Energy Accounting and Carbon Dioxide

 

A E Banner, June 2020

A great amount of work has been done over many years by brilliant climate scientists to establish the details of the Greenhouse Gas Theory, whereby increases in the concentration of carbon dioxide in the atmosphere over the last 50 years or so are claimed to have caused increases in the surface temperature of the Earth

The explanation seems to come from the well-known energy diagrams of Trenberth et al, followed by others, including NASA.  Unfortunately, the diagrams are puzzling, principally because of the large amounts of energy shown to be circulating in the atmosphere and being sent to the Earth.  The energy escaping to space is correctly shown equal to the energy received from the Sun, but it is not clear from where these extra amounts derive.  Moreover, the figure given for the power of the atmospheric window, a highly significant feature, although changed, is still not in agreement with the figure used in this work, which is based upon the reference and the theory given below. 

However, the Greenhouse Gas Theory is, in fact, correctly based upon the fundamental property of carbon dioxide and some other gases, water vapour in particular, to absorb radiative photons of certain energies, and subsequently to re-emit this energy equally in all directions.  Nevertheless, the computer models are still said to be “running hot”, and with the difficulties mentioned above, it seemed to be worth a re-evaluation of the problem.

When considering the possibility of global warming, there is a fundamental principle which applies.  This is the requirement for energy balance: the power emitted to space by radiative emission from the Earth’s surface must be equal to the power received by the Earth’s planetary system from the Sun.  This is generally taken to be 239 Wm^-2.

The Earth’s surface emits electromagnetic energy as photons, very small quanta of energy, with a wide range of wavelengths from 400 microns down to 2 microns, in accordance with Planck’s Law.  It should be noted that the Planck program gives values in milliWatts per square metre per cm^-1.  This applies for emission per steradian, but for our purposes, this needs to be converted to units of Watts per square metre per cm^-1, in line with the Law of Stefan-Boltzmann which covers the full wavenumber spectrum.  This involves multiplying the Planck numbers by “pi”.

This paper is essentially about energy accounting.  

All of the energy, and therefore power, emitted from the Earth’s surface must be accounted for, and its destination determined. 

All the radiative power emitted from the Earth’s surface is shown in column A of Table 1 as the Planck distribution for the full range of wavelengths for a surface temperature 288 K, duly converted into units of Wm^-2.  The emission between wavenumbers 25 and 714 cm^-1 is absorbed by the greenhouse gases, and is entered into column C.  

The power between wavenumbers 715 and 4500 cm^-1, flows through the atmospheric window and some of it escapes directly to space, but allowance must be made because the proportion transmitted is well short of 100%, depending on the actual wavenumber.  The successful, transmitted proportions are shown, together with the other window data, in green italics in the table. 

Insert Fig 1 Planck

Ref (1) has enabled initial, approximate visual estimates to be made of the proportions of power transmitted in each micron section within the window.  Later calculation from theory, see below, suggests that the overall power of the window is 96.26 Wm^-2, and the successfully transmitted powers are shown in column B.  The differences between columns A and B represent the powers which are not successfully transmitted, because of absorption, and are entered into column C.

To summarize, photons which do not escape directly to space are subject to absorption by the greenhouse gases in the atmosphere.  These are within the wavenumber range from 25 to 714 cm^-1, and comprise the power in column C, which is the value in column A minus the value in column B, ie (A – B), so for these wavenumbers the entry in Column C is equal to that in column A.  However, the energy which is not transmitted directly to space through the window, which is value in A minus value in B, must also be entered in column C. 

In accordance with the Greenhouse Gas Theory, half of this power in column C escapes to space, and is given in column D.  So, the total power to space is the sum of columns B + D, that is column E. Note, also, that the value in column D is also equal to the power returning to the Earth’s surface, where it is added to the power received by Earth’s planetary system, 239 Wm^-2, to make the required value of 381.74 Wm^-2, the amount emitted by the surface.

It is seen that for the value of window used, 96.26 Wm^-2, and 288K, the total power to space is equal to 239 Wm^-2, as required.  This means  that all the power which is not directly transmitted to space has been absorbed by the greenhouse gases and then re-emitted; and the energy flowing directly through the window has also been taken into account.  Therefore, the total energy radiated by the Earth’s surface has been accounted for.  Therefore, no further increase in carbon dioxide can cause any further increase in temperature.

The 32 deg C Rise

Consider the atmosphere without any greenhouse gases.  Planck results show that the total radiative emission from the Earth’s surface at about 256 K is 239 Wm^-2. 
Suppose greenhouse gases are added so that some energy absorption can occur.  Let the amount be x, and in a new table this would be added into column C, and the corresponding increase in column D would be 0.5x. 
But, absorption of x in C means a decrease of x in B. 
So E increases by (0.5x – x), which is a decrease of 0.5x, and this causes the total output to space to fall below 239 Wm^-2.  This continuing process is corrected by increase in temperature until stability is achieved at 288 K with all the power absorbed, apart from the window power.

Table 1
Micron range  400  14
Planck energy           196.23 Wm^-2     A
Transmitted power       0                        B
Absorbed power       196.23                   C  =  (A – B)
50% Absorbed pow.   98.12                   D = 0.5*C
Power to space            98.12                  E  =  (B + D)

Window
Micron range
   14   2
Planck energy           185.51 Wm^-2     A
Transmitted power      96.22                  B
Absorbed power          89.29                  C  =  (A – B)
50% Absorbed pow.    44.65                  D = 0.5*C
Power to space           140.87                 E  =  (B + D)
Total Power to space    239 Wm^-2


The Atmospheric Window
The power transmitted directly through the window without absorption by greenhouse gases is of critical importance in determining surface temperature.  The values in Table 1 were calculated for the theoretical window power of 96.26 Wm^-2, derived below.

Let P be the power emitted from the Earth’s surface into the atmosphere.
      w be the power transmitted directly through the window to space.
Then the power in the atmosphere equals ( P – w )
But, by greenhouse theory, half of this is emitted upwards, and half is emitted downwards.
So the power emitted to space is 0.5( P – w ). This adds to the power through the window, w.

Therefore, the total power to space = 0.5(P – w ) + w
                                                         = 0.5P – 0.5w + w
                                                         = 0.5( P + w )

This must be equal to 239 Wm^-2.
So 0.5( P + w ) = 239
             P + w = 478
                    w = 478 – P
At 288 K, P = 381.74 Wm^-2
So          w = 478 – 381.74
              w = 96.26 Wm^-2     This is the value used in Table 1 and in the figure.

The figure below shows the total emission of power to space for a range of surface temperatures, and for selected values of window power, confirming the window of 96.26 Wm^-2 is consistent with the required emission of 239 Wm^-2 at 288 K. 

A counter-opinion will doubtless be expressed by the confirmed believers in the greenhouse gas theory that this work is nevertheless wrong because not all the Earth’s emitted energy has been absorbed and that there is still insufficient carbon dioxide to accomplish this absorption.  This is clearly a false idea.  If some “excess” energy were not absorbed, it would have to be removed from columns C and D in the table, and put instead into column B.  But a decrease of 0.5x in D means in increase of x in B, so the total B increases, and so does the total emission to space, column E. 

An increase in B can be considered as an increase in window power as in Fig 2, which shows that the required 239 Wm^-2 would have already been attained at a lower temperature, and so the counter-opinion is groundless. 

Although it has been shown that the usual GHGs cannot cause any further temp increase, there remains nevertheless a potential problem with hydrochlorofluorocarbons HCFCs an chlorofluorocarbons CFCs being admitted into the atmosphere.  Also, sulphur hexafluoride.  These compounds have very high absorption cross sections at wavelengths within the atmospheric window, and so could reduce the window power, thereby causing an increase in temperature to be needed in order to achieve the required 239 Wm^-2 to space, as in the figure. 

Summary

With the present conditions of 288 K Earth surface temperature and the required 239 Wm^-2 power outflow to space, all of the power emitted by the surface is transferred to space, apart from that returned to Earth. Therefore, no increase in greenhouse gas concentration can cause a further rise in surface temperature.

A warning: fluorocarbon compounds can cause serious temperature rises. 

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