Earth’s Atmospheric Window and Surface Temperature

 Eddie Banner 19/11/2019

As a physicist, I have long had doubts about the Greenhouse Gas Theory for global warming, and I recently came across a paper on this matter which enhanced my difficulties.  This paper seems to base the theory on consideration of the global annual mean energy budget of Earth’s climate system, and shows a diagram by Trenberth and Fasullo.(1)

Several other figures can be found on this topic on the internet, with similar values. (2) (3)

There are a number of items in this diagram which concern me, but the obvious one is the claim for the downwelling back radiation of 333 Watts per square metre of Earth’s surface, (Trenberth).  All three diagrams show this feature, at the extreme right side.

The downwelling back radiation is shown coming from the Greenhouse gases in the atmosphere.  However, it is well known and fundamental in GHG theory that photons of energy emitted by GH gases are radiated equally in all directions; therefore there is equal energy radiated upwards and downwards.  This upwards energy flux is SIMPLY NOT SHOWN in any of the three figures, but it cannot be ignored.  It must be 333 Wm^-2 in Trenberth’s case. 

Moreover, the upwards energy would add to the 239 already shown emitted, making a total of 572 Wm^-2 emitted to space, which clearly is nonsense.  So the GHG theory has a lot of explaining to do.

Another, less obvious but nevertheless crucial, item is the amount of power shown escaping directly to space through the atmospheric window.  Trenberth and others have tried different, fixed values; I have seen 22 or 40 Wm^-2 in their energy balance diagrams.  My earlier work suggests more like 100 Wm^-2, but more recent calculations leading to my current posts indicate that at 288 K the window is 95.7, and at 289 K it is 90.4 Wm^-2. 

So the window is of critical importance, and we need to know what, if anything, can change its value.  Anthropogenic effects particularly, but it seems unlikely that carbon dioxide can have any effect here. (later) 

 My own approach has been to start with the explanations for the GHG effect given in numerous posts; two examples below, (4) and (5).

The diagrams clearly show that the theory is based upon the Stefan-Boltzmann equation for the emission of radiant energy photons from a “Black Body”.  This is acceptable for emission from the surface of the Earth, because it does approximate to such a body.  Atoms and molecules are vibrating and rapidly changing direction, and so can emit photons within the relevant range of wavelength.

BUT STEFAN-BOLTZMANN IS  !!!! NOT !!!! OK for the diffuse spread of molecules of the atmosphere.  The S-B equation does not apply in these conditions, however many layers of the atmosphere are considered, as in the Greenhouse Gas Theory.

A different approach.
At the surface, Stefan-Boltzmann does indeed apply.
Let P = output power radiated from the surface per square metre.
e = emissivity of surface
s = S-B constant = 5.67*10^-8
T = surface temperature in Kelvin 
w = power emitted through window to space
Take T = 288 K and e = 0.98 (not quite a black body; e would be 1.0 )
This gives P = 382.2778 Wm^-2 ———————-(a)
P = e.s.(T^4)
= 0.98*5.67*(10^-8)*(T^4) ———————–(b)
This is the power emitted from the surface into the atmosphere. 
Some of this power, w, escapes directly to space, as in Trenberth’s diagram. (No greenhouse gas absorption in the window). 
Let P1 be the power remaining in the atmosphere.
So        P1 = P – w
But there are greenhouse gases effective in the wavelengths outside the window.
Therefore, 0.5(P – w) is radiated upwards, and 0.5(P – w) is radiated downwards by the very action which is fundamental to the GHG theory.  ( ie. greenhouse gases emit photons equally in all directions).
So the total power into space = 0.5(P – w) + w
And for Earth’s energy balance, this must equal 239 Wm^-2
ie.        0.5(P – w) +w  =  239
0.5(P + w)  = 239
Hence  w = 478 – P
Therefore, from (b)    w = 478 – 0.98*5.67*(10^-8)*(T^4)
                                            = 95.7
The value of w has been calculated for a range of surface temperatures, and the results are shown in the graph below.

The energy flux returning to the surface from the atmosphere is 0.5(P – w).
For 288 K, we find from the equation that w = 95.7 Wm^2, and we have seen from (a) that P = 382.2778 Wm^-2, so this downward power to the surface = 143.15 Wm^-2.
In addition there is (161+78) ie 239 from the Sun, according to Trenberth, making a total of 382.2, as required for energy balance.

Finally, the effective emissivity of the Earth’s system is the output 239 to space divided by the input 382 from the surface, that is 0.98*239/382.2778 = 0.6127

The graph shows a Planck curve for 288 K and the window as calculated above.  The range of wavelengths for the window is generally taken to be from 8 to 13 microns, so our 95 Wm^-2  window lies satisfactorily within this range, from 8.2 to 12.0 microns.

Summary
The thoughts presented here rely on well accepted facts.
(a) The Stefan-Boltzmann equation for black body radiation. 
This is valid for radiation from the Earth’s surface, but not for any of the multiple atmospheric layers as claimed by the GHG theory.
(b) The emission of infrared photons is equal in all directions. 
This fact is used here and also in the GHG theory, but this does not mean any acceptance here of the GHG theory.
(c) These two facts, together with a little algebra, explain the role of the atmospheric window in governing the surface temperature of the Earth.  By definition, the window wavelengths do not include the absorption wavelengths of carbon dioxide and water, and so there is no effect from increases in concentration of carbon dioxide.
(d) There remains the question of possible reduction of window “width” by added anthropogenic pollutants in the atmosphere, eg aerosols and (? plastic) particulates. Any such effect could not be by absorption of energy, but possibly by photon scattering.

If such potential problems do, in fact, exist, and manage to close the window completely, then the surface temperature could rise to a maximum of 304.5 K, ie. 31.5 deg C.

References
(1)http://www.cgd.ucar.edu/staff/trenbert/trenberth.papers/BAMSmarTrenberth.pdf
(2)http://iacweb.ethz.ch/doc/publications/StephensLiWild_etal_NatureGeoscience.pdf
(3)https://earthobservatory.nasa.gov/features/EnergyBalance/page6.php
(4)https://www.acs.org/content/acs/en/climatescience/atmosphericwarming/singlelayermodel.html
(5)http://acmg.seas.harvard.edu/people/faculty/djj/book/bookchap7.html

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