A guest post by Dr. Weinstein in response to a post by Dr. Spencer.
EFFECT OF ATMOSPHERIC MASS AND CO2 ON GROUND TEMPERATURE
Leonard Weinstein, ScD
January 17, 2012
The issue has been raised about the effect of the mass of an atmosphere on surface temperature. It can be shown that if no optically absorbing gases, aerosols, or clouds are present in the atmosphere, that the average surface temperature will be determined by albedo, absorbed surface solar radiation (ignoring small radioactive heating effects), and outgoing thermal radiation at the ground level, and I will not discuss that issue here. The present discussion only considers an atmosphere with greenhouse gases, and for simplicity only looks at the effect of CO2 as a greenhouse gas. Only long time average global average values of temperature are considered, and only at long term constant CO2 levels (i.e., transient responses are ignored).
Some simplifications are made here, as the complete analysis is complex, and requires accurate experimentally measured data values and assumptions that are not well settled. The main simplification I make is the ground temperature sensitivity of the Earth atmosphere to increases in atmospheric CO2 levels. Values from less than 0.5 C/doubling to over 4 C/doubling have been suggested as the result of CO2 increase plus all feedback effects, However I am only describing the CO2 effect independently here, and this has been shown in most studies to give a surface increase of about 1.2 C/doubling of CO2, ignoring all other effects.
I use here is a mean virtual temperature, Tv~250 K that is based on an average temperature between sea level and approximate average location of outgoing radiation to space. This is an approximation, but its exact value has little effect on the comparison shown later. In addition, I use the wet lapse rate (as found in our lower to mid Troposphere) of -6.5 C per km height, even though I ignored the feedback effect of water vapor and clouds in the atmosphere to simplify the analysis.
In an atmosphere, the height from ground to a particular pressure level can be found from the following equation:
The value of H, which differs somewhat for different lapse rates, is called the scale height, and is the height where the pressure decreases by a factor of 1/e. I am using here the value of H ~29.3Tv for Earth’s atmosphere (based on the actual measured average atmospheric gas properties and Earth’s gravity), so combining this with the value of Tv selectedgives H~7.33 km. Changes in this value would be small enough for different assumptions that it would not change the basic result shown here.
I now examine two simplified cases:
- The case of a surface pressure of 1 bar (Earth’s actual value), with present amounts of CO2 (390 ppm), and with the effect of other greenhouse gases, aerosols, or clouds having a constant effect that is independent of atmospheric mass or changes in CO2, and assuming the same albedo as at present.
- The case of 2 bars surface pressure, with the same total amount of CO2 as case 1, but with an added equal amount of a mix of N2 and or O2, so that the average specific heat and molecular weight of the atmospheric gases are the same for 1 and 2. The greenhouse heating effect of other greenhouse gases, clouds, and aerosols are considered to be exactly the same as case 1 to separately show the effect of CO2 alone, and the albedo is still the same as in case 1.
The total effect of the present amount of CO2 alone on an increase in temperature above the no-greenhouse gas for case 1 is not accurately established, with estimates for CO2 alone from 5 C to 15 C as compared to the 33 C estimated total greenhouse effect with all gases, clouds, and aerosols. Since some of the CO2 absorption and radiation wavelengths overlap some of the water vapor wavelengths, the effect of CO2 in the presence of water vapor is even less addition than if considered alone. I am examining the effect of only CO2 here. I use an estimated value of the total CO2 effect of 10 C for the present amount as being reasonable (the exact amount is not important as long as is significantly larger than the effect of one doubling). If case 1 has the same mass atmosphere as the present atmosphere, except the concentration of CO2 was 0.5 times that of the present (195 ppm), this would have resulted in a reduction of surface temperature of 1.2 C for the lower concentration, ignoring feedback. Case two does has half the concentration of case 1, but also has twice the atmospheric mass, so the total mass of CO2 is the same for both case 1 and 2, and the only difference is atmospheric mass (and corresponding thickness) of the total atmosphere. The question is: what does this do to surface temperature?
The atmosphere is considerably thicker for case 2 than case 1 due to having twice the mass of gases, and this raises the altitude of some of the (assumed well mixed) CO2 a considerable amount. A simple analogy to see the effect is that if a thin unmixed layer of CO2 containing all the present CO2 mass in the present atmosphere were forced to lie close over the surface, and most of the atmosphere above it had none, the greenhouse gas effect would only raise the location of outgoing radiation a short distance above the surface. Multiplying the average outgoing altitude by the lapse rate would result in surface temperature increasing only a fraction of the 10 C presently possible for mixed atmospheric CO2. While the gases would mix eventually up into the atmosphere, this point shows the effect of altitude of the greenhouse gas as also being a factor.
The equation for the relation between pressure and height for p1/p2=2 gives a value of (Z2-Z1)=5.08 km. Thus the pressure at 5.08 km for case 2 matches the surface pressure for case 1. The fact that a 0.5 change in CO2 would only change surface temperature 1.2 C implies that it only changes the average location of outgoing radiation by 1.2/6.5= 0.18 km if that were the only factor considered. However, the total change of 10 C possible for all of the CO2 alone implies the average altitude of outgoing radiation to space for all the CO2 alone was about 10/6.5=1.54 km. This is nearly an order of magnitude larger than the change due to a 0.5 change of CO2 (i.e., it is the result of the exponential response).
We thus have case 2 with only 0.5 the CO2as case 1 in the lower 5.08 km of atmosphere, but where it has the same total mass of the entire case 1 atmosphere. However, we have on top of that, additional atmosphere with the same total mass of atmosphere as all of case 1, and also with 0.5 the CO2 as all of case 1. This upper layer would be as thick as the entire present case 1 atmosphere. If the upper layer absorbed and radiate all portions of wavelengths absorbed and radiated in the lower 5.08 km, this upper portion alone would have a location (for CO2 alone) 1.36 km above the 5.08 km level where outgoing radiation occurred. The actual solution of the resulting average altitude would require a full radiation analysis, and is not as high as that oversimplified version. However, it is clear that a thicker atmosphere, even without increasing total greenhouse gases over the thinner case, would have increased surface temperature due to the increased average altitude of outgoing radiation. It is also true that it is not the mass or pressure of the atmosphere alone that causes the increase, it is the combination of average altitude of outgoing radiation and lapse rate, and the increase in mass of atmosphere would raise the average location of outgoing radiation by virtue of thickening the total atmosphere. The final increase in surface temperature is the product of average outgoing altitude (including from the ground, greenhouse gases, clouds and aerosols), and lapse rate.