Great Steve, We are getting somewhere now
Only little left
You write:
"The atmosphere gets cooler as you go up. Each part of the atmosphere radiates according to its temperature. As well as the radiation from the surface being absorbed, the radiation from any part of the atmosphere can be absorbed too."
My comment: In all direction, but the net transfer is temperature dependent and always to cooler part inside the atmosphere.
You write
"The earth's surface receives more than 240W/m^2 from the atmosphere. About 320W/m^2 is the usual average figure given."
My comment: Because of clouds yes. I calculated for clear sky conditions.
"Sentence 2:
Outgoing LW radiation is 200 from the atmosphere and 40 from the surface.
LW radiation from the atmosphere to the surface, though, is 320W/m^2. All from the atmosphere."
Se above: clear sky
You write:
"You can't just add the two numbers and then plug into the BB formula. The BB formula is W/m^2 so adding the radiation to space and radiation to ground also doubles the area."
My comment:No.
Make a thin slab with a temperature and tell me that this only radiate from one side. It radiate in all direction but with slabs is the "sides" neglect able.
So for total HEAT balance is it right to add all of the outgoing radiation caused bu its temperature. The NET is then the difference with the incoming which must be in balance.
You write:
"Looking up from the earth's surface, the atmosphere is radiating 320W/m^2 toward you which is the equivalent of a black body at 274 Kelvin."
My comment: Perfect proof that the atmosphere absorb close to all IR within just a few hundred meters. I thought that would be my argument.
But that is with clouds again. se below.
Then You write:
"Common experience is that the earth's surface is warmer than the atmosphere. Even on a cold night with a ground frost, the temperature of the atmosphere only a couple of hundred meters up will be lower than the surface temperature."
No. Ever heard about inversion?
Take a look here
en.wikipedia.org/wiki/Inversion_%28meteorology%29Especially at winter night is the inversion common. Bad example
"Therefore the net transfer of heat is from surface to atmosphere, and there is a net transfer by radiation."
Yes there is a net transportation of heat but You must agree that the amount and direction of energy is depending on the temperature differences between the two interacting bodies.
Net always from warmer to colder.
But take the example You mentioned with ground frost.
That outgoing radiation from the ground is passing the warmer atmosphere since it is not absorbed. The famous atmospheric window.
But that part is far below 100W/m2. But depending on the amount of vapor in the atmosphere. More vapor less window.
So a dry winter night IS a bad example
If it is low cloud is there less chance for ground frost. Because the net outgoing is down to zero. That because the cloud have about the same temperature and also very effient close the atmospheric window. No NET heat transfer at all outward because the temperature difference is down to zero between ground and cloud. By radiation!
Sometimes is even the cloud warmer and heat the ground. By radiation. Down wards.
And here is the other proof:
Even with clear sky is the net outgoing through the atmospheric window a minor part of the total emitted heat from the earth surface.
The heat balance demand that the down going radiation is in balance and that is close to the same figures as up going. That show that the atmospheres emittance toward earth is from a low altitude because the temperature/radiance is about the same as upping radiation. That is with about the same temperature so it must from close to the surface.
Precisly as the satellites show.