Not being to figure out what the exact value of radiant forcing is in an atmosphere is ridiculous. I mean really! We have pretty good approximations of black bodies, why so much grief over a gray body, a body with a radiant active atmosphere? Sounds kind of stupid for so many smart guys working on simple problem.
Well, since we don't know, I am going to define an ideal gray body. An ideal gray body has a surface emissivity of 1, a perfect black body and an atmospheric emissivity of 0.5, exactly half of its radiant energy is convert to heat while leaving the surface. Cool huh?
So if the Earth was an ideal gray body, its emissivity would be 0.5. So looking from the surface, if the surface emission was 400Wm-2, then the average radiant layer would be 200Wm-2. Since the radiant energy has to cool above as much as it warms below, the TOA would 100Wm-2 plus half the energy from the average radiant layer, 100Wm-2 for the TOA measurable emission of 200Wm-2. Remember, half of the 200 is required to create the 100Wm-2, with half passing through. I went through this with the multi-disc model so I will add that link later.
So why this crazy brain storm? Well, the the TOA emissivity is about 0.61, the Earth is neither an ideal black nor gray body. Since it misses perfection by about 0.11 out of 0.5, about 22% of the energy transfer is not playing the ideal game. If the surface is 400Wm-2, the 200Wm-2 would be the ideal emission to space. If the TOA emission is 240Wm-2, the ideal surface emission would be 480Wm-2. 22% of these would be 44 or 52.8 depending on which is the better baseline value.
Those both make sense. The 44Wm-2 is approximately the surface to space energy through the atmospheric window and the 52.8 is approximately the cloud level to space energy through the atmospheric window. So the Earth is about 78% gray body and 22% black body.
78% of the surface flux would be 312Wm-2, since perfect gray bodies emit 50% of their surface energy, 156Wm-2 would the apparent temperature of the ideal gray body Earth viewed from space. 156Wm-2 would also be the radiant portion of the greenhouse effect.
The surface is warmer than that though. More energy would have to be transferred to the atmosphere. For the 52.8Wm-2 lost through the atmospheric window, at least half, 26.4Wm^2, would need to be transferred to the atmosphere by....? I would think conduction.