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To a first approximation, the earth currently radiates out as low frequency thermal radiation the same amount of energy as it absorbs as high frequency solar radiation. (This ignores energy generated within the earth, which is also radiated away. But that amount is constant and is not relevant to my question. It also ignores energy stored or burned as fossil fuels.)

Let's assume that global warming will not change the amount of energy received from the sun and absorbed by the earth. (I realize that's not true. Global warming melts the ice caps, which reflect solar radiation. With the ice caps melted, the earth absorbs more of the solar radiation it receives.) But if we ignore the melting of the ice caps, the earth must receive and radiate away a fixed amount of solar radiation, which is independent of its temperature.

I would have thought that a warmer earth would radiate more thermal radiation than a cooler earth. But the argument above says that's not the case. How is this explained?

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It looks to me that you are mixing up the thermal radiation itself with the reflection, these are different terms in the budget. –  Bernhard Jan 24 '12 at 7:36
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There are a lot of details that could be discussed, but the basic point is that "global warming" means that the atmosphere becomes a better insulator, which raises the the surface temperature at which outgoing radiation balances incoming radiation. –  Colin K Jan 24 '12 at 8:13
    
The troposphere is heated by the greenhouse effect whereas at the same time the stratosphere is cooled (it is not called "greenhouse effect in the stratosphere ...). Indepentent of how the absorbed solare radiaton is processed due to radiative transfer by absorption, scattering, emission, etc, the global annual mean of the absorbed solar ratiation must still be equal to the global annual mean of the outgoing long-wave ratiation at the top of the atmosphere in climatological equilibrium. –  Dilaton Jan 24 '12 at 11:58
    
A change of the amount of LW absorbers in the atmosphere can be regarded as an "external prturbation" of the current steady state which leads to a (vertical in a simple model) redistribution of temperature corresponding to the new steady state of the system. –  Dilaton Jan 24 '12 at 12:06
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Colin's comment is spot on, but to expand a bit on the "lots of details" he mentioned, heat radiated from the Earth's surface is partially absorbed by greenhouse gases in the troposphere, and because the troposphere is turbulent this heat gets redistributed throughout the troposphere instead of escaping into space.

If you e.g. double the CO$_2$ content of the troposphere it will intercept and redistribute more of the heat radiated from the Earth, so the Earth will overall radiate less heat into space. Because the Earth is now radiating less heat than it receives, it gets hotter. But as it gets hotter more heat is radiated from the surface and more escapes into space. Eventually the temperature rises until the heat radiated once again matches the heat received, and the temperature stabilises.

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This seems to be saying that a hotter earth can (given an appropriate greenhouse layer) radiate the same amount of radiation as a less insulated but cooler earth. That's what I'm confused about. If you wrap insulation around a hot ball it will initially radiate less. But won't the insulation eventually reach the same temperature as the surface and then start radiating at a higher rate? Can the insulating layer remain cooler than the surface indefinitely? Or is the argument that you can have two materials that radiate at the same rates when at different temperatures? –  RussAbbott Jan 25 '12 at 1:06
    
The CO2 in the atmosphere will scatter heat emitted by the Earth so 50% of the heat absorbed by the CO2 will be radiated back towards the Earth. –  John Rennie Jan 25 '12 at 14:03
    
Where does that leave us? (Also see comments below about CO2.) Are we saying there can be two balls, each receives and radiates the same amount of energy. But one ball is twice as hot (at least internally) as the other. Can that really be a stable state? If so, the insulator for the hotter ball must not eventually heat up to the hotter ball's internal temperature. So the hotter ball must maintain an internal heat gradient. –  RussAbbott Jan 26 '12 at 2:02
    
Well, yes. Take two balls with the same heat production at their core and put them in a vacuum. Obviously at equilibrium they must radiate the same energy as is being produced. Now wrap one of the balls in a blanket and allow it to equlibrate. The surface of this ball, i.e. the ball blanket interface, will be hotter than the unwrapped ball because you now get a temperature drop across the blanket. –  John Rennie Jan 26 '12 at 7:39
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The greenhouse effect has nothing to do with the insulating effect of the atmosphere, as claimed in some of the comments already posted. Nor does the CO2 absorb significantly more solar energy than ordinary air. It is the frequency selectivity that causes the greenhouse effect.

The CO2 is transparent to solar energy, at least the visible component: we can see through it. It is opaqe to thermal energy. The same amount of solar energy is absorbed at the Earth's surface with or without CO2, and the same amount is radiated back outwards by the earth's surface. The problem is that the re-radiated energy is now in the infrared range, so some of it is absorbed by the CO2 and re-radiated back to the surface. This inhibits the effective emissivity of the planet, so the surface has to get warmer in order to acheive thermal balance with the incoming radiation from the sun.

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How is the effect you mention any different than saying that the atmosphere with more CO2 is a better thermal insulator? –  Ron Maimon Jan 24 '12 at 22:04
    
Imagine you could manufacture a kind of magical transparent Styrofoam and cover the earth with it in a layer 100 feet deep. As long as it was truly transparent over the whole spectrum, it would not have the slightest effect on the equilibrium temperature on the surface of the earth. That is why I think it misses the point to blame the greenhouse effect on the insulating properties of CO2. –  Marty Green Jan 25 '12 at 0:03
    
I see your point, it is a greenhouse after all, it has to transmit the solar energy and capture the thermal energy. –  Ron Maimon Jan 25 '12 at 5:42
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