Does Increased CO2 Cause a Decrease in Infra-Red Emission to Space?

Dr. Roy Spencer says:

"....if you add more and more CO2, the effective radiating altitude to space goes ever higher, which is colder, which means less IR radiation, which means a warming tendency for the lower atmosphere."

Let us evaluate this statement, which Dr. Spencer made in a post criticizing this post by Andre Loftus at American Thinker.   Dr. Spencer says that Andre Loftus erred in not considering pressure broadening and therefore increased absorption of the long-wave infra-red radiation emitted by the Earth's surface and this change of infra-red emission to space in the upper atmosphere.  I am only going to address the latter issue in this post.

Now if you have a simple idea that a given number of CO2 molecules are in thermal equilibrium with the atmosphere of the upper troposphere, which cools as the altitude increases, and the increased CO2 moves the source of the final infra-red emission into space to a higher altitude, then the rate of energy emission into space will decrease per molecule.  If the total rate of heat emission to space drops, then somewhere in the Earth system there will be warming.

But let us check out whether such a simple model makes sense.  Among the things we must consider are:

1)  If we increase the number of CO2 molecules, we have more emitters and more emitters might be able to emit as much or more energy into space even if each emitter is emitting less energy.

2)  While it is true that the troposphere cools with increased altitude, if the final emissions are from altitudes such as about 11 km, according to the U.S. Standard Atmosphere this is about the altitude at which the troposphere becomes the tropopause and the atmosphere is no longer cooling with increased altitude.  According to many accounts, most of the final CO2 IR emissions into space are from this altitude or higher already, so added CO2 may not provide much additional final radiation from the below 11 km altitude.

3)  While it is true that most re-emission events of adsorbed long-wave infra-red in the lower troposphere are prevented by collisions with nitrogen and oxygen molecules and with argon atoms, so that the CO2 molecule comes to be in equilibrium with the temperature of the local layer of air, this stops being true in the upper troposphere.  At sea level there are about 6.9 billion collisions/s, while at 11km altitude the number of collisions is only about 1.8 billion collisions/s.  At sea level fewer than 0.2 of the infra-red excited CO2 molecules re-emit infra-red radiation before a collision, but at 11 km more than 0.77 will re-emit any infra-red radiation they have absorbed from lower altitude molecules before they suffer a collision.  This total re-emission number increases with further altitude.  Consequently, only a small fraction of the final emitter CO2 molecules into space will be affected by either the cooler atmosphere around them or a static temperature atmosphere around them as more CO2 molecules are added.

4)  An increase in the number of CO2 molecules in the upper troposphere may result in a warming of the upper troposphere, causing the temperature at the final emission altitude to space to warm from the current profile and making each final emitter molecule in the upper troposphere a more efficient energy emitter.

So, we basically have four cases for a final emitter CO2 molecule:

1)  The molecule is in the upper troposphere where a decreasing temperature gradient exists and was in equilibrium with the immediately surrounding molecules.

2)  The final molecule was in the troposphere, but not in equilibrium with the immediately surrounding molecules.

3)  The molecule is in the tropopause where there is no temperature gradient, but it simply re-emits the radiation it received from a molecule in the top of the troposphere, so it changes nothing.

4) The final emitting molecule is in the tropopause and in equilibrium with the surrounding tropopause molecules.  Only the increase in the number of such cases relative to those of Case 3 will result in any decrease in the efficiency of energy transmission into space per molecule.  This decrease is not proportional since the energy that was transferred to collisions goes into increasing the static temperature of the tropopause.

Let us consider the case which best lends itself to Dr. Spencer's argument.  The final emitter CO2 molecule is in the upper troposphere and in equilibrium with its immediate surrounding layer of air.  This is actually not a very common case, because according to reports, the mean free path of CO2 emissions at the principle absorption wavelength of interest is between 25 and 48 m at sea level.  We will take the greater length of 48 m, since that is the better case for Dr. Spencer's argument.  The mean free path (mfp) is proportional to the atmospheric density assuming a well-mixed CO2 case.  That implies the mfp is 142 m at 10 km according to the relative densities at sea level and 10 km according to the US Standard Atmosphere.  This in turn makes it clear why most final CO2 emitters are in the tropopause and not in the upper troposphere.  Even with a chance of only about 0.2 or less of an absorbing molecule in the tropopause of coming into equilibrium with the surrounding molecules of the tropopause, most emission chains will have many chances to do so.  So most final emitters are already in equilibrium with the tropopause and are not going to change their energy emission efficiency to to an increase in numbers.

Now let us double the number of CO2 molecules in the atmosphere.  Let us assume that half of the present final emitters are at 10 km altitude and half are in the tropopause.  Then we will double the number of CO2 molecules and the mfp becomes half what it was, or about 71 m.  As a result, we will assume that all of the final emitters are now in the tropopause to minimize their temperature.  Let us compare the rates of energy emission into space for these two cases assuming an emissivity of 1 (since it does not matter for the comparison) and ignoring the fact that CO2 emits only a small fraction of the total black body spectrum:

Case 1:

P = (0.5) σ (T at 10 km)⁴ + (0.5) σ (T of tropopause)⁴

P = (0.5)(5.6697 x 10^-8) [(223.25)⁴ + (216.65)⁴] = 132.87 W/m²

Case 2 with doubled CO2:

P = 2(5.6697 x 10^-8) (216.65)⁴ = 249.82 W/m²

Of course these emission powers are proportionally exaggerated for simplicity, since water vapor still plays a final emission role and the emission is not black body emission.  Doubling CO2 results in a 1.88 times increase in the cooling rate of the Earth attributable to CO2 with the ballpark reasonable assumptions made.

The doubling of CO2 causes an increase in radiation into space and hence a cooling of the Earth system.  This is not to say that the surface temperature is proportionally cooled, but the complete system would be cooled.  It is difficult to see what set of assumptions on the altitude of final CO2 emitters would lead to a decrease of CO2 radiant cooling into space.  Even putting all of the present final emitters in the troposphere and keeping them there after doubling CO2 is not going to result in a reduction of infra-red emission by them into space.

The real effect of doubling CO2 is not as dramatically cooling as these calculations show because the upper troposphere and tropopause would surely warm up relative to their present temperature profile.

The quoted argument by Dr. Spencer does not hold up to examination.  There are many reasons, as I have argued frequently, to believe that carbon dioxide has a net cooling effect on surface temperatures and even on the heat of the Earth system as a whole.  In reality, its effect on surface temperatures is very small, for reasons I have discussed elsewhere.