First, I want to apologize for the typographical errors. The little netbook I am using locks up at every opportunity and I have to retrieve autosaved versions and attmept to remember what I previously corrected. Not fun. Hence the donations for a new computer request.
The 330 Watt Weirdness
Climate Sensitivity is defined as the amount of warming that the Earth would experience with a doubling of the amount of CO2 in the atmosphere. Nothing we can do about that, it is the definition.
The Greenhouse gas effect would logically be the amount of warming due to greenhouse gases in the atmosphere. In order to determine the defined Climate Sensitivity, we need to know how much of the atmospheric effect is due to greenhouse gases and how much of that is due to CO2.
The Earth is estimated to be 33 degrees C warmer than it would be without an atmosphere. Is that the greenhouse effect? I don’t think so. Part of that 33C is due to radiant interaction of outgoing long wave radiation and some is not.
In determining that the Earth without greenhouse gases would be 33C colder, it was assumed that 30 percent of the incoming sunlight was reflected, just as thirty percent is reflected on average today. Of that 30% most is due to clouds, snow and ice, all due to water in its phases and more is reflected by liquid water on the surface of the Earth. Clouds, Ice and Snow are part of the atmospheric effect that may be impacted by a change in CO2, but are not directly caused by CO2. The assumption goes that their reflective effects are fully considered, that they cause the colder temperatures that CO2, Water Vapor and other radiant absorptive gases compensate for in warming the Earth to its current average of 288K degrees.
Without this reflective property of water, the Earth would absorb approximately 96% of the 340Wm-2 average energy we receive from the sun on an annual basis. 326Wm-2 absorbed in a stated state would result in a temperature of approximately, 275 K degrees. Using the same ratio of surface and atmospheric albedo 26% reflected by clouds and 4% reflected from the surface as today, we have 238Wm-2 absorbed by the surface and at steady state, that would result in an average surface temperature of 254.5K degrees, via S-B emissivity =1. Clouds and atmosphere depress the surface temperature by 20.5 degrees. Without greenhouse gases, there would be an atmosphere and clouds are required for approximately 87 percent of the reflected sunlight.
Why would there be an atmosphere? Because the Earth is not at absolute zero and there is a wealth of nitrogen and oxygen available with thermal conductive and convective properties. Not including this simple physical reality makes it impossible to determine how much impact CO2 has on surface temperature. By definition, that is what is required to be determined. And by definition it has to be determined at the surface.
Co2 is estimated to cause between 5 and 30 percent of the greenhouse effect, with water vapor responsible for the majority of the rest. Since we have a model now, let’s allow water vapor to become a greenhouse gas and not add any CO2 or methane. Since the atmosphere depressed the temperature by 20.5 degrees, let’s say that water vapor interaction with OLR returns it to 275K degrees. We can assign the rest, 13C to non-condensable greenhouse gases. We still have the full atmospheric effect including Greenhouse gas effect of 33.5 degrees, Slightly more than the classic 33C published in the literature with 390-238=152Wm-2 the total amount of the additional flux created by the greenhouse effect. Dividing that 152/33.5 yields 4.53Wm-2 per degree of warming.
The measured emissivity of the atmosphere from the very top of the atmosphere (TOA) is 0.61, meaning 61% of the surface radiation calculated at 390Wm-2 for a perfect black body is measured in space by satellites. 61 percent of 390Wm-2 is 238Wm-2. So these values correspond to both the conditions at the surface and at the TOA. Note: 0.64 is a commonly used value for TOA emissivity. Then the values used for solar in and OLR also vary.
4.53W/m-2K-1 can be re-written as 1/(4.53W.m-2K-1) or 0.22Km^2/W for convenience. This is sometimes called the Planck response or Planck parameter. This does not include any adjustments and assumes perfect blackbody relationships between temperature and radiant energy flux.
Since we are assigning water vapor two thirds of this value and CO2, other greenhouse gases the remaining third, this can be re-organized as Pr=0.67Wv+0.33CO2=0.22Km^2W-1.
The doubling of CO2 will produce an increase in forcing or resistant to OLR based on the Arrhenius equation equal to Delta F=Alpha ln(Cf/Co) = Alpha * 0.69. The constant alpha is based on the Stefan-Boltzmann relationship for black body radiation. Since I am using the perfect black body approximation I will use 5.67 for alpha. Generally, 5.67 times the 0.926 or ~ 5.25 is used for alpha. Since CO2 is assigned 33% of the greenhouse forcing or .33*152=50.16, a doubling of CO2 would increase that to 55.8 which with 4.53Wm-2 per degree would be 55.8/4.53 = 12.3 degrees versus 50.16/4.53=11.07 or 1.23 degrees of warming at the surface.
The only reason I did all this is to show that the S-B relationship for both water vapor and non-condensable greenhouse gases is needed when determining the Planck response and in determining the impact of double CO2. The Earth is of course not a perfect black body.
With the equation for Pr and the initial value of 0.22, we should be able to determine the range of error possible in the estimate of 33% impact of non-condensable GHGs. If the non-condensable GHG are responsible for 50% of the Greenhouse effect, the small increase in forcing due to doubling would have a smaller impact. If the ncGHGs are responsible for only 5% of the greenhouse effect, they would have a larger impact. Since the lowest estimate is about 5%, then 95% of 0.22 or 0.21 is due to water vapor and only 0.01 is due to ncGHGs, quadrupling that value or eliminating that value has little change on the impact of a doubling of CO2 as measured from the surface using perfect black body approximations.
Why? Because the impact of CO2 is greatest where is does not have to compete with water vapor as a greenhouse gas.
This is why the K&T over estimation of DWLR is so humorous. If a doubling of CO2 is going to increase DWLR 3.7Wm-2 estimated by the IPCC, the increase would be 3.7/330 or 1.1%. If we use the entire 33.5 degrees 1.1% of 33.5 is 0.38 degrees. After allowing for estimates of emissivity, Dr. Richard Lindzen determined that per the K&T data that doubling of CO2 would cause 0.5 degrees of warming. Dr. K. Kimoto estimated 0.5 degrees warming. Anyone that wants to will estimate lower than expected warming can use the K&T literature, because it is wrong. I find that absolutely hilarious. It is amazing how many contortions the defenders of the cartoon will go to “Prove” it is right when it implicitly states there is little warming possible due to a doubling of CO2.
The truth is that a doubling of CO2 will cause more warming than indicated in the K&T cartoons. How much though is a little more complicated. To do that you have to start with an accurate model that is a little more involved than the cartoons.
5.67e-8 is the main constant in the Stefan-Bolzmann equation. I say main, because emissivity has to be considered and the average for space, 0.926 J/K varies as, well, the energy per degree with respect to the radiant properties of the source. One of the things we need to know is the combined emissivity of the various sources and the transmittance of the media between them.
For a CO2 doubling we can assume an object in the atmosphere is a S-B energy source. If it emits energy F at T1, then its difference in energy emitted at T2 would be,
dF/dT=[ 5.67e-8(T2)^4 – 5.67e-8(T1)^4]/(T2-T1),
Instead finishing a formal derivation, let’s just select a reasonable T2=255K and T1=254K then,
5.67e-8(255^4-254^4)/(255-254)=5.67e-8(42.3e8-41.6e8)/1=5.67*(0.7)/1, note that the e-8 and the e+8 can be added which simplifies the result to 5.67*0.7 equals 3.969= dF/dT @ dT254.
Using the same relationship, dF/dT@288K=5.44
So it should be obvious that the change in flux F is dependent on the temperature of the body and the emissivity of the body. For a point source of CO2 forceing, if the source temperature is ~255K, 4 is a good approximation of the change in flux with respect to a small change temperature and for a point source at 288K 5.44 is a good approximation for the change in flux with respect to change in temperature.
You can simply graph this relationship on appropreate paper, interpolate between the ranges or plug the whole shebang into your computer if that blows wind up your skirt.
The main point is that the surface to TOA relationship is 5.44/4 or 1.35 and for a TOA source with respect to the surface 4/5.44 or 0.74.
In our next installment, we will discuss the impact of the Tropopause temerature variations from ~-50C to -100C on what should be a simple calculation.