Most good textbooks introduce simple models to help the student gain a conceptual understanding.
In Elementary Climate Physics, Prof. F.W. Taylor does the same.
Now the atmosphere is mostly transparent to solar radiation (shortwave) which is centered around 0.5 μm, and quite opaque to terrestrial radiation (longwave) which is centered around 10 μm. Note that absorptivity is very wavelength dependent, especially for radiatively-active gases.
So in this first model, which is very common in introductory books on atmospheric physics, three things are assumed - and none of them are true:
- the atmosphere is isothermal – a slab of atmosphere all at the same temperature
- the atmosphere is completely transparent to solar radiation
- the atmosphere is completely opaque to terrestrial radiation
As many people know, climate scientists introduce things that are not true in climate science books because either they have no idea what they are talking about, or because they are trying to deceive their readers.
Surprisingly, despite the incompetence and mendacity of these awful people the model is quite illuminating.
How do we calculate the surface temperature?
With apologies for the lengthy explanation that follows – necessary because of the confusion frequently spread on this subject. To grasp the essence of the simple model you don’t need to follow every point here.
When a body is a perfect absorber of radiation it is also a perfect emitter. If this is true at all wavelengths, the body is called a blackbody. In practice, no real bodies, or bodies of gases, are blackbodies but many come close. Especially, many come very close at certain wavelengths or bands of wavelengths.
If a layer of atmosphere has an optical thickness = 10 across a band, then its emissivity in that band = 1.0000.
This means, in this band it is.. still not actually a blackbody because its emissivity has not really reached 1, it is actually = 0.9999546. And if the optical thickness = 20 across a band, then its emissivity in that band = 0.9999999979 – still not a blackbody. For all practical purposes we can say it is a blackbody at these wavelengths because within the limits of accuracy we need, emissivity = 0.9999546 is the same as saying emissivity = 1. Nothing special or magical happens in the equations of heat transfer when we transition from 0.99 to 1.00. And assuming 0.9999546 = 1 introduces a 0.005% error.
The equation for the emission of thermal radiation is Planck’s law which describes how the intensity varies with wavelength for a perfect emitter (i.e., a blackbody), e.g.:
For any real surface (or body of gas) this Planck curve is multiplied by the emissivity curve (vs wavelength) to get the actual thermal emission vs wavelength.
To calculate the flux (W/m²) we can instead use the Stefan-Boltzmann equation (which is just the integral of the Planck curve over all wavelengths):
E = εσT4
where E = energy emitted in W/m², ε = emissivity, σ = 5.67 x 10-8, T = surface temperature
Because ε is a function of wavelength, and because increasing temperatures shift the emission to shorter wavelengths we need to use the value of emissivity for the temperature in question.
Notice, in figure 2, that the emission is very low below 4 μm.
Now, for our incorrect assumption that the atmosphere is completely opaque for all wavelengths greater than 4 μm (longwave) then the equation for emission from the atmosphere will be:
E = εσT4, and as ε=1 at these wavelengths,
E = σT4
Now we have that out of the way..
The (incorrectly assumed) optically thick atmosphere emits radiation to the surface at σTa4 and out to space at σTa4 (where Ta is the temperature of the atmosphere). The radiation from the earth’s surface is (incorrectly assumed) completely absorbed by the atmosphere.
In equilibrium, as the general rule:
Ein = Eout
Therefore, the absorbed solar radiation = energy emitted to space from the atmosphere.
Absorbed solar radiation = (1-0.3) x S/4, where S = solar constant of 1367 W/m². The “0.3” is the reflected radiation due to the albedo of the earth and climate system. So only 70% of solar radiation is actually absorbed on average. The term 1/4 appears because solar radiation is not directly overhead all points on the globe at all times. For the detailed explanation see The Earth’s Energy Budget – Part One.
Therefore, for the energy balance of the whole climate system:
(1-0.3) x S/4 = σTa4 
And for the surface energy balance, where Ts= surface temperature (and refer to figure 1):
(1-0.3) x S/4 + σTa4 = σTs4 
So,  -> 
2σTa4 = σTs4
Re-arranging, we get: Ts = 21/4.Ta 
And from , Ta = (239/5.67 x 10-8)1/4 = 255 K
Therefore, Ts = 303 K
So we have a solution to the problem for our simple model with three totally incorrect assumptions. Compare the calculated value with the observed 288 K average surface temperature.
As Taylor says:
This calculated greenhouse enhancement of 48 K is rather larger than the observed 33 K, not surprisingly in the light of the simplicity of the model.
This model helps us see how temperature of the atmosphere and the surface are related under the simplest of assumptions.
In practice, the atmosphere is not completely opaque to terrestrial radiation and therefore, does not emit like a blackbody. The atmosphere is not completely transparent to solar radiation, and therefore, the atmosphere is also warmed directly by the sun. The atmosphere is not isothermal and, therefore, emits differently to the surface compared with its emission to space.
And everyone in climate science knows this. Real climate models are slightly more sophisticated.
When you read examples like this and like the “multiple shell” model, they are for illumination and education. Simple models teach beginners more than complex models. Who can understand a GCM if they can’t understand this model?
When you read people writing that climate science assumes the atmosphere radiates as a blackbody you know they didn’t make much progress in their elementary climate science textbook. That is if they even picked one up.
Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations – the actual equations of radiative transfer (no blackbodies or Stefan-Boltzmann equations to be seen)
CO2 – An Insignificant Trace Gas? Part Five – the radiative-convective model with a couple of solutions
The Amazing Case of “Back Radiation” -Part One – for measurements of radiation from the atmosphere