This is great! I havent entirely gone through it but he gets the same figures.
0.8 W/m^2 since 1950 should have produced 0.15 K warming, he says. Extrapolating to 3,7 for CO2 doubling gives 0.69.
Thanks for sharing!
Ok so you mean like following?
I = (S/4)*(1-a) – eoTs^4
Old equilibrium:
I = 1362/4*(1-0.304) -0.603535*5.670373E-8*289^4 = 236.988 - 238.729 = -1.741
New equilibrium:
I = 1362/4*(1-0.304) - 0.597948*5.670373E-8*289.6756^4 = 236.988 - 238.739 = -1.751
Well look at that, it doesnt match. The horror!
Note the consistency in the mismatch though.
Now why doesn't it match in the first place?
Well several reasons.
First of all we first gave a solar radiation of 236.307 W/m^2, later when we analysed the emissivity we used a value of 239.
We also used a value of 396 W/m^2 for Earth radiation which is not the blackbody radiation value of a 289K body but instead the radiation value of a blackbody with Ts = 289.0819K
So lets see what happens when we implement these values.
First the initial emissivity changes to 236.988/396 = 0.5984545, the second emissivity changes to 236.988/396 = 0.5929147.
A delta of -0.0055398.
This now gives for Old equilibrium:
I = 1362/4*(1-0.304) -0.5984545*5.670373E-8*289.0819^4 = 236.988 - 236.988 = 0
Yes! will the new one be as picture perfect? Spoiler, I dont think so.
For the new equilibrium we have to recalculate our de/dTs value first:
de/dTs = -(1362/5.670373*10-8)*(1-0.304)*289.0819^(-5) = -0.00828076
So given our emissivity change we get a dTs of 0.669
This now gives for new equilibrium:
I = 1362/4*(1-0.304) -0.5929147*5.670373E-8*289.7509^4 = 236.988 - 236.975 = -0.13
Wait, but why?
Well de/dTs is dependent on Ts so as you heat this factor will change and you will propagate a small error in your calculations.
If we would have gone over the calculations 10 times with each time only an 0.37 W/m^2 increase, we could balance the equation with a smaller error.
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That said, this doesnt "prove" it.
The main dispute here is if you can model the earth as a 289K greybody with a certain emissivity or a 255K blackbody.
Since at the TOA you only have 237 W/m^2
Lets do it with a blackbody (no emissivity)
I = 0 = 236.988 - 1*5.670373E-8*T^4
gives for T = 254.2604K
Now the original emissivity is 1 so what will happen to the emissivity after doubling the CO2?
Well nothing! It is defined as a blackbody and it will stay a blackbody (emissivity = 1) because beyond it there is no matter to absorb or reflect the radiation from.
And it will stay at the blackbody temperature that balances the equation at TOA.
The IPCC and its peers are handling the equation as if all CO2 will be emitted not in the atmosphere but beyond the TOA like a second atmosphere around the original one.
The radiation at the edge of space will stay the same, it's what happens underneath that gets impacted by the greenhouse effect.
Adding CO2 will trap and absorb heat and heat the surface of the Earth, the real Earth, the greybody Earth. It will not deliver that heat right at the edge of space.
And that kids, is why you should never make wrong assumptions for you physics problems.
Unbelievable.