# Need Help: Can You Model CO2 as a Greenhouse Gas (Or is This Just Wishful Thinking?)

by Ray001
Tags: carbon dioxide, experiment, global warming, greenhouse gas, temperature
P: 1,750
 Quote by Evo To both sylas and chris, claiming to be on the "right side" and determining who is not on the right side isn't going to fly here. Either post without claiming superiority or you will be deleted. Same goes for anyone claiming to have superior knowledge to anyone else.
All my posts have consistently been based on physical arguments. I refer to experiment, measurement, scientific paper and texts, and basic theory and calculation. I don't argue from credentials, for me or others. I have, given the above concerns by various individuals, noted a bit of my own policy for the record, but as far as I am concerned, this entire page of discussion has gone way off track; and it was not driven in that direction by me.

In response to some questions recently about claims for greenhouse gases leading to cooling of the atmosphere, I have explained why those claims are incorrect from straightforward physics, and with reference to an easily accessible online text book used in teaching atmospheric physics. THAT is the basis of my argument.

There was a citation to a paper which claims the reverse. I was asked explicitly about it. I have stated where it goes wrong, as requested, with reference to the actual content of the paper. That paper is not by a climate scientist -- since YOU are the one asking about credentials here for some reason -- but any argument from anyone stands or falls on its own actual physical merits.

It is entirely proper given conflicting claims to determine which claim is right with reference to the actual physical merits of those claims. That is what I have done.

We get excellent input here on many topics from people who are not professionals, and from people who are professionals, and in this and other forums contributions here are based on their merits and not on who writes them. Professionals often give the best responses, but that is never merely presumed.

Chris has given a really first rate summary of background issues above. He's quite well known on this topic, and maintains one of the better blogs on this topic, that deals with the technical science of climate science. He has received some public recognition of his ability and thoroughness in the technical side of climate science from practicing climate scientists; but he himself is a student. A damn good one, it seems to me.

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I appreciate people's concerns, and find them baffling. My aim has always been been to give argument based on explicit physics, with explicit and cited reference to experiment, measurement, theory and calculation. I aim to not only abide by the rules and conventions of this forum, but to go above and beyond them. I welcome any official input from mentors; and you may do that either privately or publicly as you see fit.

If I am told that I have to declare my credentials, I will object. I do not claim credentials. I have some background and I would guess I know more about the relevant climate science than most contributors here, but I don't presume on that or ask others to presume on that either.

If I am told I must not even mention the fact that a cited paper is incorrect, or that the author is writing outside his field, I will object. I accept that empirical or scientific claims must be supported, and I do support them.

The topic of climate science is a hot button topic these days. It is pretty much impossible to tackle the subject without running into accusations of being a fraud or an incompetent, no matter what case you are presenting. I've had it myself, and I prefer to ignore it, mostly.

I understand all your concerns. I have aimed, always, to keep to precisely the position you are arguing for. That is, to avoid argument based on credentials, or authority, or anything of that kind. I will mention when someone uses a reference that has no standing by the rules of the forum; this is a sensible rule in an area like this. In this case, we had a reference which IS permitted by the rules of the forum, but is nevertheless in truly fundamental physical error. In such a case, I give a substantive response to the claims. But it is also relevant in that case to note that the paper does not have much standing, and it is entirely sensible and useful to say so. But the major basis of response remains the actual physics.

I am working on a post which gets back to that in a big way, but it is hard work. I am pretty much having to write a tutorial on some of the relevant physics. I think it will be useful.

Sylas
P: 59
 Quote by Evo To both sylas and chris, claiming to be on the "right side" and determining who is not on the right side isn't going to fly here. Either post without claiming superiority or you will be deleted. Same goes for anyone claiming to have superior knowledge to anyone else. Chris, are you a climate scientist? A search doesn't bring up any affiliations for you.
This is not appropriate, and I would be happy to discuss this with another (impartial) moderator privately.

In fact, neither sylas or myself have ever claimed any special authority, have never claimed to be "on the right side," have welcomed substantive challenges and corrections insofar as they are in accord with forum guidelines, and sylas has specifically said that he doesn't expect anyone to take his (or my) word on anything on its own merit. Sylas has admitted to minor errors where applicable, and has done an excellent job laying out the basic thermodynamics behind the greenhouse effect and the radiative-convective balance which constrains the global climate. He has openly stated he is not a climate expert, and for disclosure, neither am I...I'm actually a student of the atmospheric science, and like him, have a strong interest in the science (as a hobby) and in discussion of the relevant physics.

Much of what we discussed has been done properly and with suitable references where possible, or is basic undergraduate-level physics which can be found in standard textbooks. No one goes to the homework forum to tell people "they are acting superior" for instructing others how to take derivatives or how to calculate the net force on an object, since physicsforums is an outlet to share knowledge and ideas. I remain more than happy to address disagreements or questions pertaining to what I've written.

I don't want to speak too much for sylas...but from my observation, the worst thing we've done is made our opinions known about the quality of certain references (e.g., Khilyuk and Chilingar; Gerlich and Tscheuschner), and others have expressed friendly disagreement with our approach, and I don't mind. But aside from andre, the only real scientific replies have been from someone telling us the greenhouse effect is a hoax, and another who continues to insist no experiment exists to substantiate a CO2 greenhouse effect (and is therefore a hoax) in the face of numerous correction. As such, I find it an odd situation that you choose to target us for violation of guidelines or inappropriate dialogue.

Edit-- I did write this after sylas and independent of him.
 Mentor P: 25,928 That's all that is needed. We have already discerned that no posters here have proper credentials. If you did, we would have gladly recognized you as such. We have been debating if we should just close this forum down due to lack of anyone knowledgeable enough to moderate it.
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 Quote by chriscolose This is not appropriate, and I would be happy to discuss this with another (impartial) moderator privately.
Feel free to send me a pm.

Welcome to our hornets nest, Chris.
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 Quote by chriscolose I'm not quite sure I understand. This diagram (from Trenberth, Fasulo, and Kiehl) shows global energy flows. If you're referring to the surface, it loses heat both through radiation and convection.
No, what I mean is this. Suppose that you have a gas layer (atmosphere), "cold empty space" above it, and a black (or grey) "surface" beneath it. Now, suppose that you heat the surface, and you want to know what is the relationship between power lost by the surface and temperature of the surface. If there weren't any gas layer, this would be Stefan's law. If the gas layer is entirely transparant, this would also be the case. So we consider the case where the gas contains absorbers. You could consider Stefan's law as a kind of "thermal resistance of space".

Now imagine that the gas layer is stationary - no convection. We "glue the gas in place" so to say. We now have a purely radiative transfer through the different layers of the gas, and this will give us a greenhouse effect: the thermal resistance is higher now, we need a higher surface temperature to radiate away the same amount of heat, or, equivalently, for the same temperature, you radiate away less heat. We added a series resistance representing the radiative thermal transfer through the atmosphere.

Intuitively, I would have thought that if there was now on top of that, convection, that the thermal resistance would lower with respect to the previous case. I picture that as having put in parallel to the radiative transfer thermal resistor, a convective resistor. An extra path for heat to go from the surface to the empty space.

Other possibility, we consider only convection, and no radiative transfer. We could think of putting "reflecting foils" between the gas layers, so that there is no radiative coupling between layers. We only consider material transport (a convective flow) to do the heat transport. Intuitively, I would expect the thermal resistance to be higher than in the previous case, as now the radiative resistor is taken away.

As I said, this is how I would intuitively picture things, before doing any calculation. I'm not saying things are like this, I'm just saying that if things aren't like this, it wasn't intuitively clear to me. I went in deeper details to explain you my puzzlement, because you asked me.
 P: 59 vanesch, Unfortunately the circuit analogies are a bit over me, so I'm still unclear as to what you're getting at, especially with the concept of "thermal resistance." It may be useful to consider the top of the atmosphere energy balance as a separate entity as the surface energy balance. The former, at equilibrium, is the solar constant*co-albedo*0.25 = sigma T^4 The planet does not lose heat by convection to space. However, the latter involves not only radiative transfer but also the sensible and latent heat fluxes which couple the atmosphere to the surface. The former is essentially the driver of planetary climate change, while the surface budget serves to regulate the gradient between the surface and overlying air. With no convection, the surface would be considerably warmer and the atmosphere much colder
P: 1,750
 Quote by vanesch Now imagine that the gas layer is stationary - no convection. We "glue the gas in place" so to say. We now have a purely radiative transfer through the different layers of the gas, and this will give us a greenhouse effect: the thermal resistance is higher now, we need a higher surface temperature to radiate away the same amount of heat, or, equivalently, for the same temperature, you radiate away less heat. We added a series resistance representing the radiative thermal transfer through the atmosphere. Intuitively, I would have thought that if there was now on top of that, convection, that the thermal resistance would lower with respect to the previous case. I picture that as having put in parallel to the radiative transfer thermal resistor, a convective resistor. An extra path for heat to go from the surface to the empty space.
Your intuition here is probably correct; but it's not quite that simple.

The purely radiative case in a "glued" atmosphere will have have a certain "greenhouse" effect, and you get a certain temperature gradient.

Now add convection... this will alter the lapse rate. Whether the greenhouse effect is enhanced or reduced will depend on whether the radiative gradient is more, or less, than the adiabatic lapse rate which convection gives you. For Earth, I think convection will tend to reduce the greenhouse effect.

On the other hand, previous discussion has taken this the other way around. Suppose you have a radiatively neutral atmosphere. That will develop an adiabatic lapse rate, from convection. Now add radiative transfers. That will have only a small effect, if any, on lapse rate. The adiabatic lapse rate will be maintained by the effects of convection. But the effective radiating altitude will increase, and bring in a greenhouse effect with a warmer surface and warmer atmosphere.

I'll go into this a bit more, with reference to the text on planetary climate I have mentioned, when I finally get my next major technical contribution complete. I have to crunch some numbers to be sure of what I am doing as well.

Cheers -- sylas
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 Quote by chriscolose vanesch, Unfortunately the circuit analogies are a bit over me, so I'm still unclear as to what you're getting at, especially with the concept of "thermal resistance." It may be useful to consider the top of the atmosphere energy balance as a separate entity as the surface energy balance. The former, at equilibrium, is the solar constant*co-albedo*0.25 = sigma T^4
That's maybe not so obvious, as the atmosphere is partially transparant. I agree with you that convection by itself won't cool anything to outer space ; the only way to do so is of course radiation.

But the way I picture it in my head is that each layer, even the earth surface, can partially emit directly to outer space, and partially transmit heat to other layers. This last process can be radiative, but also convective. It is not only the upper atmosphere which radiates into outer space, I would think, because the opacity is not total (it would be, if the atmosphere were totally opaque, which it is for certain wavelengths ; then for others, the radiation depth is probably rather large - I don't know these numbers by heart).

So in my idea, any process that "gets heat easier to the upper layers" lowers the thermal resistance (allows for a higher heat flux for a given surface temperature).

 With no convection, the surface would be considerably warmer and the atmosphere much colder
Eh, yes. That was what I was intuitively trying to say. I had, erroneously probably, understood from sylas' post that convection didn't affect (or affected aversely) the heat transport, and that was against my intuition - which is limited, I grant you that.
P: 1,750
 Quote by vanesch Eh, yes. That was what I was intuitively trying to say. I had, erroneously probably, understood from sylas' post that convection didn't affect (or affected aversely) the heat transport, and that was against my intuition - which is limited, I grant you that.
The lapse rate (rate at which temperature falls with altitude) is independent of thermal emissivity. Almost. There will be small second order effects.

Hence, if the radiative transfers are small, the net vertical energy transport in the atmosphere will be small. You can't have a sustained trend up or down, because there's no source or sink for the energy. Hence, without radiation transfers, convective heat transport works to maintain a lapse rate, but it does so being sometimes with energy flowing up, and sometimes down, and with no sustained trend.

Now add radiation transfers. Because of the lapse rate, the immediate effect is an upwards flow of energy, by the second law, from warmer parts to colder parts; and there is energy being lost altogether out from the top of the atmosphere. But it's not completely clear whether there is heating or cooling at a given level. Each level of the troposphere is warmer than the level above, and colder than the level below. Every level is emitting both up and down, according to its temperature. So any level will on balance lose energy by radiation to the level above, and gain it by radiation from the level below.

If there is an imbalance at any level, additional convection will apply to oppose the heating or cooling at that level, and move towards the adiabatic lapse rate again.

At the very bottom of the atmosphere, of course, the upwards radiation is from the surface; and surface has emissivity close to unity. The immediate effect of radiant transfers in the atmosphere, therefore, is a flux of thermal radiation (called backradiation) coming down to the surface which wasn't there before. And the surface will heat up, and the radiant fluxes will increase as well all up and down the column; with convection always working towards the adiabatic lapse rate.

End result; an atmosphere with a lapse rate very close to the adiabatic rate (observed) and hence an atmosphere which is hotter by a similar amount as the surface. Any changes in lapse rate, whichever way they go, have less impact that the fact that the whole thing is hotter at the start point. The net flow of radiant heat is up. The net convective flow is also observed to be upwards, and so I guess this means the net effect of radiant fluxes tends to be towards cooling upper levels and heating lower ones by comparison with prevailing temperatures; but that can't mean that adding radiant transfer gives a cooler atmosphere. The whole thing is a response to heating, and temperatures don't just depend only the lapse rate, which doesn't actually change much anyway. It's crucial that the bottom of the whole stack heats up to shed the atmospheric backradiation. Humidity feedbacks impact lapse rate, but a feedback can't change the sign of the net effect.

Cheers -- sylas
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 Quote by sylas Hence, without radiation transfers, convective heat transport works to maintain a lapse rate, but it does so being sometimes with energy flowing up, and sometimes down,….
Energy flowing down? But how? Convection brings more air to upper levels, disturbing the normal atmospheric pressure distribution. So obviously on other places air has to descend for counter balance balancing anywhere else. Now Isn’t it that a parcel or air descends because of its buoyancy is less than its surroundings? Hence isn’t its relative temperature/ energy lower than the not descending air around it. So it would seem that this descending air returns less energy than the convecting air is withdrawing? Hence a net energy flow up incomplete convection cell with both up and downdraft? Under which conditions could that be different?

 …Each level of the troposphere is warmer than the level above, and colder than the level below. …
I would be busted for misinformation if I said something like that.

Especially in subsidence inversions temperatures aloft can be considerably higher than below.
P: 1,750
 Quote by Andre Energy flowing down? But how? Convection brings more air to upper levels, disturbing the normal atmospheric pressure distribution. So obviously on other places air has to descend for counter balance balancing anywhere else. Now Isn’t it that a parcel or air descends because of its buoyancy is less than its surroundings? Hence isn’t its relative temperature/ energy lower than the not descending air around it. So it would seem that this descending air returns less energy than the convecting air is withdrawing? Hence a net energy flow up incomplete convection cell with both up and downdraft? Under which conditions could that be different?
The quantity you want is "potential temperature". It's described in chapter 2 of the text on planetary climate I mentioned for you last time. There's more to the energy of a packet of air moving up or down than its measured temperature. You also need to consider the pressure difference, for example.

But the case we are speaking of here is particularly simple. It is what you described earlier as a radiatively inert atmosphere. That is also explicit in the very first sentence you have quoted in my extract. In that case, the only energy flows to worry about are convection and latent heat (sometimes bundled together). And that has to add up to zero by conservation of energy. Do you agree?

Now of course, it will vary from time to time, but the net will be zero. Sometimes the energy flow is up, sometimes it is down. First law of thermodynamics, applied to the case you proposed.

Even on Earth, you can sometimes get local convective energy transport downwards; although the net is upwards, estimated at about 17 W/m2, plus 80 latent heat, in the energy flow diagrams that have been cited.

 I would be busted for misinformation if I said something like that.
I hope not! It's not illegal to make errors (which means misinformation.) You'll be picked up for errors by other posters, but it's not against forum rules to be wrong about information.

However, in this case you have simply failed to look sufficiently carefully at the specifics of the case described. If you had quoted the entire paragraph, this was explicitly in the context of the standard lapse rate.

The main conclusion of the description I gave was that it's not immediately clear whether there is a net radiative heating or cooling at a given level. On Earth, on balance, it is generally a radiative cooling effect I think, but in the context of a net upwards radiant flow and a significantly raised overall temperature from what you have without the radiant transfers. Upwards convection is also strengthened when radiant transfers are present. The whole response of the planet to an atmosphere that interacts with thermal energy is that it has to work harder to get rid of the same amount of energy, and on Earth this results in surface temperatures that are, on average, about 33 degrees higher than the effective radiating temperature of the planet into space, and about that increase also up through the atmosphere as well; though not uniformly.

Cheers -- sylas
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P: 5,450
 Quote by sylas The quantity you want is "potential temperature". It's described in chapter 2 of the text on planetary climate I mentioned for you last time. There's more to the energy of a packet of air moving up or down than its measured temperature. You also need to consider the pressure difference, for example.
Right, but it still requires more density for air to descent, if that air is containing more energy it must at an higher ambient temperature and/or a higher pressure. If it is at an higher pressure, it will expand and decreases in density increasing it's bouyancy, stopping the downdraft. I still can't see how in a complete convection cell the net energy flow can be downwards.

There is still more air convecting up also transfers (heat) energy into potential energy, which process is reversed in descending air

I would be busted for misinformation if I said something like that.

 I hope not! It's not illegal to make errors (which means misinformation.) You'll be picked up for errors by other posters, but it's not against forum rules to be wrong about information
Not if one is a declared crook according to the moral panic principle.

 However, in this case you have simply failed to look sufficiently carefully at the specifics of the case described. If you had quoted the entire paragraph, this was explicitly in the context of the standard lapse rate.
Here is the full quote

 Each level of the troposphere is warmer than the level above, and colder than the level below. Every level is emitting both up and down, according to its temperature. So any level will on balance lose energy by radiation to the level above, and gain it by radiation from the level below. If there is an imbalance at any level, additional convection will apply to oppose the heating or cooling at that level, and move towards the adiabatic lapse rate again.
A subsidence inversion is the norm above the deserts, in the downdraft regions of the hadley cells, as the descending air increases in density and heats up adiabatically, additional convection would be extremely rare and certainly not the norm.
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Thanks for the explanations. I will try to digest them "out loud".

For all clearness, I'm not talking about earth's atmosphere (yet), only about "plateworld"s atmosphere: a black hot plate, a gaslayer on top of it, and outer space. Some magic to do things with the gas, which is normally not possible, such as switching on and off convection, radiation, and a few other things, to get an understanding of the different mechanisms and their interplay.

We start by giving our gas layer (with some magic, or gravity) a pressure profile, with pressure decreasing with altitude. I will try to see where I get.

 Quote by sylas The lapse rate (rate at which temperature falls with altitude) is independent of thermal emissivity. Almost. There will be small second order effects.
Ok, so what's understandable from this, is that if the pressure profile is given, the relative temperature curve is given if there is sufficient convection. A balloon with some gas at temperature T1, lifted in this atmosphere, will cool down adiabatically (because the pressure lowers, and the balloon expands). A balloon going down will heat up (compression). If the atmosphere is not heated or cooled (no radiative stuff in it), it would reach a certain equilibrium given by this adiabatic.

Let's play a bit with this non-radiative atmosphere. Let us say that at the surface, I've 10 degrees, and at 20 km, I have -50. (making numbers up here).
Now, suppose that with electric heaters, I bring the layer at 20 km at -30. This would mean that the less cold air at 20 km, going down in the convective stream, will now bring the surface layers to a much higher temperature (say, 40 degrees, following the adiabat from -30 and compressing). The whole atmosphere will now settle to a new equilibrium, again with an adiabat, but with the top layer now at -30, and the surface at 40.

Right. This is something I didn't realise that convection could transport heat down against a temperature gradient.

Let's play another game (this is fun!). Suppose that thermal conductivity of our atmosphere is very bad but not 0. Still no radiative stuff, we're just looking at the transparant atmosphere. We switch off the EM field (I told you we had magic!).

Now, we do the following: our initial surface is at 10 degrees, the top of the atmosphere is at -50, and there is this adiabatic equilibrium due to convection (which is driven also by magic).

Suppose now that we build a huge heat exchanger at 20 km height, and another at the surface. Suppose that the surface has a thermostat that keeps it at 10 degrees, but heat can be supplied or extracted. It's a thermal reservoir. Now, we connect our two heat exchangers with some or other liquid. We take heat from the soil at 10 degrees, and bring it to the upper layers to heat the upper layers. This is possible, because up there, it is -50.
We do this until the upper layer is now at -30. We are in the same situation as before, so now the lower part of the atmosphere is hotter than the surface !

There is something wrong. We violated the second law here: we took heat from the surface at 10 degrees, delivered it to our gas at -50 (still ok), and this heated the air to 30 degrees just above the surface. So the whole cycle took heat at 10 degrees and delivered it at 30. That's against the second law (unless we do work). So, the problem was that I introduced too much magic, and introduced convection even when the upper temperatures were above the adiabat. That forced convection (my magic) did work on the gas.

I guess that if upper layers, in one way or another, are hotter than they should be according to the adiabat, convection simply stops.

So it seems that you can't heat "downwards" using convection, no ? Violates the second law, no ?

So where's the culpritt ? I would guess that it comes from thinking that a hotter gas can convect down in a cooler gas. It will be less dense, so it will have tendency to go up, not down.

So, convection cannot really take heat down, can it ? In other words, the adiabat is defined by the temperatures in the lower layers, not in the upper layers. Am I right here ?

 Hence, without radiation transfers, convective heat transport works to maintain a lapse rate, but it does so being sometimes with energy flowing up, and sometimes down, and with no sustained trend.
So, is this true ? What about my above example ?

Or is it rather: convection will transport heat up, and if it should transport heat down, it stops.

I will stop here already (didn't know it when I started typing) because I'd rather sort this out clearly before going on.

edit: I was typing this independently from the discussion with Andre, but it seems he butted on a similar difficulty after reading the exchange...
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 Quote by Ivan Seeking Feel free to send me a pm.
He said impartial Ivan.
P: 59
 Quote by vanesch That's maybe not so obvious, as the atmosphere is partially transparant. I agree with you that convection by itself won't cool anything to outer space ; the only way to do so is of course radiation. But the way I picture it in my head is that each layer, even the earth surface, can partially emit directly to outer space, and partially transmit heat to other layers. This last process can be radiative, but also convective. It is not only the upper atmosphere which radiates into outer space, I would think, because the opacity is not total (it would be, if the atmosphere were totally opaque, which it is for certain wavelengths ; then for others, the radiation depth is probably rather large - I don't know these numbers by heart). So in my idea, any process that "gets heat easier to the upper layers" lowers the thermal resistance (allows for a higher heat flux for a given surface temperature). Eh, yes. That was what I was intuitively trying to say. I had, erroneously probably, understood from sylas' post that convection didn't affect (or affected aversely) the heat transport, and that was against my intuition - which is limited, I grant you that.
Vanesch,

What's happening with the greenhouse and convection can be understood at a wide variety of levels. In actuality, constructs like S(1-a)/4 = sig T^4 are not actually used in sophisticated GCM's, but such simple formulas valuable way of explaining the basic physics of radiation balance, and serve as a bridge between grey-gas models and more realistic models.

One classical "layer model" which is often employed at a lower level allows one to think of several imaginary "panes of glass" floating in the atmosphere which are perfectly transparent to visible and perfectly opaque to infrared radiation.

One can then proceed to set up multiple equations and go about solving for the temperature at each layer. Generalizing, the surface temperature will end up being the top temperature (the emission layer) multiplied by (n + 1)^0.25 where n is the number of layers. So, a two layer atmosphere will have a surface temperature of 335 K. This suggests that radiative equilibrium is not a good approximation for the surface temperature, which loses substantial heat by convection and conduction as well. With radiative equilibrium, the lapse rate of temperature too large in the troposphere, the stratosphere is approximated pretty good, but the surface is too hot. With other forms of heat transfer now-- The whole troposphere is well mixed in heat, and is more or less constrained by convection to stay near the moist adiabat. In that sense, the vertical structure is largely fixed by convection and the IR heating simply sets the intercept (e.g. the lower tropospheric temperature).

In actuality, the atmosphere is semi-transparent to a differing degree at different wavelengths. The radiative transfer issue is best addressed numerically with sufficient number of vertical layers to resolve the atmospheric temperature and absorber distributions, and with sufficient resolution to pin down the spectral dependence of individual gases. Looking down from space you would indeed see radiation coming from various levels of the atmosphere, but the bulk of it comes from some location (determined by the atmospheric greenhouse composition) where opacity is strong. This is often called the $$\tau = 1$$ level and much of the radiation below here is absorbed before getting to space and much radiation from above is a small term as emissivity is weak. The effect of adding CO2 is to raise $$\tau = 1$$ to higher altitude (lower pressure) thereby warming the whole troposphere. So while a real "effective layer" doesn't exist, it's a usefuil concept for thinking about the radiation balance of the planet.
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 Quote by chriscolose One can then proceed to set up multiple equations and go about solving for the temperature at each layer.
Yes. However, if you take "black sheets" each time, with no radiation that transmits through a layer without being "thermalized", then you have actually a series of independent "resistors" (except that "ohm's law" is not linear but for small temperature diffs we can linearise).

If we take one such element, with on one side T1 and on the other, T2, we have a net power transmission between them of sigma (T1^4 - T2^4).

Assuming small temperature differences, we have approximately 4 sigma T^3 (T1 - T2), and we can roughly say that one such "interlayer" corresponds to a thermal resistance of
$$\frac{1}{4 \sigma T^3}$$

(current = 1/R x potential difference)

The different successive layers are series connections of these resistors.

So the more of these the radiation has to cross, the higher the total resistance, and hence the higher the temperature difference for the same thermal flux ("current").

The more radiatively absorbing gasses you have, the more of these "black" layers we have.

In fact, you also have to count the last "gap" towards outer space as a resistor of the kind, but here you can for sure not linearize anymore as T2 = 0 (or 4 K if you want to).

So this explains intuitively the greenhouse effect in a layered, static atmosphere.

I'm trying to wrap my mind around what is the influence of convection in this picture.
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(In what follows, my main reference is an online undergraduate level textbook, Principles of Planetary Climate, by Ray Pierrehumbert. You can find the same material in other texts; this one has the advantage of being easily available as a shared reference by anyone who wants to look it up. If I refer to page numbers or equation numbers; they are from this text. I take full responsibility for any errors in my answers, and you can check the methods I apply with this reference.)
I have made an error in my posts; I spoke of convection working to maintain the lapse rate. That is incorrect; I should have said convection works to decrease a lapse rate towards the adiabat, or near it. A weaker or negative lapse rate is stable against convection, so convection in the atmosphere only heads towards this point from one direction.

We've been considering the case of an "optically thin" atmosphere, with minimal thermal emissivity. If there is no energy exchange with the surface, then the atmosphere in this simple case would be isothermal (one temperature) at 2-0.25 = 0.84 of the surface temperature. (p 142) This is called the "skin temperature". But because there is an energy exchange where the atmosphere is in contact with the surface, the bottom of the atmosphere is heated by the surface; and this proceeds up the atmospheric column to establish a temperature gradient, up to the point where the "skin temperature" is again established, and from there you get an isothermal stable stratosphere (p 143). Radiant energy transfers make life more complex; but the radiatively inert case in this example is simpler. Here's a diagram from the book.

The lapse rate is therefore directly linked to the height of the tropopause, given a surface temperature balanced with solar input, and a stratosphere at the skin temperature. If the tropopause is at low altitude, the mean lapse rate is large; and unstable. This leads to heating, by transfer from the surface and then by convection; that raises the tropopause, until you get to the adiabatic lapse rate, which is now stable.

Given the small loss of energy from the atmosphere, on the assumption of being optically thin, or radiatively inert, this equilibrium state has negligible net flow of energy up or down, and that does mean convection processes will sometimes transport energy up, and sometimes down, however this is presuming that you have a bit variance in conditions, rather than always being right at an equilibrium.

I think you really only get energy flux downwards by "forced convection", or a mechanical result of wind or other movements that do work. But I'll accept guidance from others on this point. The major point is that the net upwards energy flux into a radiatively inert atmosphere is zero.

 Quote by Andre Right, but it still requires more density for air to descent, if that air is containing more energy it must at an higher ambient temperature and/or a higher pressure. If it is at an higher pressure, it will expand and decreases in density increasing it's bouyancy, stopping the downdraft. I still can't see how in a complete convection cell the net energy flow can be downwards.
Net isn't downwards. In the radiatively inert case, the net is effectively zero. But that suggests that as you introduce a bit more complexity, like horizontal wind and so on, there's going to transient periods of a downward flux of energy, occasionally, with other periods of a net upwards flux.... sometimes up, sometimes down. I know that a downwards energy flux is not a stable situation.

And in particular, any implication I gave that you get a spontaneous movement of energy downwards against the temperature gradient of the conventional lapse rate was my mistake. Such movement may occur, I believe, as a result of mechanical work from winds, but they are not sustained.
 Not if one is a declared crook according to the moral panic principle.
I don't panic, and will be happy to back you up if you get unfairly disciplined for simply being presumed to be in error. I do appreciate your substantive engagement, whether I agree with it or not.

Quote by Andre
Here is the full quote

 Quote by sylas Now add radiation transfers. Because of the lapse rate, the immediate effect is an upwards flow of energy, by the second law, from warmer parts to colder parts; and there is energy being lost altogether out from the top of the atmosphere. But it's not completely clear whether there is heating or cooling at a given level. Each level of the troposphere is warmer than the level above, and colder than the level below. Every level is emitting both up and down, according to its temperature. So any level will on balance lose energy by radiation to the level above, and gain it by radiation from the level below. If there is an imbalance at any level, additional convection will apply to oppose the heating or cooling at that level, and move towards the adiabatic lapse rate again.
Um... with respect, you continue to omit the initial sentences of the paragraph which describe the lapse rate being assumed. I have taken the liberty of inserting the rest of the paragraph in bold, where the lapse rate is mentioned explicitly; and adding the link to the post through the quote tag.

It's quite true that you can get inversions within the troposphere. They tend to be of a limited depth; less than a kilometer. The majority of the troposphere in a real planet is still with the positive lapse rate (falling temperature with altitude) and this is the case in the simple example I was explicitly discussing.

The point is that in the troposphere, any layer tends to be gaining radiant heat with respect to lower levels, and losing it with respect to higher levels, by virtue of the temperature gradient that occurs in the troposphere, and so you can't presume net heating or cooling immediately.

However, I can't quibble too much here, because the big error here in my post is in the second paragraph you've quoted. Convection does not necessarily work to oppose heating or cooling. It is only an overly large lapse rate above the adiabatic rate that is unstable to spontaneous convection.

This doesn't alter the main point that increasing the capacity of an atmosphere to interact with thermal radiation will give higher temperatures at the surface and in the troposphere; even though the normal equilibrium at those higher temperatures may show radiative cooling at that level, balanced by the special heat flux.

But it's still another screwup, and I am glad to acknowledge it and fix it. Thanks.

A nod to vanesch as well for picking up the problem also; you get the kudos for being first, and I sneak in third.

Cheers -- sylas
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You're too kind Sylas

 Quote by vanesch So the more of these the radiation has to cross, the higher the total resistance, and hence the higher the temperature difference for the same thermal flux ("current")....The more radiatively absorbing gasses you have, the more of these "black" layers we have.
Would the band saturation decrease that effect? If that frequency band is 'saturated' it appears that it won't make that much difference anymore how many times radiation energy is absorpted and re-emitted.

 I'm trying to wrap my mind around what is the influence of convection in this picture.
Ah let's try some ideas, especially with wet convection, involving latent heat. So, as a wet surface heats up, water evaporates (latent energy -which reduces the temperature increase). Conduction and radiation heat up the lower layer(s) of the troposhere, causing the well discussed convection. Heat- and latent energy -water vapor- are now transported up. Due to expansion the updraft cools adiabatically and water condenses forming clouds and releasing the latent heat again. Clouds are good radiators as they radiate on all water IR- frequencies. So this energy is radiated outwards in al directions as it would have done on the Earth surface without convection. But the difference is that energy -on water frequencies) emitted upwards will find less water vapor molecules because the upper levels are much drier than the surface levels. Evidently, the CO2 frequency bands are also less relevant here. Consequently the energy emitted by clouds (tops), on water frequencies, has more chance to escape into space than energy emitted by the surface in all bands including the CO2 frequencies.

Now if the greenhouse gas concentration was to increase then the heating of the lower atmosphere by radiation was also to be increased, this would enhance the convection rate, transporting more energy upwards, where more energy can radiate into space. Consequently it appears that convection acts as a negative feedback on greenhouse gas variation

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