Bill Illis said:
Anyone ever put Myhre's forcing estimates into the Stefan-Boltzmann equations.
Yes, they have. What the Stefan-Bolzmann equations give you is called "non-feedback response". But note that the forcing calculation by Myhre is not based on Stefan-Bolztman. It is based on radiative transfers up and down a transparent atmosphere. The calculation you are doing is a kind of check to see how the known forcing of CO
2 matches up with the observed changes in temperature.
The difference with the real climate on Earth is that as temperature increases, so also do other aspects that bear upon energy balance. Ice melts (increasing albedo), humidity increases (more greenhouse, but less lapse rate) cloud changes (altering both greenhouse and albedo in complex ways); plus also changes in vegetation and land cover.
All this becomes feedback, either positive or negative, which bears upon "climate sensitivity" to either amplify or damp the base non-feedback response.
I mentioned this in my earlier post, as an aspect of climate that is NOT well known. The base non-feedback response is pretty well constrained, but it does not suffice to let you infer the temperature impact.
If the total forcing increase from GHGs is 1.7 W/m2, the Stefan-Boltzmann equations predict very little temperature change from an increase this small.
Surface TempK Today = (390 W/m2/5.67E-08)^.25 = 287.98K = 15.0C
Surface TempK Pre-Ind = (388.3 W/m2/5.67E-08)^.25 = 287.66K = 14.7C
So either Myhre's estimates are not really the traditional watts/metre^2 measure we use normally or the Stefan-Boltzmann equations aren't even being used.
There are several omissions in your application of Stefan-Boltzmann, even ignoring the effects of all the feedback processes. Most importantly, you are mixing up a surface temperature change with a forcing that is defined with respect to the top of the atmosphere.
Here is a better way of doing it. The Earth is not a black body emitter. You must consider both albedo and emissivity.
The Earth has an albedo of about 0.3, which means that about 30% of the solar input is reflected, and does not give any heating. Also, the radiation into space from the Earth corresponds to an effective temperature of about 255K. This is because of the greenhouse effect, which gives Earth an effective emissivity. Surface temperature is about 288K, because of the additional blanketing effect of the atmosphere.
You get into the right ball park by using a grey-body approximation relating surface temperature to the absorbed solar energy. For a grey-body, the Stephan-Boltzmann relation is
Q = \sigma \epsilon T^4
Q is the thermal emission flux from the top of the atmosphere, and ε is an effective emissivity. T is the surface temperature.
The solar constant is about 343 W/m
2 over the surface of the planet. 30% of this is reflected, leaving 240 W/m
2 energy absorbed. This is what Earth emits as thermal radiation; it is the value for Q. T is the surface temperature, about 288K.
The value for ε is about 0.6; but in fact we can cancel it as follows.
\frac{dQ}{dT} = 4 \sigma \epsilon T^3 = 4Q/T
The rate of change of energy emitted with temperature is about 4 * 240 / 288 = 3.3 W/m
2/K. This is a non-feedback response. For every 3.3 W/m
2 of forcing (change in Q) we should get about a degree of temperature change -- ignoring feedbacks.
The 1.7 W/m
2 forcing corresponds to a bit over half a degree.
You obtain a slightly smaller value because you are using Stephan-Boltzmann at the surface, with Q=390 as the emission of radiation from the surface. But that's the wrong comparison for a forcing, which by definition is the change in energy balance at the top of the atmosphere. The method with a non-unit emissivity will give you a closer value for the proper application of Stefan-Boltzman to a forcing. There are in the literature more detailed calculations considering the emissivity of the atmosphere and the fact that the Earth is not a uniform sphere. The value obtained is around 3.2 W/m
2/K; close to the simple approximation I used above.
There are a number of important provisos with this number.
- It ignores all feedbacks. This is one of the major open questions in climate research. Most of the available evidence indicates a net positive feedback, to give a substantial amplification of the base response. A small minority of researchers propose a zero or negative feedback, but over all they have not been made a good scientific case for this, and nearly all estimates suggest that the base response is increased by a factor of around 2 to 4 times.
- This is an equilibrium response. Since it takes a long time for the ocean to heat up in response to a new forcing, you don't get all this temperature increase realized at once. Research indicates at present a steady flux of energy flowing into the ocean. This represents a forcing that is not yet realized as a surface temperature increase.
- It ignores all other forcings; there's more than CO2 involved.
As it turns out, these additional considerations mostly cancel out. The effect of time delay with heating of the ocean works in the short term rather like a negative feedback, and so the observed amplification short term is smaller than than full equilibrium response. The other additional forcings are both positive and negative, and of comparable magnitudes each way.
The net effect is that the base response is not that far off what we should experience; generally estimated to be a small amplification over base response.
So... a naive application of Stefan-Boltzman to CO
2 levels alone gives about half a degree of heating. And the actual temperature rise we have observed is in a similar ballpark... around 0.7 degrees.
For more discussion in other threads:
- Feedback is now being discussed in thread [thread=360877]The AGW climate feedback discussion[/thread].
- Base response is discussed, with references, in [post=2199572]msg #69[/post] of thread "Physics of Global Warming", and [post=2225918]msg #47[/post] of "Ocean heat storage". The values given for base response from the literature are about 3.2 W/m2/K.
Cheers -- sylas