make it closer to the black body. So I guess what I’m saying is that arguing with 255K as a baseline is probably something only cranks, pikers, and pedants find unsatisfactory for most purposes.
is of course massless and superconducting. I prefer to at least say, for laymen, that the ideal grey sphere is spinning so fast the temperature is equal at every point. An ideal gray body is pretty darn basic physics. High school level stuff int the NYS regents science course I took in the 1970’s anyway. It’s been described here on Watt’s Up With That many times too.
The problem is the body is painted with all kinds of colors other than levels of grey (albedo). And even the grey changes in a very poorly characterized manner. Then there’s like an orchestra playing with different frequencies of light instead of sound and a fairly large assortment of different arrangements of matter that interacts with it to figure out just the radiative part. Then, as long as this remains a water planet, the three phases of water and other unique properties like high latent heat capacity and solid form lighter than liquid to deal with. Then there’s convection up the wazoo which doesn’t happen on grey bodies and has a large effect on temperature stratification. The grey body is an anchor and of course it must be understood what a grey body is and everything that can act to change things.

Thank you Dr. Brown. For us non physicists a lucid description. Like all good scientific investigations it raises way more questions then answers. It also illustrates better then I have ever been able to do how dependent any of these numeric models, even the ones we have confidence in, are to voracity of the underlying empirical measurements.

Stephen Wilde

Excellent article.
One question:
How do we know that the Earth is any warmer than it would be without greenhouse gases if the standard assumptions are so obviously inappropriate and/or incomplete ?
I don’t believe there is any model anywhere that is based on empirical data rather than flawed guesswork.


If we absorb radiation as a disk and radiate away as a sphere the cards are already stacked in favour of cooling. Without the oceans we would be icicles.

Larry Kirk

Deliciously put! With that many variables (ie. more than about two), nobody could ever HOPE to model the system, or the effect on it of changing a single, relatively minor factor. Anybody who thinks that they can has got to be kidding themselves. (Nobody but an economist, that is. But they do not have a particularly good track record, knighted or otherwise)


Re: Francois

You know there are a few books which might help you understand how the system works.

List a few for me please. I would like to read them.

Joe Soap

Just out of curiosity what is the geothermal input?
Or heat from magnetic flux eddy-currents?
Or the energy input from lunar gravity moving the oceans?
Maybe these are very small but it would be good to see some estimation of them.


One question I have had is the impact of changes in solar UV on the troposphere by indirect means via changes in stratospheric heating. For example, if there is an increase in UV, the stratosphere experiences more heating. If the stratosphere is warmer than this means the top of the troposphere is warmer. If the tropopause is warmer, then if I due the adiabatic lapse to the surface, the surface will be warmer. If the temperature at the tropopause cools, then figuring the temperature down the column to the ground also cools. Unless — the affect from stratospheric changes is a change in altitude of the tropopause. If the stratosphere warms, the tropopause happens at a lower altitude — it finds the temperature inversion “sooner” as stuff is convecting upwards. In that case, since the troposphere is now “thinner” and the stratosphere is “thicker” the change is compensated for and the temperature at the surface is unchanged.
We are currently experiencing less UV than usual with a cooler Sun. That would be reflected in a cooler stratosphere. That might be reflected in a rise of the tropopause (or might not if the troposphere also cools, we are talking about temperature deltas here, not absolute temperatures). The whole thing is like squeezing a water balloon.
Nice article. I think the models currently assume an infinitely thick atmosphere with no convection, radiative heat transfer through it with no evaporation, condensation, clouds, etc.

seems to me that the 33 degree greenhouse effect is wildly overstated …
Well, or understated. I thought I had a relatively simple argument that would have suggested that the true baseline should be more than 33 degrees; temperature differentiation favors faster cooling, so both a static perfectly insulating sphere and a rotating sphere with heat capacity and with poles would respectively cool relative to the superconducting sphere and warm relative to the non-rotating sphere. But then I head that voice — dooo theee integrallll. Assume make an ass outa u and me. The problem really is — does the rotating sphere with heat capacity warm relative to the superconducting sphere? Intuitively, I’d say no, it still cools. But at this point I want to do the integrals. Which means first I have to derive them, which I can’t do right now because I’m about to be ass-deep in alligators teaching (really, I already am).
So it might take me weeks or even months to do so, although I don’t think it is that hard. I can probably use e.g. octave/matlab to do them numerically, although the rotating sphere with heat capacity technically requires the solution of a set of time dependent ODEs as a point at a given latitude rotates, through enough rotations to approach a steady state. Basically you have something like dQ/dt for a surface element equals dP_in/dt – dP_out/dt, = CdT/dt, where dP_in/dt is \vec{S}\cdot \hat{n} dA for incoming Poynting vector from sun, dP_out/dt is blackbody power out of dA, and C is the heat capacity of dA. \hat{n}(t) is an outward directed normal (as a function of time as the sphere rotates). The solar flux is modulated by a periodic square wave so it is zero as the point goes darkside.
One should be able to start this from any temperature distribution and spin forward to equilibrium as a function of theta, and only have to do this for the upper half sphere as it is symmetric. Then one has T(\theta, \phi, etc) and one can plot, integrate to find averages, and so forth, for different values of this and that. I think it would be very educational to do this and would take the guesswork out of the question “what does water do” or “what does an atmosphere do” to the sphere (relative to superconducting or insulating static sphere). With a bit more work, one could probably add in depth and do the vertical heat equation as well (only conduction for some conductance) — I think octave would still solve it in less than eternity, although it might well be hours per sphere. A small cluster and you could do a whole range of spheres in a day of compute time and generate pretty pictures without resorting to C coding.
So I don’t pretend to be able to guess the answer. Time to do the work instead.


Dr Brown beautifully demonstrates the complexity of the problem. I don’t think we know how much of that complexity is embodied in the IPPC models but I suspect not enough. I also know from experience that you can reach a point of complexity in models that you cannot really understand what your own models are doing. Then you start running sensitivity analyses until they become so complex you don’t understand them.
It is a wonderful world of uncertainty. It is fun to play with. Just don’t bet the farm on the results.

A physicist

Professor Brown, please let me express my appreciation for this outstanding WUWT guest post!
Regarding the physics of radiation transport, please allow me to recommend the web page that the American Institute of Physics maintains on this topic, titled Basic Radiation Calculations.
One merit of the AIP’s page is that it covers the history in parallel with the physics: definitely it’s taken a long time and a huge effort to work through these details.
It made me LOL to read, halfway down the Basic Radiation Calculations web page, this passage:

“Dear reader: You have made your way into one of the most difficult corners of this experimental site, and it would be very useful to know why. Would you take just three minutes to answer a few questions? Please click here”

Professor Brown, you have earned everyone’s appreciation, and please accept my sincere thanks, for working to bring a comparable level of understanding to the skeptical community here on WUWT. Your post shows rational skepticism at its finest and best!

Joules Verne

@rgb (con’t)
So I do agree the ideal grey body baseline is horrible. I prefer to start with the moon as the baseline. That gives us an actual sphere made out of the same rocks as the earth at the same distance from the sun. The earth spins faster than the moon which makes it several degrees warmer which is fairly straightfoward.
So here’s what I say. The next biggest difference between earth and moon is that earth basically presents as a ball of brine instead of a ball of rocks. There’s a lot of radiative heating/cooling implications in just that alone. Water is very much different from rocks in physical properties.


Bravo Dr. Brown.
The 33 K figure is based on treating the earth as a spherical cross section, which is obviously completely wrong. The AGW’ers are off by over 100 K to start with and they are trying to make predictions accurate to tenths of degrees : )

When I see those magic words “lapse rate”, I hope that what you are talking about makes sense.
It does. The lapse rate is just the rate that the temperature of the atmosphere drops off with height. As the Earth’s surface cools via radiation, the vertically stratified volume over a given piece of surface (including the surface) radiates at different temperatures if/when the frequencies associated with those temperatures become unblocked by “high albedo” GH gases that have a large absorption cross section there. CO_2, for example, tends to radiate from near the top of the troposphere where it is much colder. Colder means that it radiates less power (in this part of the spectrum) than the warm surface underneath would have due to BB radiation alone. Radiating away less heat creates a differential warming (less cooling). In Caballero’s nice online book, you can look at figure 5.15 and see- the measurements of the IR spectrum from the Sahara Desert that perfectly illustrate the point, and the book itself defines and derives e.g. the adiabatic dry air lapse rate.
It’s complicated, in other words — very complicated when you add clouds, varying albedo, convection, wet air, global circulation — but not senseless or unreasonable.

Joel Shore

(1) Arthur Smith has already tackled the issue of the temperature variation on a planet with some rotation rate and some uniform heat capacity:
(2) While the issue of average temperature is complicated, it is not quite as bad as you seem to think. First of all, we have Holder’s Inequality, which tells us that the 4th root of the average of T^4 is greater or equal to the average of T for any distribution of temperature T. (The equality occurs in the case of a uniform temperature distribution, and it turns out that the current Earth is close enough to having a uniform temperature that the difference between these two ways of computing average temperature is small.)
Second of all, what is probably the best way to characterize the radiative greenhouse effect is by considering the average emission (in W/m^2) of the planet’s surface (given its actual temperature distribution) vs the average intensity (in W/m^2) that is absorbed by the planet and its atmosphere. So, for example, for Earth we know that the average intensity absorbed is ~240 W/m^2 and the average emission is ~390 W/m^2 or so. We also know that the average surface temperature of the Earth is ~288 K. In the absence of a radiative greenhouse effect, an Earth (with everything otherwise the same, including the albedo) would have to an average temperature such that it is emitting 240 W/m^2. The highest average temperature that a blackbody can have and emit 240 W/m^2 is 255 K (by Holder’s Inequality). To the extent that the temperature distribution is non-uniform, the average temperature could be lower…even considerably lower for very non-uniform distributions. (To the extent that the Earth is not a perfect blackbody emitter in the infrared, the average temperature could be a little higher…but in fact the emissivity of the Earth is very close to 1 over the relevant wavelengths.)

So here’s what I say. The next biggest difference between earth and moon is that earth basically presents as a ball of brine instead of a ball of rocks. There’s a lot of radiative heating/cooling implications in just that alone. Water is very much different from rocks in physical properties.
I think you’re right, although it isn’t just water — water alone is just a matter of changing the heat capacity per unit area of the surface in the model series I suggest. It has to be water with circulation, I think. And as I said, until I formulate and do the integrals I won’t try to guess the answer. Too difficult, and there is Kirchoff lurking behind Arrhennius and Stefan-Boltzmann. Minimizing temperature differences warms, but only up to where there is none and you’re back to the superconducting limit, I think. But I want to do the integrals and see.


Great article.
I like your ‘well, lets try to figure it out from first principles approach’.
Most others slide past the awkward bits leaving the reader feeling inadequate.
A sign of a good teacher!
I thought you might be interested in a new peer reviewed paper by Gerhard Kramm and Ralph Dlugi.
Its a review of the current understanding of the atmosphere effect.
Of particular interest
A detailed application of Keplers Laws to Earth/Sun situation.
Equation 2.17 which includes a ground flux contribution as well as the radiative contributions.
The energy reservoir diagram Fig 11
The irradiance overlap area Fig 5 which contradicts some IPCC presentations.


Read up Misckolczi’s paper: he’s done a lot of the maths.

Larry Kirk

What one could in theory do, would be to simply change one of the variables of this immensely complex system and then sit back for an appropriate length of time and observe the effect that this change had had.
Not in the case of this planet however, as we are already dealing with a dynamic, ever-changing system, of which we do not know the detailed history, nor the full nature or history of the numerous factors that acted upon it to date.
So, looking forward: as we do not know how this constantly changing system would have changed anyway, without our intervention, then we can never deduce the effect that our interference with one single variable has had.
And that is the flaw in global warming modelling. We do not understand the pre-existing system, let alone the effects of our intervention. And we never could, because it is too complex.
Economics seems equally flawed, and equally subordinate to political expediency and the strange human desire to always have been ‘right’, regardless of the facts.

Joules Verne

I don’t think there’s much argument that building a climate model from the bottom up is hard. Climate boffins do it from the top down. This drastically simplifies things as it’s all radiation at that point and these days you can measure it with satellites. Believe it not though we still can’t get satisfactory agreement between different methods of global average albedo better than a few percent accuracy. About the the only thing all the methods agree on is it’s not constant from year to year. The more sophisticated models take obvious seasonal variation into account but from year-to-year it holds constant. So it becomes a prime candidate for a fudge factor which can be tweaked to make your model more agreeable with history. A variation in global average albedo of just 1% is equivalent to the net of all anthropogenic forcing estimates. Experiments measuring albedo, those few that have been running for long, find albedo variation of that amount from year to year.

Joel Shore

Stephen Wilde says:

How do we know that the Earth is any warmer than it would be without greenhouse gases if the standard assumptions are so obviously inappropriate and/or incomplete ?

Because we know that the Earth + atmosphere are absorbing ~240 W/m^2 from the sun and that the Earth’s surface is emitting ~390 W/m^2 or so. The only way that this can happen without rapid cooling is if the Earth’s atmosphere absorbs some of the radiation emitted by the surface, which means there is a radiative greenhouse effect. (People like you have desperately tried to explain it other ways, but alas none of those ways use correct physics principles.)
This notion that the atmosphere is absorbing the surface radiation is in fact seen to be empirically-correct by looking at the emissions from the Earth as seen from satellites in space. They see an Earth emitting only ~240 W/m^2 and the spectrum shows that the reason it is emitting less than the 390 W/m^2 that the surface is emitting is because certain wavelengths are getting strongly absorbed. These wavelengths correspond precisely to those wavelengths at which the various greenhouse gases (and clouds) absorb radiation.

Viv Evans

Yep – a damn hard problem, and that’s without sticking stuff into the model which mess up things like volcanoes or plants (forget SUV-driving humans for the moment, they mess up everything anyway!!!) and all the small critters in the oceans doing their things as well …
I thoroughly enjoyed reading this post!

Start from the definitive facts, of my Venus/Earth temperatures comparison, and don’t lose sight of them, ever. At least half of what is written in this article as unquestioned assumption, I consider rank speculation. The Venus/Earth comparison demolishes all climate models with the simple facts. Someone earlier likened the problem to the Gordian (not “Gorgons”) Knot, and looks for an Alexander to cut it. Well, you cut it by not jumping on the radiative transfer theory as holy writ, and by not ignoring the Venus/Earth facts. I submitted the following comment to tallbloke’s web site just this morning:
In the context of public debate, the “greenhouse effect” is not about radiative equilibria with and without an atmosphere, or even with or without “greenhouse gases” in the atmosphere. It is about whether atmospheric temperature at the surface (or at any given pressure level) increases with an increase in atmospheric carbon dioxide. I am astonished that even skeptics cannot focus upon this obvious fact in the real world; everyone can’t seem to stop themselves from launching into radiative transfer theory arguments. And in that context, of the public and political debate — that irreducible, unarguable reality — the FACT (not theory) is, my utterly simple and transparent comparison of the temperatures in the atmospheres of Venus and Earth demonstrates there is no such greenhouse effect, whatsoever. All the supposedly learned theorizing by one and all is precisely worthless, because everyone uses it to ignore the simple, definitive fact that disproves the tyrannously-promulgated carbon dioxide greenhouse effect, and reveals the radiative transfer theory as unconnected [or better, disconnected] from the real thermodynamics of the atmosphere. (And that last should be obvious, since the radiative theory ASSUMES a fixed temperature distribution, with every incremental layer of the atmosphere at thermal equilibrium. So the radiation levels in the atmosphere are, by that assumption, the EFFECT, not the CAUSE, of the thermodynamics — the radiative EFFECT of the gravitationally-imposed tropospheric lapse rate.)
Where I stand at the moment (and don’t even bother trying to change my mind, I am still learning, and I will develop my understanding on my own, unless and until I see “experts” recognize what I have done, and do better than I have already done): You can’t consider the Earth’s surface a blackbody, or each differential layer of the atmosphere as contributing as a “gray body” (blackbody times emissivity) without the assumption of detailed thermal equilibrium (radiative transfer theorists, are you listening?). More specifically, you can’t do it in the presence of convection and conduction of heat energy between the layers, and expect to get the thermodynamics right. That’s why the blackbody is traditionally described in terms of an enclosed cavity, held at constant temperature, with just a small hole to allow transfer of radiation (no convection, no conduction) into and out of it (Christopher Monckton followers, are you listening?). And even with such detailed thermal equilibrium as the radiative transfer theory imagines, you can only recover the observed radiation levels, and pretend you know the thermodynamics from them; you can’t predict their thermodynamic effect, because the cause-and-effect works the other way — the thermodynamics (in the presence of incident solar infrared irradiation) gives the temperatures and those determine the measured radiation levels. And the presence of CO2 and other IR-active gases only funnels the radiation portion (but not the convection and conduction portions) of atmospheric heat transfer through them, it doesn’t necessarily heat the atmosphere (in particular, in passing from the surface upward, as my Venus/Earth analysis demonstrates). The Gordian Knot does not exist.

Harry Dale Huffman,
Dont worry, your comment was noted. Just been too busy at work to reply.
Call back soon.

I’m surprised at your surprise. Anyone who’s been paying attention to this blog knows that the 255K baseline is an ideal grey sphere which has no mass and superconducts. The only difference between gray and black body is albedo. The only difference between sphere and uniformly lit plane is angle of incidence adjustment.
Well there you have it. I haven’t been “Paying attention to this blog” or reviewing climate physics in detail until, well, yesterday. Having a topical textbook really helps. All the meandering above was my attempt to mentally organize some of what I’ve learned so far from it — I didn’t really intend for it to be a toplevel post but rather the last post I was going to make in a thread.
Now I may have to make this one the last. I”ve got a ton of work to do (of the teaching variety) before I can come back to WUWT, although it is addictive.
I’ll make one last comment. Do not assume (and this goes for everybody) that all of the physics in the standard models is ill conceived or overtly wrong. The real differences between well-educated skeptics (where I am not one, not yet, not in the right physics in the right context) and dilettantes is that the former tend to think that the GCMs are “mostly right” in their physics, but are getting some pretty subtle stuff wrong, notably climate sensitivity. None of them doubt the greenhouse effect, most of the good ones include the ocean and so on. Perhaps not at the right level of detail — not all of the work done is of high quality — but it is probably mistaken in small details with large effects, with certain notable possible exceptions.
So I’m pretty “skeptical” that current climate science gets the kind of stuff I’m considering above in the first few steps at all wrong — I just want to work through it on my own as “homework”, to learn it properly. Far better to derive things yourself than to trust or rely on a textbook, better still to do both, derive it yourself and learn to understand it and then check your work against texts and resolve the differences, learn from your errors (and look for possible errors in the accepted literature — they are not unknown:-).
Believing that you know the answer before you work it through is OK, right up to the point where it becomes self-fulfilling prophecy and you make choices that make your beliefs work out. Confirmation bias (and its close cousin, cherrypicking) are often just as seductive to skeptics as they are to non-skeptics — they are just polarized the opposite, contrary way. The best thing to do is just plain work through it. When I have time, now, I will.

Mac the Knife

Excellent summary!
You ‘painted’ a very complex thought experiment with sufficient fidelity to reality and attention to detail that I could visualize contributions from each added element as you constructed the global warming picture. I have a much clearer mental picture of the problem and a better grasp of the daunting complexities associated with any ‘modeling’ attempts, as a result. I’m going to read this through several times more, to reinforce my understanding and appreciation of your ‘modern physics masterpiece’. As I’m malingering at home with a minor bout of ‘flu’ today, this will provide appropriate distraction, education, and entertainment.
Thank You!!!

At Earth orbital distance from the Sun the total incoming energy provided by the Sun is 1366 / 4 = 341.5 W m-2 .
(solar constant divided by 4 to take day/night and spherical shape into account).
Without an atmosphere, reflection and absorption of the solar irradiance would only take place at the Earth surface, composed by ice, rocks and sand. With an assumed albedo of 0.4 (reflecting power of a surface) the earth surface would absorb 341.5*(1-0.4) = 204.9 W m-2 and re-emit this energy to the outer space.
According to the Stefan- Boltzmann law, the mean temperature at the earth surface would establish itself at:
T= (E / ε σ)^¼ = (204.9/(1 x 5.6710-8))^ ¼ = 245.2 K (-28°C)
Obviously, large differences would be observed at different latitudes and wide changes would take place during every day-night cycle, including some ice melting and water freezing (but with water evaporating an atmosphere would exist… no water is left on the surface of planet Mars).
Also, the assumed albedo is quite hypothetical since we have no clue about what would be the Earth’s surface if no atmosphere were present.
Actually, the mean Earth surface temperature is approximately 15 °C, or 43 K higher than the assumed naked planet. If you assume 33 K it is also within the ballpark.
This is the warming contribution of the atmosphere, mostly by absorbing some of the INcoming sunlight (UV/visible) AND some of the OUTgoing long wave radiation from the Earth surface (by so-called greenhouse gases such as water and carbon dioxide).
Actually the mean surface temperature doesn’t make any physical sense in this hypothetical calculation. The surface temperature would swing between extremes over the night/day cycle (incoming radiation between 0 and 1366 W m-2), with the heat capacity of the material at the surface contributing to a dampening effect.
For a simple 2 layers atmospheric model look at:

DeWitt Payne

Hölder’s Inequality requires that the average temperature on a non-isothermal sphere at radiative equilibrium, assuming the same emissivity and incident power level, must be less than the average temperature of an isothermal sphere. That’s something that G&T do correctly ( Section 3.7.4 ).

ask scaffeta, he models all that with one function.
sounds like you are on your way to building a GCM

Thermodynamic professor, power plant manager

I’m sure that thermodynamic atmosphere effect explains everything much better, it’s the only method that is based on physical facts. If you try to model earths temperature with this very clear Anthonys description you can argue everything until you are dead or used 1 bn$;) Backradiation is bullshit from cooler gas to warmer surface, physically impossible! Gases can’t heat up with infrared radiation without incredible W/m2, only surfaces.

David L.

All this reminds me once again why we talk about “average temperature”? The number, as has been mentioned on this site many times, is meaningless. Let’s talk about my house. I have a wood stove in the living room and it pretty much heats my entire house. I have a grate in the ceiling above the wood stove and being a stone farmhouse there is convection throughout the main part of the house. The kitchen is on the opposite side end of the house from the woodstove. Our TV room is perpendicular to this arrangement and the bedroom is above the TV room on the second floor. Now I like it cool and my wife likes it warm so we are constantly changing the draft on the stove or kicking on the oil burner for extra heat, especially in the bedroom which has it’s own heating zone since it gets very little convection from the woodstove.
Now my wife and I never talk about the average temperature of our house. Typically when it’s 20F outside, the equilibrium temperature distribution in my house is: The stove is 400F, the living room is 80F, the kitchen is 64F, the TV room is 70F, the bedroom is 62F, and the second floor is 67F. Oh, by the way, inside the fridge is 30F and the freezer is 10F. The basement is 56F. The attic is 32F. Some of the house is insulated by stone, some by modern R33 fiberglass…some by 1940’s R8 insulation.
So when I’m cold in the bedroom I kick on the furnace. When I’m too warm in the living room I close the woodstove damper. When the milk goes bad in the fridge I turn down the thermostat in the fridge. My point is that no one would ever think of this system as an average temperature. It has no meaning. There are different temperatures and gradients all over that house for real physical reasons…a good physicist may actually even attempt to model it and get it close…maybe with a degree or two?
Why does the average not matter? Why should it? If I did somehow calculate an everage home temperature what would it mean? I can’t act on that information. If the average temperature of the house increased because the fridge broke and warmed up, then that’s not helping my wife who says she’s freezing in the living room. I can’t say “hey, the average home temperature is actually going up so you can’t be freezing in there”. What’s important is knowing the actual temperatures at the various locations and how they change or their distribution changes over time.
One last point: I couldn’t ascertain an average temperature of my home to better than a few degrees. I certainly couldn’t tell if that average changed by a few degrees or even if it mattered. And they think they can 1) get a handle on the average temperature of the entire planet to +/- 0.1 and 2) that it provides any useful information?


This may be simplistic, but would it be fair to say that the 33K of temp attributed to CO2 is really 33K of temp attributed to the non simple BlackBody?
What I mean is if we assume the black body gets us to 255K, then all the complexity (including, but not limited to rotation, circulation, CO2 ppmv, atmosphere and water ingeneral, etc) makes up that 33K? Or did I miss something.
I’m wondering if each comonnent mentioned in the article can be thought about AFTER a baseline is used. I know that PV=nRT is for ‘ideal gasses’ and non of them exist, but it is a decent enough guide to teach in intro Chem classes. is the T_b=255K derived in a similar fashion?

jim hogg

A roadmap to a possible approximate answer from someone who knows where to start: our messy and incredibly complex reality. A pleasure to read a first rate and honest mind at work.

Great Greyhounds

Ahhh! Those fine words… “Do the Integral!”
Thank You for the explanation of how a model should be developed, but the big question for me is: How do we hope to test the output of the Model?
I worked for a number of years doing E-M simulations for Antennas and Radomes, and at least we had the ability to build real models to test our simulations against!

RE: R. Brown, 9:00am

But then I head that voice — dooo theee integrallll. Assume make an ass outa u and me. The problem really is — does the rotating sphere with heat capacity warm relative to the superconducting sphere? Intuitively, I’d say no, it still cools. But at this point I want to do the integrals.

Another big complication is the heat capacity effects of water turning into ice. First, there is the simple heat of fusion where heat is released while maintaining a temperature of 0 degC while water changes from liquid to solid. But then you form a crust of ice whose surface can be much colder than the water under the ice. I cannot imagine the differential equations describing these effects filled with discontinuities, much less attempt to integrate them.
I very much appreciate you contributions here over the past week. I envy your students at Duke.