vintage cessna
Cleared for Takeoff
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retro
It is NOT an assumption, that's the point. We live on the surface of the Earth, we can measure the properties of the materials that make it up. The emissivities are measurable and have been measured. They're close to 1 around 10 um for nearly all of them that are present in appreciable quantities. Including water, I might add.
Where the trapped heat goes is a different question, along with what the dominant mechanisms are for transferring it to the surface. I don't know the breakdown, conduction vs. radiation, but I have to believe that radiation is not negligible given how large the back radiation is - it is less than but comparable to the flux radiated from the surface.
Transfer of heat to the deep ocean does NOT affect whether the surface radiates as a blackbody. (Read that again, be sure you understand what I'm saying there. I'm talking about the S-B law and not saying anything about what the surface temperature actually is.) It DOES, of course, affect the surface temperature. Go back to our discussion of how heat capacity and thermal diffusion affect the daily temperature cycle on a world like the Moon. The oceans have a huge heat capacity and the lag time between changes in forcing and changes in surface temperature could be of order years or longer. In the meantime the OLR from the planet might be less than the net incoming solar flux (incident minus reflected), i.e. the planet would be out of equilibrium.
But the surface, and the TOA, would still be radiating very nearly as a blackbody.
(In fact the energy budget charts show that the planet IS not quite in equilibrium, there is a net absorbed flux of something like 0.6 W/m2. I'm not sure whether that is really significant or whether it could be due to measurement uncertainty in the other terms.)
I think I will stop beating this dead horse. But the next time I'm cruising at 7500 ft I will look at the OAT, and my mind will fill with wonder and think :Isn't Mother Nature clever, all that magical back radiation bravefuly heading back down to Earth out of this frigid air to keep us toasty.