Revised Greenhouse Gas Theory

 A E Banner March 2020

Introduction

The Greenhouse Gas Theory is intended to explain the increase in the temperature of the surface of the Earth in the last few decades, due to the effects of the actions of human-kind.  This is known as “anthropogenic global warming”.  The theory depends upon the property of the so-called “greenhouse gases” in the atmosphere to absorb electromagnetic energy of certain wavelengths and then to re-emit this energy into the atmosphere, when the process can then be quickly repeated.  The energy is absorbed/emitted in quantum amounts called “photons”, and is specific to the particular gas concerned.  A fundamental fact is that the energy is emitted equally in all directions, and so energy emitted upwards is equal to that emitted downwards.  The energy emitted downwards warms the Earth’s surface.

The most important gases are water vapour and carbon dioxide, and it follows that more carbon dioxide in the atmosphere will cause more warming. 

The theory also requires that the Earth should be in energy balance, and so the power emitted to space must be maintained equal to the power received from the Sun.  This is achieved by changes in the surface temperature, in line with the Stefan-Boltzmann law for “black body” radiation.   

These two tenets are not in dispute, and are generally accepted by the scientific community. 
They are included in the following revised treatment.    

Calculation of the numerical relation between the surface temperature increase and the concentration of atmospheric carbon dioxide is very complicated because there are many factors involved.  Although HITRAN provides the absorption cross section for carbon dioxide, this may be modified by the pressure, and there may be cross contribution from other gases at particular wavelengths.   

Again, the power received by the planet depends upon the surface reflectivity, the albedo, which in turn might be affected by deforestation.  Yet another problem is the effect of aerosols emitted into the atmosphere.  So, altogether, the determination of the total “radiative forcing” is an extremely difficult problem.  A huge amount of work has been done over many years by brilliant climate scientists to produce models to emulate these processes, but there are still claims that the models are “running too hot”.  That is, calculating temperatures noticeably greater than observed figures.

Therefore, the following revision of the GHG theory of recent years is approached in a different way.  Rather than trying to improve on the radiative forcing calculations, and so to produce a quantitative theory, this revised method starts instead with the known effects of the greenhouse gases in the atmosphere, and then proceeds to include the requirements of energy balance. 

It is generally accepted that in the absence of anthropogenic effects the power emitted to space is 239 Watts per square metre, and so this must also be the power received by the planet from the Sun.  This power emitted to space comprises the energy of the upwards flowing photons having gone through absorption/emission by the ghgs, together with the power of the Atmospheric Window.  This is found to be a critical feature of the global warming problem and seems not to have been adequately addressed in previous explanations of the GHG theory. Trenberth et al have previously suggested 40, and again, 22 Watts per square meter for the power transmitted to space through the window, but these figures are shown here to be serious under estimates, and this has important implications.

The following revision enables correct determination of the required values for the temperature of the Earth’s surface both for an atmosphere with, and also without, greenhouse gases and water vapour, (or indeed, no atmosphere), and hence quantitatively explains the “32 deg C rise”.  It also shows that with the recent value of window power determined below, increased greenhouse gas concentrations can have only a limited further effect of about 1 degC on surface temperature. This, in turn, offers a credible explanation for the “temperature hiatus”, starting around 1998.

Any significant reduction in the available window power can have serious results.

The Atmospheric Window
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 of the appropriate wavelengths emitted from the surface can ass through the window to space with only moderate absorption because there are few greenhouse gas wavelengths within the wavelength range of the window.  This range is generally taken to be from 8 to 14 microns.                                                       

Reference (1) shows the observed 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 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 of the appropriate wavelengths emitted from the surface can pass through the window to space with only moderate absorption because there are few greenhouse gas wavelengths within the wavelength range of the window.  This range is generally taken to be from 8 to 14 microns.                                                       

Reference (1) shows the observed 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 for each micron segment within the range, and allowance has been made for the amount absorbed.  It provides a total value of 90.2 Wm^-2 for this 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 revised approach
In the following treatment, the term “Greenhouse Gases” includes water vapour and clouds, in addition to carbon dioxide, methane and all the other radiative energy absorbing molecule
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. 
This power emitted to space is represented by w.
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 revised approach.

Fig 2 The energy in the atmosphere

It 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^-1
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, because there is little greenhouse absorption in the window.
Therefore, the power remaining in the atmosphere is (P – w).
But there are greenhouse gases 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 as shown, 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 to 289K could occur, or has already occurred, if greenhouse gas concentrations increased f to 1.0   But no further rise in temperature would happen.
This may be an explanation for the “temperature 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 could continue.

Summary
The Revised Greenhouse Gas Theory, is much simpler than previous versions.  It certainly depends upon the effects of the several greenhouse gases, and Stefan-Boltzmann radiative emission from the Earth’s surface, but it avoids the complicated problems of a multi-layered atmosphere. 
It gives the correct values for the two “fixed points” of Earth’s surface temperature, namely for the conditions of “no atmosphere”, 256 K, and for the atmosphere with pre-anthropogenic concentrations, 288 K.   And hence the “32 deg C rise”.

It shows the importance of the greenhouse gases, and clearly demonstrates that the effect of carbon dioxide is almost exhausted, and so more CO2 would not cause any further rise.  Indeed, this condition may already have been reached around 1998, with the ensuing temperature “hiatus”.

Further temperature increases since then might be attributable to increased atmospheric concentrations of fluoride gases, HFCs, because of their absorbing wavelengths within the window.  This would reduce the available window power, and so cause surface temperature increases as explained above.  Unfortunately, information on these concentrations is not immediately available. 

Ref (1) https://en.wikipedia.org/wiki/Infrared_window#/media/File:Atmosfaerisk_spredning.png

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