Let us examine a current NASA depiction of the huge role the catastrophic AGW hypothesis gives to back-radiation:
Now notice that the emitted surface radiation is 117% of the top of the atmosphere average radiation from the sun and back-radiation is 100% of the top of the atmosphere incident solar radiation. Notice also that only 48% of the total solar radiation was ever absorbed by the Earth's surface. Of the 48% of the total solar incident energy absorbed by the surface, 25% is lost by evaporation of water and 5% by convection loss, leaving only 18% to be emitted by radiation. Now the proponents of the large greenhouse warming effect claim that the 18% is just a net difference between a larger surface radiation emission rate and a large back-radiation rate. In this case the difference of 117% - 100% = 17%, which is about 18%. We are going to evaluate this claim.
Consider the surface of the Earth for a moment. The density of atoms per cubic meter in the surface is about 1 gram per cubic centimeter for the water that covers about 71% of the Earth's surface and even greater for land materials or if considering the salts in the oceans. Expressed as a density per cubic meter of water, this is 1000 kg per cubic meter. A cubic meter of water has 3.34 x 1028 water molecules. Infra-red emission and absorption in a surface occurs in the outer 2 micrometers of the surface. Consequently, there are about 6.7 x 1022 water molecules emitting infra-red at most from the surface. When infra-red radiation is incident upon the surface, these are the same water molecules that would be able to absorb that radiation.
The density of the atmosphere near the surface at sea level is 1.225 kg per cubic meter in the U.S. Standard Atmosphere at a temperature of 288.15K. Adding water vapor very slightly decreases that density. The number of molecules per cubic meter at sea level in the U.S. Standard Atmosphere is 2.55 x 1025/m3. The important infra-red active gas near the surface is water vapor and its density per cubic meter is commonly between 10 g/kg to 14 g/kg of air as shown in Fig. 1. below. At a specific humidity of 12 g/kg, the number of water molecules/m3 of air is about 4.9 x 1023. Therefore, there are more water molecules in the first cubic meter of air above 1 m2 of water surface emitters by a factor of 7.3. This should mean that radiation that can be absorbed by water vapor will be absorbed in the first meter of air above the surface at a humidity near the Earth average humidity. There is a report that the average mean free length for water vapor absorption in the atmosphere is as long as 8 m. If it actually is that long, then the result I get below on the amount of back-radiation can be multiplied by a factor of 8. That number will still be most unimpressive.
Fig. 1. The average specific humidity of air is shown as a function of latitude. The specific humidity is the weight of water in grams in one kilogram of air.
Returning to the back-radiation caused by water vapor, we find that the temperature difference over the 1 meter absorption range according to the U.S. Standard Atmosphere is only about 0.0065 K/m. For humid air, the temperature gradient is even less. So if the water surface and the water vapor in the first meter of air above the surface are treated as gray bodies, we have a power transfer from the surface to the water vapor in the atmosphere of PW
PW = σ (εs Ts4 – εa Ta4) ,
for the surface s and the atmosphere water vapor a and if we take the emissivities to be a high value of 0.95 often used by the catastrophic man-made global warming promoters, this is equal to 0.033 W/m2.
This is actually an over-estimate because we have ignored the fact that some of the heat energy in the surface is used to evaporate water and some is lost to the air by means of air molecule collisions with the surface. Much of the solar radiation absorbed by the surface is actually transferred out of the surface by evaporation and by conduction. Of the 48% of solar radiation absorbed by the surface according to the above NASA schematic, a total of 30% is removed from the Earth's surface by evaporation and convection/conduction. This leaves only about 18% to be radiated away from the surface. So if the Earth's surface was an interface directly with vacuum, the 48% of solar radiation absorbed would be radiated in its entirety. But other processes, evaporation and conduction, remove most of this energy, leaving a fraction of only 18%/48% = 0.375 of the total energy flux to be removed by radiation.
But water vapor does not absorb infra-red across the entire surface emission spectrum. It actually absorbs only about 65% of it. At least 28% of the surface emission is into the atmospheric window where no atmospheric gases absorb the infra-red radiation. See the transmission spectrum of the Earth's infra-red surface emissions taken by Nimbus IV in 1970 near Guam below:
This points out one of many flaws in the NASA depiction of the Earth Energy Budget above. Taking the two atmospheric windows at higher wave numbers than the peak of the Earth emission spectrum, the infra-red radiated into them is about 28%. 28% of the 117% surface infra-red emission claimed by NASA is 33%, not the 12% quantity their schematic assigns to the atmospheric window radiation lost directly into space from the surface. The direct loss to space of surface emission with the NASA depicted 117% surface emission should be 2.75 times what NASA puts in its Earth Energy Budget schematic diagram at the start of this article. This inconsistency suggests that the surface emission radiation is actually much less than the 117% claimed. If that emission is much less, then the back-radiation would also have to be much less.
So the actual water vapor absorbed infra-red radiation is only about
PW = (0.375) (0.65) (0.033 W/m2) = 0.0080 W/m2.
But because of the 6.9 x 109 collisions/s in air near sea level, most of this energy is transferred to non-radiating nitrogen, oxygen, and argon. Only about 20% is re-radiated and half of that is radiated toward space. Consequently, the total back-radiation which can be absorbed by the surface, PB, is about
PB = (0.2) (.5) (0.0080 W/m2) = 0.00080 W/m2
Thus, the absorbed back-radiation has an upper limit of about 0.00023% of the average solar insolation at the top of the atmosphere (342 W/m2)! For all intents and purposes, the absorbed back-radiation is zero. This value should be multiplied by the mean free path for absorption by water vapor, which may be as large as 8 m on average.
I just proceeded down the true path of analysis dictated by the use of Occam's Razor. Let us assume that NASA is right that 117% of the solar incident radiation at the top of the atmosphere actually is emitted from the surface due to some unknown extra input energy that lifts the 48% of the solar absorbed energy flux at the surface by 117% + 25% + 5% - 48% = 99% so that one can have 117% radiation, 25% evaporation, and 5% conduction cooling mechanisms at work. Is it possible for that 99% warming source to be back-radiation?
The 117% radiation is greater than the 18% upon which I performed the Occam's Razor version of the calculation. In fact, the back-radiation per meter of mean free path is then 117% / 18% = 6.5 times greater than the value I calculated. So, if we assume that the average water vapor mean free path is 8m, we get 8 (6.5) (0.00023%) = 0.012%, which does not look at all like the 100% claimed by NASA.
Because back-radiation to the surface is insignificant compared to the claims made by the proponents of catastrophic man-made global warming, the mechanism upon which that theory stands is wrong. Indeed, adding carbon dioxide to the atmosphere will actually cause more incoming solar radiation to be absorbed by the atmosphere before it reaches the surface. This results in a cooling of the surface. In addition, more CO2 in the atmosphere near the surface will also cause the temperature gradient in the atmosphere to become slightly smaller, just as infra-red absorbing water vapor makes it smaller. This is because radiation transport effects operate at the speed of light, which is faster than evaporation/condensation transport or conduction/convection transport of energy. Since all remove energy from the surface, they are all cooling effects. As a result, adding CO2 to the atmosphere actually causes a very slight cooling of the surface, contrary to the claims of a substantial warming effect.
The primary sources of CO2 in the atmosphere are warming oceans, decaying plant life, and heat vents and volcanic emissions. Since CO2 in the atmosphere creates a slight cooling of the surface, it acts as a negative feedback to the warming oceans that cause it to increase and it slightly cools the decaying plants to slow down the further generation of CO2 from that source. Of course volcanic and heat vent sources of CO2 are also providing heating, so carbon dioxide as an surface coolant acts to stabilize the Earth's temperature much as water vapor does. It has negative feedback effects. It is no more subject to the sort of tipping point catastrophes that global warming alarmists put out than is water vapor, though its effects are much, much weaker.
This article was last updated on 29 July 2014.
[The fact that back-radiation itself is insignificant in explaining the 33C warming of the surface claimed by the proponents of catastrophic AGW is not at all to say that the effects of water vapor on the surface temperature are without significance. During the day the evaporation of water is an important surface cooling effect and during the night its condensation can be an important warming effect. During the day, the infra-red active gases absorb a portion of the incoming solar radiation and prevent it from being absorbed by the surface. At night, fog and clouds scatter, reflect, and absorb the infra-red radiation emitted from the surface. Also at night, the small true back-radiation effect is not off-set by the absorption of incoming solar radiation by the same infra-red active molecules in the atmosphere. There is a very important moderation of the surface temperature between day and night due to these effects. This note was added on 13 November 2014 to remind people of the context in which my criticism of the usual greenhouse gas theory is framed.]