Guest post by Willis Eschenbach
There is a lot of misinformation floating around the web about how the greenhouse effect works. It is variously described as a “blanket” that keeps the Earth warm, a “mirror” that reflects part of the heat back to Earth, or “a pane of glass” that somehow keeps energy from escaping. It is none of these things.
A planetary “greenhouse” is a curiosity, a trick of nature. It works solely because although a sphere only has one side, a shell has two sides. The trick has nothing to do with greenhouse gases. It does not require an atmosphere. In fact, a planetary greenhouse can be built entirely of steel. A thought experiment shows how a steel greenhouse would work.
Before we start, however, a digression regarding temperature. The radiation emitted by a blackbody varies with the fourth power of the temperature. As a result, for a blackbody, we can measure the temperature in units of radiation, which are watts per square meter (W/m2). For objects with temperatures found on the Earth, this radiation is in the range called “longwave” or “infrared” radiation. See the Appendix for the formula that relates temperature to radiation.
This means that we can denote the temperature of a blackbody using W/m2 as well as the traditional measures (Fahrenheit, Celsius, Kelvin). The advantage is that while temperature (degrees) is not conserved, energy (W/m2) is conserved. So we can check to see if energy lost is equal to energy gained, since energy is neither being created nor destroyed by the climate.
For our thought experiment, imagine a planet the size of the Earth, a perfect blackbody, heated from the interior at 235 watts per square meter of surface area. How is it heated from the interior? Doesn’t matter, we’ll say “radioactive elements”, that sounds scientific.
The planet is in interstellar space, with no atmosphere and no nearby stars. The equilibrium surface temperature of this planet is, of course, 235 W/m2. To maintain the equilibrium temperature, it constantly radiates this amount of energy out to space. Coincidentally, this is the amount of solar radiation that makes it past the clouds to warm the Earth. If we convert 235 W/m2 to one of our normal temperature scales, it is -19 Celsius (C), or -3° Fahrenheit (F), or 254 Kelvins (K). It’s a possible temperature that the Earth might have if there were no greenhouse effect. That is to say … cold.
Now imagine that the planet gets completely surrounded by a thin black steel shell, located a few thousand meters above the surface, as shown in a cutaway view in the picture above, and in Figure 1 below. What happens to the surface temperature of the planet? (To simplify calculations, we can assume the shell has the same outer surface area as the surface. For an earth-sized planet with a shell two kilometers above the surface everywhere, the difference in area is only six-hundredths of one percent. This assumption makes no difference to the argument presented.)
In order to maintain its thermal equilibrium, including the new shell, the whole system must still radiate 235 W/m2 out to space. To do this, the steel shell must warm until it is radiating at 235 watts per square meter. Of course, since a shell has an inside and an outside, it will also radiate 235 watts inward to the planet. The planet is now being heated by 235 W/m2 of energy from the interior, and 235 W/m2 from the shell. This will warm the planetary surface until it reaches a temperature where it radiates at 470 watts per square meter. In vacuum conditions as described, this would be a perfect greenhouse, with no losses of any kind. Figure 1 shows how it works.
Figure 1. Building a steel greenhouse. (A) Planet without greenhouse. Surface temperature is 235 W/m2 heated from the interior. (B) Planet surrounded by steel greenhouse shell. Shell radiates the same amount to space as the planet without the shell, 235 W/m2. It radiates the same inward, which warms the planet to 470 W/m2 (29 C, 83°F, 302 K). [Clarification added] Note that the distance from the shell to the planet is greatly exaggerated. In reality, it is close enough to the planet that the difference in the areas can be neglected in practice.
The trick can be repeated again, by surrounding the planet and the shell with a second outer shell. In this two shell case, the planetary surface temperature (in W/m2) will be three times the initial surface temperature.
In nature, planets have atmospheric shells, of course, rather than steel shells. The trick works the same way, however. Rather than being warmed from the inside, the Earth receives 235 W/m2 from the sun. Solar radiation passes through the atmosphere. Outgoing longwave radiation is absorbed by the atmosphere, just as it is absorbed by the steel shell shown in Fig. 1.
So that’s the trick of the greenhouse. It has nothing to do with blankets, mirrors, or greenhouse gases. It works even when it is built out of steel. It depends on the fact that a shell radiates in two directions, inwards and outwards. This radiation keeps the planet warmer than it would be without the shell.
Now, it is tempting to think that we could model the Earth in this manner, as a sphere surrounded by a single shell. This is called a “two-layer” model, with the two layers being the surface and the atmospheric shell. In fact, a number of simplified climate models have been built in this way. Unnoticed by their programmers, however, is that is not physically possible to model the Earth as a two-layer system. Figure 2 shows why. Note that in all cases, the system has to remain in thermal equilibrium. This means that the surface must radiate as much as it absorbs, and the shell must also radiate as much as it absorbs. In addition, radiation from the shell to space (upwelling longwave radiation or ULR) must equal radiation from the shell to the Earth (downwelling longwave radiation, or DLR)
Figure 2. Single-shell (“two-layer”) greenhouse system, including various losses. S is the sun, E is the Earth, and G is the atmospheric greenhouse shell around the Earth. The height of the shell is greatly exaggerated; in reality the shell is so close to the Earth that they have about the same area, and thus the small difference in area can be neglected. Fig. 2(a) shows a perfect greenhouse. W is the total watts/m2 available to the greenhouse system after albedo. Fig. 2(b) is the same as Fig. 2(a) plus radiation losses Lr which pass through the atmosphere. Fig. 2(c) is the same as Fig. 2(b), plus the effect of absorption losses La. Fig. 2(d) is the same as Fig. 2(c), plus the effect of thermal losses Lt.
Figure 2(a) shows the same situation as Figure 1(B), which is a perfect planetary greenhouse. In this case, however, it is heated by an amount of energy “W”, which is coming from the sun. The planet receives solar radiation in the amount of “W” from the sun, and longwave radiation “W” from the atmospheric shell. The surface temperature is thus 2W. All energy flows are in Watts/square metre (W/m2).
Figure 2(b) adds two losses. The first is the reflection from the Earth’s albedo (Wo – W). This is energy which never enters the system and is reflected back into space. We are still getting the same energy entering the system (W). The second loss Lr is from radiation which goes from the surface to space through the “atmospheric window”. Because of the second loss, the surface of the Earth does not get as warm as in a perfect system. In a perfect system, the temperature of the surface is 2W. But including the radiation loss Lr, the surface temperature drops to 2W – 2Lr.
Figure 2(c) adds another loss La, which is the solar radiation which is absorbed by the shell. This cuts the radiation hitting the surface down to W – La. Including this loss, the surface temperature is 2W – 2Lr – La.
Figure 2(d) adds the final loss. This is Lt, the thermal loss. It is the sensible energy (energy you can feel) and latent energy (evaporation) which is transported from the surface to the shell by convection. Including all of these losses, the surface temperature of the Earth is 2W – 2Lr – La – Lt.
Now, why can’t the Earth be modeled in this manner? A look at the size of the various losses shows why. Here is the canonical global energy budget, called the “Kiehl/Trenberth” budget, or the K/T budget.
Figure 3. The Kiehl/Trenberth Global Energy Budget. This is a “two layer” representation, with the surface and the atmosphere being the two layers. Lr, the radiation loss, is the 40 W/m2 labeled “Atmospheric Window”. La, the absorption loss, is the 67 W/m2 labelled “Absorbed by Atmosphere”. Lt, the thermal loss, is 102 W/m2. This is the sum of the 24 W/m2 labelled “Thermals” and the 78 W/m2 labelled “Evapo-transpiration”. W, the energy from the sun, is the incoming solar radiation of 342 W/m2 less the 107 W/m2 that is reflected by the surface and the clouds. This means that W is 235 W/m2. SOURCE
What’s wrong with this picture? Note that the temperature of the Earth is 390 W/m2, labeled as “Surface Radiation”. This is 15 C, or 59°F. But from Fig. 2(d), we know that the surface temperature of a greenhouse system with a single shell, as shown in the drawing, is 2W (470) – 2Lr (80) – La (67) – Lt (102) = 221 W/m2. This is far below the known surface temperature of the Earth. In other words, a single shell greenhouse system simply isn’t efficient enough to give a surface temperature which is warm enough to allow for the known losses.
So where is the problem with the K/T budget diagram? The hidden fault is that the upward radiation from the atmospheric layer does not equal the downward radiation. There is 195 W/m2 going to space from the atmospheric shell, and 324 W/m2 going down to the surface.
In order to get enough energy to allow for the known losses, the simplest model requires two atmospheric shells. A perfect greenhouse with two shells would give a surface temperature of 3W, or 705 W/m2. This is enough to allow for the known losses, and still give a surface temperature which matches that of the Earth. Figure 4 shows one such possible system.
Figure 4. Simplest greenhouse global energy budget capable of representing the Earth. Note that all major flows in the K/T energy budget have been maintained. There are two shells, which must be physically separate. These are the lower troposphere and the lowest stratosphere. They are separated by the tropopause.
This budget fulfills all of the requirements for thermal equilibrium. The same amount is radiated upwards and downwards by each layer. The amount absorbed by each part of the system equals the amount radiated. Further verification of this energy budget is the 147 W/m2 emission from just above the tropopause. This converts to a Celsius temperature of about -50 C, which is a typical temperature for the lowest part of the stratosphere.
I have written a simplified Excel radiation/evaporation/convection model which encapsulates the above system. It is available here. Click on the “File” menu on the webpage and select “Download”.
I invite people to play with the model. It is what I term a “Tinkertoy” model, which is what I call the simplest possible model that can represent a particular reality. One of the interesting results from the model is that there can be very little thermal loss between the inner and the outer atmospheric layers. If there is more than a trivial amount of leakage, the surface cools significantly. Among the other insights yielded by the model is that a change equivalent to a doubling of CO2 (an increase of 3.7 W/m2 downwelling radiation at the top of the atmosphere) can be canceled by a 1% increase in the upper and lower cloud reflections.
Experiment with the values, it is an interesting insight into the energy flows in the simplest possible climate model that can represent the Earth’s greenhouse system.
APPENDIX
The formula that relates the temperature to radiation is called the “Stefan-Bolzmann” equation:
R = sigma * epsilon * T^4
where r = radiation (W/m2)
sigma = the Stefan-Boltzmann constant, 5.67 * 10^-8
epsilon = the emissivity of the body, which for a blackbody = 1
T^4 = absolute temperature in Kelvins (which is Celsius plus 273.15) raised to the fourth power
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Myrddin Wyn (04:09:21) :
Perhaps a refinement of the thought experiment would help. Imagine the planet without a shell. Now, in an instant, we add a cold shell. As you point out, it starts to warm. Because it has twice the surface area as the planet, as it warms it radiates energy both inwards and outwards.
The downwelling radiation from the shell, however, warms the planet. So rather than radiating 235 W/m2, it starts radiating more and more.
Equilibrium is finally reached when the situation is as shown in Fig. 1(B). At that point, the planet is receiving 470 W/m2. Of this, 235 W/m2 come from the core, and 235 W/m2 come from the shell. It radiates that same amount, 470 W/m2, so it is in thermal equilibrium.
The shell, as you point out, divides that in half, with 235 W/m2 radiated outwards, and 235 W/m2 radiated inwards. Thus the shell is in equilibrium as well.
Hope this clarifies it,
w.
“”” PSU-EMS-Alum (05:01:09) :
You know what needs to be removed from this page?
Any comment that uses the term “re-radiate”.
This is a pet peeve of mine …. it demonstrates a complete lack of understanding of the concept of “radiation”. “””
How so ? Are you for banning any term YOU don’t understand ?
For a body to “re-radiate” is a verty common occurrence. For example, road tar surfaces receive radiation (solar spectrum) from incident “sunlight”; some of which is absorbed and some reflected, so the surface warms, and re-radiates in a completely different thermal spectrum that depends on the surface temperature and its spectral emissivity.
Yes I agree you could just say “radiates”. The point is that a body can be at a fixed stable temperature and be re-radiating all of the energy it is receiveing in the form of some other radiation; with the incoming and out-going spectra being quite different.
Where I see a lot of confusion, is when people talk about the atmosphere , with its GHGs “reflectign surface emitted LWIR radiation back to the surface.
Reflection is an optical process that returns the incident spectrum only modified by the spectral reflectance of the surface; and is quite independent of temperature. Well if you want to be pedantic, the spectral reflection coefficient might vary with temperature.
Some might argue that the term “re-radiate” should be reserved for cases where a molecule or atom absorbs a photon of a given energy, and later emits a photon of the same energy, as the excited state returns to normalcy.
since that is a rarity in earth’s lower troposphere, it is not of much interest in climatology.
Marcus (04:20:51) :
Marcus, you are a gentleman and a scholar. Would that all people on both sides of the climate discussion were as gracious.
w.
Vincent (04:22:19) :
No energy is created or destroyed. It is simply trapped inside the system, which raises the temperature. See my post at Willis Eschenbach (11:20:27), which may assist.
PSU-EMS-Alum (05:01:09) :
In theory you are correct. The energy which is absorbed is not “re-radiated”. Energy is absorbed. Energy is radiated. They are not the “same energy”.
However, like my pet peeves of “baited breath” and “tow the line” and the constant misuse of “begs the question”, I fear it is ingrained in common usage, so I can’t help either of us.
Stephen Goldstein (05:02:24) :
Aaaahhh, very funny, yonason, I missed that entirely. I should get out more …
Willis Eschenbach (20:50:08) :
1) Thanks for the explanation. Between that and your reply to Lord Monckton, I think I understand (though my head still hurts a little).
2) Thanks for the idealized “greenhouse” model. It really helps clarify the whole issue, plus it takes me back to my school days of weightless pulleys and frictionless surfaces (where the IPCC apparently still dwells).
3) Thanks for sticking around for the discussion. The apparent confusion over the model, in this relatively sophisticated readership, does not bode well for educating the general public. Definitely an uphill battle (hopefully not on a frictionless surface).
Charlie (10:09:01), great clarifications. Kudos.
Here’s what I think. I think the greenhouse metaphor is broken. It never really was a simple model.
In 1906 American experimental physicist R. W. Wood published the
results of an experiment that demonstrated that a glass greenhouse was
not heated by trapped long-wave (infrared) radiation. In fact, the
glass windows excluded more infrared energy entering the greenhouse
than they trapped inside the greenhouse, which actually lowered the
temperature inside a glass greenhouse when compared with a comparable
quartz greenhouse.
Ok next question. How come you can fry an egg on the sidewalk? …. How come on a sunny day at the beach sand feels so hot on your feet?
Willis Eschenbach (11:11:01) :
The greenhouse effect itself is something in an atmosphere that radiates LW and in real atmospheres that something is GHGs and that fact makes your statement very misleading.
The fluxes involved would seem amenable to recreation in an actual, physical model.
A sixty watt light bulb and a basketball come mighty close to the correct proportions.
R.W. Woods words on the greenhouse effect:
http://www.wmconnolley.org.uk/sci/wood_rw.1909.html
TomVonk (06:54:18) :
Here is the definition of the Stefan-Boltzmann equation from Wikipedia, which is a good place to start:
Note that what is being calculated is the energy radiated per unit surface area. You can measure this in square meters, or (for a sphere) in steradians. (A steradian is the solid version of a radian. A radian measures the angle of a circle based on pi. A steradian measures the area of a sphere based on pi.)
However, I know of no such measure as watts per square meter per steradian. The K/T budget given above does not use such a measurement.
The units of W/m2 is also derivable from the equation. The Stefan Boltzman constant is 5.67E-8 J s^-1 m^-2 K^-4. In english this is Joules per second per square metre per Kelvins to the fourth power. Multiplying this by Kelvins to the fourth leaves Joules per second per square metre.
But a Joule per second is a Watt, so we are left with Watts per square metre.
I still do not buy it, assigning present surface temperature to “greenhouse effect”.
Mars has 15x more CO2 than Earth. This is effectively the same “greenhouse” as on Earth (Mars has no water vapor). But still, calculated and actual temperature on Mars is the same : 210K. Hint: Mars atmospheric pressure is just 600 Pa, Earth pressure is ~101,000 Pa.
“Greenhouse” effect is obviously non-existent without presence of bulk atmosphere, which can retain heat, absorbed from the surface. So it is the atmosphere itself, working as heating blanket. Question is, how much heat is removed from the surface by radiation, air convection and evaporative cooling.
If earth surface is “heated by back radiation”, then covering my face against night sky should be felt as lack of warming, as it does on sunny day. It does not happen, since my face is warmed by ambient air – nitrogen and oxygen, not some hypothetical arrow painted in scheme, made so to get the ins and outs into balance.
I believe clouds have some measurable effect, but for the rest call me a denier.
Doesn’t the steel shell have twice the surface area of the ball in the middle.
You know – two sides rather than one.
P Wilson (07:57:56) :
As the night-vision glasses used by armies show, everything emits infra-red rays. The amount and frequency of the waves is determined by their temperature. Air both absorbs and emits infrared rays. Why is it suddenly “improbable” that radiation from the air warms things around it?
SteveBrooklineMA (09:48:33) :
My point is that unless the greenhouse system has more than one shell, with the two shells separated so that there is minimum thermal loss between them, it does not concentrate enough energy to reproduce the earth’s conditions.
Willis,
You haven’t replied to my objection above that you can’t heat something above the temperature of a blackbody using an external energy source.
Your sphere must have a hole in them to let light in. We now have a spherical cavity with a hole in it. This is the typical example given to approximate a blackbody. It doesn’t matter how many additional shells are inside. It doesn’t matter what happens inside. The temperature inside can not exceed the temperature for a blackbody.
I assume you will claim the Earth is different because these are special steal spheres that are transparent to visible light, but opaque to IR, so you don’t need a hole for light to get in. But this doesn’t matter. This is just a different kind of hole. A hole in frequency spectrum.
A hole in frequency spectrum is not a way to ‘cheat’ thermodynamics and attain an internal temperature higher than a black body. The distribution of energy inside the sphere will continuously readjust to pour out the frequency hole, just as heat energy will continuously bounce around to pour out a physical hole.
If you disagree, then can you describe an experiment that would demonstrate this effect? Maybe a series of glass spheres inside each other with a black sphere in the center? I feel confident the temperature will not exceed that of a black body.
Willis;
There is no doubt your steel sphere analogy is correct. In fact one can buy home insulation based on it. The insulation consists of a stack of aluminium foils and is used where space is at a premium. However, with regard to your two shell model consider one simple issue. The tropopause is colder than both the rest of the troposphere and the stratosphere, how can you model explain that fact. How can the region between two shells be colder than either shell? For a region to be colder it must be losing energy to a heat sink colder than it is but the only thing colder than the tropopause at -57C is outer space at -269C. That means the tropopause has to be losing energy directly to space. How is that possible if there is an opaque shell above it?
BY the way with my comment about the tropopause being too cold to radiate 165 watts/sqM as required by Trenberth if it were a black body which most certainly it is not, by Stafeans law it would have to be at a temperature of -40.7C yet it is actually at a temperature of about -57C so indeed it is significantly too cold. When you factor in that at the very least it cannot radiate between 8 and 14 microns (the atmospheric window) because if it did it would be opaque at these wavelengths it is very very much too cold to radiate 165 watts. The Trenberth data is extemely suspect.
Willis, in the same spirit as Marcus, I will also chime in. Through Fig 2, I’m agreeable to your effort. Haven’t looked at the rest carefully yet. Along with Charlie above, I see the same misunderstandings repeatedly through the comments. So, some comments/suggestions:
-Doubly emphasize in the appendix that any surface with a temperature T will radiate at sigma*epsilon*T^4. This is true, no matter what the ambient or surrounding objects are. The surroundings only come into play if you want to consider the net heat transfer.
– People are missing that the two sides of the steel shell are at the same temperature. Perhaps it’d help them if I said the thermal conductivity of the steel shell is infinite; there is no temperature gradient across the shell.
– It is good that you drive home that a single-slab atmosphere isn’t representative. In reality, the radiation to space is from a colder temperature than the temperature at the bottom of the atmosphere.
-Emphasize that this is an equilibrium model, not a kinetic one. The temperatures and flows don’t change instantaneously to the final values if you add a steel shell.
– You might want to make it more clear in the first figures that the internally warmed planet is NOT the Earth. I was reading quickly the first time, and missed that you were making a fake planet there.
Ian Schumacher (12:47:38) :
I fear I don’t understand this at all, which is why I didn’t reply. What do you mean by “heat something above the temperature of a blackbody”?
If you mean that no matter what you do, the sun can’t heat something above the temperature of a blackbody heated by the sun at the same distance, the Earth itself shows that this is not true. Blackbody temperature at 235 W/m2, the amount of incoming solar radiation entering our planetary system, is 255K, or -19 C. Thus the earth has “an internal temperature higher than a black body”, something which you claim is impossible under any conditions.
However, this is not “cheating thermodynamics” as you say, any more than using a Thermos bottle cheats thermodynamics to keep your coffee warm.
Really sad to see this kind of rubbish published on this great site. The physics is completely wrong. Anthony, please do not let this kind stuff published on this site, it will take the value of the whole site down.
You know Willis’ model of a steel greenhouse is functionally the same, even if you postulate that the planet is a perfect sphere, and the steel shell is one micron thick, and is separated from the planet by a one micron gap.
So all of the discussion about the relative surface areas is somewhat off target; the analysis Willis gave doesn’t really depend on the gap. Now if the gap is in fact large so the shell OD is somewhat larger than the planet OD, the 235 W/m^2 number will change, inversely as the area of the shell. but his basic concept is unchanged by the shell spacing from the planet.
They system is a steady state, only because Willis asserted that the 235 W/m^2 and the original 254K surface temperature are maintained by internal heating as say from nuclear decay.
In reality, that energy source decays so the thing would eventually die out.
But it is not strictly correct to say the system is in equilibrium; the continual loss of energy to space would not occur if the system was in equilibrium.
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jt (21:14:58) :
“As a result the quantum of energy exchanged is never actually in a state of “free flight” between the two interacting electrons. ”
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Wow, no wonder they had to get rid of the cathode ray tube type TV. If electrons are never in free flight, it would be impossible to to bend their path with a magnet, like most old TVs do.
BTW – Carver Mead’s work isn’t widely accepted. Does not mean it’s wrong.
If you want to know what Earth would be like without a greenhouse effect, there’s no need to look as far away as Mars or Venus.
We have a moon right here, in the same orbit. Temps average over 100 Celsius in the day and drop below -150 Celsius at night. Since earth’s average daytime temp is lower, our atmosphere is obviously shielding us from solar energy during the day. And since earth’s average nighttime temp is higher, our atmosphere is obviously retaining some latent heat capacity through the night.
“Since I can build a steel greenhouse, clearly the greenhouse effect itself has nothing to do with greenhouse gases.”
It’s called the greenhouse effect, not the greenhouse gas effect. The greenhouse gasses are named after it, not the other way around.
And no, you have not built a steel greenhouse. As I pointed out earlier, your figure 2a clearly shows 2W inwards and 3W outwards, for a net of 1W outwards. This does not balance to zero, so you have a second source of heat W within your system.
Willis;
I think much of your post is valid and is a worthwhile contribution to the debate about AGW but I see the the point about the cold tropoause as a massively important key to our understanding of the action of green house gases. Starting from the point of a cold tropopause one can readily show that the Kiehl and Trenberth data is almost certainly wrong and we need to remember that this data forms the basis of the models which are predicting dangerous warming. The K&T data exaggerates the impact of GHG because it claims atmospheric absorption which is much too high and also because it claims that atmospheric absorption and radiation significantly modulates the energy loss to space whereas in fact the concentration of GHG is so high in the atmosphere that they largely block radiation to space at certain wavelengths.
This is easily shown by looking at the energy spectrum as seen from space looking down towards the planet (Nimbus data). What you find is that in the region between 8 and 14 microns the equivalent radiation temperature is the surface temeprature of earth (which means the radiation comes from the surface – nothing else is warm enough) whereas at the GHG absorption wavelengths the equivalent radiation temperature is that of the tropopause. One can even see the comb effect where there are a number of absorbing lines close together (look below 8 microns) and the equivalent radiation temperature varies rapidly with wavelength between surface and tropopause temperature giving a very jagged plot until the lines get so close together that the interferometer cannot resolve them and one gets a very noisy average.