Earth’s Atmospheric Window and Surface Temperature

 A E Banner Dec 2019

The Atmospheric Window has a vital role in governing the temperature of the Earth’s surface.  Without it, the temperature would be too great for life.  Radiative energy emitted from the surface can pass through the window to space with only moderate absorption because the are few greenhouse gas wavelengths within the wavelength range of the window. 
This range is generally taken to be from 8 to 14 microns.

The following hypothesis gives the required values for the temperature of the Earth’s surface both for an atmosphere with, and also without, greenhouse gases and water vapour, and hence quantitatively explains the “33 deg C rise”.  It also shows that with the current value of window power, increased greenhouse gas concentrations have only a limited effect on surface temperature, but any significant reduction in the window power can have serious results.

The Atmospheric Window
Reference (1) shows the transmission of radiative energy emitted from the surface of the Earth through the window.  The transmittance is between 80% and 70%, but reducing to zero through the 13 and 14 micron sections.  A computer program for the Planck distribution at 288 K has enabled the transmitted power to be calculated, and allowance has been made for the amount absorbed.  It provides a value of 90.2 Wm^-2 for the range of microns. 

The Planck curve for a surface temperature of 288K, and this window power of 90.2 Wm^-2 is given in Fig 1.  It must be pointed out that this data is the total energy in Joules per square metre per second, radiated across all wavelengths.  This is in line with the Stefan-Boltzmann equation. 

The Planck figures, however, apply in terms of the steradian, and must be multipied by ‘pi’ to achieve agreement with the S-B figures.

A new approach
In the following treatment, the term “Greenhouse Gases” includes water vapour and clouds, in addition to carbon dioxide, methane and all radiative energy absorbing molecules.

The greenhouse gases have very little effect within the window and so photons with wavelengths within the window pass through to space with only a little absorption.  Radiation from the Earth’s surface is absorbed and re-emitted by the greenhouse gases with wavelengths outside the window.

Fig 2 shows schematically the basis of this new approach.

The hypothesis starts with the emission of radiant energy from the Earth’s surface in line with the equation of Stefan-Boltzmann for a “black body”.  This is acceptable for Earth with an emissivity taken to be 0.98

Let   P = output power from the surface in Wm^-2
e = emissivity of the surface
s = Stefan-Boltzmann constant, 5.67*10^-8 Wm^-2K^-4
T = surface temperature in K
w = power emitted through the window to space, Wm^–2
Take T = 288 K and e = 0.98
P = e.s.(T^4)  …………………..(1)
= 0.98*5.67*(10^–8)*(T^4)
= 382.28 Wm^–2
This is the power emitted as photons from the surface of the Earth into the atmosphere.
Some of this power, w, escapes directly into space through the window.  ( little greenhouse absorption in the window)
Let   P1 be the power remaining in the atmosphere.
So    P1 = P – w
But there are GHGs effective in the wavelengths outside the window, and so absorption and emission occurs here.
Now, it may be that not all of the energy (P – w) is absorbed/emitted.  This might be due to insufficient greenhouse gases in the atmosphere, or too small a molecular cross section for absorption.
So let  f  be the energy absorption factor combining these effects.
If all the radiative power is being absorbed, then  f = 1.0
If none of the radiative power is being absorbed, then  f = 0.0
Therefore, the power absorbed and then re-emitted is (P – w)f.
Since greenhouse gas molecules emit photons equally in all directions, the power radiated upwards is 0.5(P – w)f , and this is equal to the power radiated downwards, 0.5(P – w)f.
However, if the energy absorption factor f is less than 1.0, there is energy still left unaccounted for in the atmosphere.  Let this remainder be R.
Therefore, it follows that  R = (P – w)(1.0 – f)
So the total power into space  = w + R + 0.5(P – w)f
And for Earth’s energy balance this must equal 239 Wm^-2.
So           w + R + 0.5(P – w)f  =  239
Hence,   P(1.0 – 0.5f) + 0.5wf  =  239
So           P = (239 0.5wf ) / (1.0 – 0.5f )
Substituting for P from eqn (1), this gives
T^4 = (239.0 –­ 0.5wf ) / (e.s.(1.0 – 0.5f ))
T^4 = 0.179966*(239 – 0.5wf )*10^8 / (1.0 – 0.5f )

The value of  T  has been determined for a range of energy absorption factors f,  and for specified values of window power w; the results are given in Fig 3.

For the current window power  w = 90.2 Wm^-2 , it shows that the surface temperature of 288K is obtained with an energy absorption factor  f = 0.981

If there were no greenhouse gases in the atmosphere, or indeed, no atmosphere, this would be equivalent to zero energy absorption factor, f = 0.0.  This gives a temperature of 256 K, which is correct for an emissivity of 0.98
This provides the temperature rise of 32 deg C.

The energy flux returning to the surface from the atmosphere is 0.5(P ‒ w)f.
For T = 288 K, P = 382.28 Wm^-2, and the value of  w = 90.2 Wm^-2, the downward power to the surface is 143.27 Wm^-2.  In addition, there is 239 from the Sun, making a total of 382.27 Wm^-2, as required for energy balance.

Fig 4 shows the critical role of the window.

For any value of w, the temperature cannot exceed that given by the curve for f = 1.0, because with f  = 1.0 all of the radiant energy in the atmosphere is already being absorbed and emitted by the greenhouse gases.  Further increases in greenhouse gas concentrations will, therefore, have no effect. 
With the current window of 90.2 Wm^-2, the temperature of 288 K is obtained with f = 0.981
It is clear that an increase of 1 K could occur, or has already occurred, if greenhouse gas concentrations increased f to 1.0   But no further rise in temperature would happen. Perhaps this could explain the “hiatus”.
However, if the window power were to be reduced, the results would be serious. 
Fig 5 shows the temperature increases for convenience. Even without any more carbon dioxide, the temperature rise with w = 0 could be 15 deg C.

The potential problem is due to the increasing use of compounds of fluorine; particularly the CFCs and the HCFCs.  Also, sulphur hexafluoride.  These substances have very significant wavelengths within the window, and so are very dangerous.   Fortunately, these are man-made substances, and so in principle it should be possible to exert some control on their use, in accordance with the Montreal Protocol.
However, these ozone destroying substances are being superseded by HFCs which also have high radiative absorption wavelengths within the window. And so the problem continues.

Ref (1)

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