An interesting paper that asserts the use of “Back Radiation” is in error. ( H/t to JWR in comments on WUWT
It’s long, but I think it is worth it. The author uses a lot of exclamation points, which is a common habit in non-native speakers of English from some particular cultures, but do try to get past that. The math looks right to me.
He also uses a ‘shorthand’ where several terms in T^4 are presented as a single symbol. Yes, it makes the writing more compact, but for those of us who do ‘visual thinking’ we can’t just see the 4th power in the equations and have to keep remembering it…
Some selected quotes:
It has been argued by Claes Johnson that what is called in climate circles “back-radiation of heat” is not a physical phenomenon. [1, 2, 3] Heat flows and radiates spontaneously only from warm to cold. The fact that for some cases between two plates e.g. the surface of the earth and the representation of the atmosphere by one single slab, the formulation for heat flow with the non-physical back-radiation of heat gives the same temperature distribution as the formulation with the correct interpretation of the law of Stefan-Boltzmann (SB), makes some people to believe that backradiation of heat exists and heat can flow spontaneously from cold to warm! It will be shown in this paper that in the one-slab model for a semitransparent atmosphere with the so-called two-stream formulation and thereby back-radiation of heat, a spurious absorption occurs, twice as high as compared to the one-stream formulation from warm to cold according to the Second Law of Thermodynamics.
A simple proof of the fact that back-radiation of heat does not exist!
But hasn’t ‘back radiation’ been measured?
Still others claim that they have measured back-radiation of heat!
Claes Johnson  has argued that pyrgeometers do not measure heat fluxes but rather frequencies which can be related to temperature. A micro-processor converts the signal by means of SB to Watt/m^2: it is called “back-radiation of heat” from the colder sky to the warmer surface of the planet! See also Douglas Cotton . In a blog by John O’Sullivan  are reported discussions between IRthermometer manufactures and Alan Siddons and the experiments of Nasif Nahle.
We conclude that back-radiation has not been confirmed experimentally.
About 1/2 way down the paper the author finds a way to calculate heat flow that happens by ‘means other than radiation’ in a semi-transparent atmosphere. (It’s a somewhat impressive feat… but one where I’m still looking at the formulas to try to find an error as it just seems a bit counter intuitive that you can calculate that from a radiation oriented model… but I’ve not found an error… yet…)
Note the scale difference of the x-axes in the two figures: the IR-sensitive trace gases emit to higher levels and to outer space about 10 times the amount they absorb! The difference is transferred by molecular collision from the 99% bulk of the atmosphere to the IR- sensitive trace gases. That heat arrives at the bulk of the atmosphere by “mechanisms other than radiation”! As we will see later, about 60% from the surface by convection of sensible and latent heat and the other 40% by absorption of SW sunlight by aerosols at different heights of the air column (see also appendix 2). If there were not an atmosphere with a lapse rate defined by thermodynamics
All in all, an interesting paper.
And one that, should it stand up to close inspection, will likely become a bedrock paper of the “convection and evaporation does it” folks.
The whole notion of “Back Radiation” always annoyed me. First off, there’s that Second Law problem of heat flow from cold to hot. Then there’s just this feeling that the wave function of a particle trying to emit a photon is not going to be happy doing an overlap with an incoming photon / wave function of ‘back radiating’ heat without just rejecting the whole thing. Yes, you can end run around that with the assertion that the outgoing comes from one molecule while the ‘back radiation’ is absorbed into some other molecule and then ‘they talk’… but that just ignores the implications of the ‘two slit’ experiment that the wave functions have more geographic coverage than that ( i.e. go through both slits with wide physical separation… at least until ‘observed’.)
So that whole ‘back radiation’ idea just seemed wrong to me. The flux of photons / wave functions would be too high (especially in a multi layer system) and just the flux density ought to lead to paradox issues or some kind of ‘symptom’… (unless one does a lot of ‘hand waving’…) Intuitively it just feels right to think that nature works on a ‘net net’ basis and doesn’t shovel vast numbers of wave functions back and forth waiting for a solution to be observed… Then again, Q.M. can be very non-intuitive…
At any rate, the paper deserves inspection.