Absorption of Earth’s Radiated Energy

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A E Banner November 2021

Introduction

The present climate change disasters are being explained by anthropogenic “global warming” caused by increases in the concentration of carbon dioxide in the atmosphere.  This is the classic “Enhanced Greenhouse Gas theory” which has been around for nearly forty years, but has ever been flawed by trying to cope with the whole planet on an average basis. 

A new treatment is offered here which deals with individual areas of one square metre at a specified latitude.  This involves the investigation of energy balance considerations, in that the energy output per second of a body must be equal to the energy input per second.  So, in fact, we are dealing with power balance, and the units are Watts per square metre, Wm^-2. 

The Enhanced Greenhouse Gas theory says that if the concentration of carbon dioxide in the atmosphere is increased, then the temperature will increase because the greenhouse gases absorb more of the energy radiated from the surface, and some is retained, so raising the temperature. 

This treatment requires two fundamental requirements to be satisfied. 
The output power to space must be equal to the input power received from the Sun.  Were this not so, the Earth would either heat up or cool down appropriately.  This requirement is also maintained by the Enhanced Greenhouse Gas theory. 

The other condition is that the Law of Conservation of Energy must be obeyed, and it applies to energy in all of its forms. It states that energy can neither be created nor destroyed, but it can be changed from one form to another. 

This law is supported by the Second Law of Thermodynamics, although this is principally concerned with sensible heat energy. 

The Law of Conservation of Energy seems to have been overlooked by the Greenhouse Gas theory.  

The radiative power output of a body depends upon the temperature and can be calculated from Planck’s Law, where the temperature is on the Absolute Scale, and so, a temperature of T degrees Celsius is (T + 273.2) Kelvin, K.  The fourth power of the temperature applies.

An attempt is made here to investigate the matter.  It is assumed here that a proportion, ap, of the absorbable power radiated from the Earth’s surface is absorbed. 

The absorbed proportion is ap, and must be between 0.0 and 1.0

The problem involves detailed consideration of the power flow in the Earth’s system, as shown in Fig 1.  The power is in units of Watts per square metre.

Theory of power flow in the Earth’s system
 
Let  ap be the proportion of the absorbable surface radiated power which is absorbed.
G be the power from the Sun arriving at the surface, in Wm^-2
r  be the energy absorption factor
i  be the angle of incidence of the input from the Sun
P  be the total output power radiated from the surface
L  be the longwave output power radiated from the surface
w  be the shortwave output power radiated from the surface
f   be the atmospheric window factor


We can now examine the power flow in the Earth’s system. 
Please refer to Fig 1.

The total power output from the land surface is shown as P, and this depends on the Absolute Temperature in Kelvin.  It can be determined from the Planck equation. 
This power comprises both longwave  L, and shortwave  w, energies. 
The longwave range of wavelength is from about 2000 microns to 14 microns. 
The shortwave range of wavelength is from 14 microns to about 2 microns. 

A proportion  f  of this shortwave range is not available for absorption, because of the lack of greenhouse molecules with absorbing wavelengths in this region.  The value of f has not been extensively determined, but it must be between zero and unity. There is some evidence that the value is about 0.493, determined by close examination of the figure, Ref (2).

This provides the atmospheric window, fw, for which the power escapes directly to space. fw …………SPACE
This leaves a power of  (1 – f )w  in the atmosphere. 
But, only a proportion ap is supposed to be absorbed because of insufficient GH gases, and so a power of  (1 – ap)(1 – f )w escapes directly to space.
(1 – ap)(1 – f )w ………….SPACE
This now leaves a power of  ap(1 – f )w  in the atmosphere which is absorbed, and in line with the Greenhouse Gas theory, half of this is re-emitted upwards to space 0.5ap(1 – f )w ………….SPACE

And the same amount returns to Earth 0.5ap(1 – f )w ………….Earth

The longwave power emitted has two components as shown. If there is insufficient carbon dioxide, a proportion (1 – ap) cannot be absorbed, and so
this this power escapes directly to space. (1 – ap)L …………SPACE
This leaves an amount ap.L in the atmosphere.
Again, in line withe the Greenhouse theory, half of this is re-emitted upwards to space 0.5.ap.L ………….SPACE

and half is re-emitted to Earth. 0.5.ap.L …………. Earth

Therefore, summing the several contributions,

Therefore, summing the several contributions,

Power escaping to Space = fw + (1 – ap)(1 – f)w + 0.5ap(1 – f)w + (1 – ap)L + 0.5ap.L
This simplifies to:-
Power escaping to Space = fw  + (1 – 0.5ap)( L + (1 – f )w  )    …………………………….(1)

Power at the Earth’s surface
Let i be the angle of incidence of the Sun’s input power to the surface, that is the angle between the Sun’s rays and the normal to the surface. 

When dealing specifically with the maximum Land Surface Temperatures (LST) at local noon in mid-June, the appropriate tilt angle of the Earth’s axis in its orbit around the Sun is 23.4 degrees.  The angle if incidence is then given by ( latitude – tilt ).

So, for example, with the tilt angle of 23.4 degrees, then a true latitude of 35.0 degrees gives an angle of incidence  i = 11.6 deg.

Let the input power from the Sun arriving at the Earth’s surface at normal incidence
be G Wm^-2.  In the calculations, the cosine is involved, so it is necessary to include the expression cos(i).

So, the power arriving at the surface at angle of incidence  0.0  is  Gcos(0) Wm^-2.
and thepower arriving at angle of incidence is Gcos(i) Wm^-2.

Not all of this power ENTERS the surface.  Some is reflected away.  The proportion which enters the surface and is absorbed is explained by the work of Fresnel (1) and it is seen that this proportion reduces to zero at 90 deg latitude.  Therefore, a new term is introduced, the energy absorption factor, r, which is changes with the angle of incidence i and so with latitude. Therefore, the power from the Sun entering the surface and being absorbed, is G.r.cos(i)

Power balance in the Earth system

The energy absorption factor r is based on the work of Fresnel, and is estimated to be approximately r=0.8

Equation (1) gives the output power to space:-
Power to space[a]  =  fw + (1 – 0.5ap)( L + (1 – f)w )       

The input power from the Sun at noon is G.r.cos(i)

There is also power input from the Earth Return.  From Fig 1:-
Earth Return = 0.5ap.L + 0.5ap(1 – f )w
                      = 0.5ap( L + (1 – f )w )

Therefore, for power balance
G.r.cos(i)
  =  fw + (1 – 0.5ap)( L + (1 – f)w )

And this simplifies to   G.r.cos(i)  =  fw  +  (1 – ap)( L + (1 – f)w )  ………….(2)

The solar constant is 1362 Watts per square metre. 
NASA says that there are losses in the atmosphere, so the power which reaches the Earth’s surface is only 48% of this.  That is G = 653.76 Wm^-2.

The chosen location for the particular square metre was somewhere to the West of Chicago at latitude 43.26 degN and longitude -93.88 degW.   The land surface temperature (LST)  was obtained from Copernicus for local noon at the Summer solstice in 2020.  The temperature was 305.69 K Various values of r, the energy absorption factor, were tried to find power balance at this temperature, with Planck giving L, w and A.  The atmospheric window factor f was taken to be 0.5  For the Copernicus LST of 305.69K, and ap = 0.0,  a power balance is found with the energy absorption factor r equal to 0.789 , as shown below. The output power to space is shown as “output”, and the Earth Return as “ereturn”.  The sum of these two values comprise the “total”.  Energy is conserved as required, even though the absorbed proportion ap has been increased, as shown.

We have LST = 305.69 K and Planck gives the figures for L, w and A

ap         r        input         T             L            w           A        output ereturn total      
0.0    0.789    484.71    305.69    228.41    256.16    356.49    484.57        0.00    484.57
0.5    0.789    484.71    305.69    228.41    256.16    356.49    395.45      89.12    484.57
1.0    0.789    484.71    305.69    228.41    256.16    356.49    306.33    178.25 484.57


Discussion

The greenhouse gas theory claims that, starting from an initial condition of power balance, increased concentration of such gases in the atmosphere will cause more power to be absorbed from the power radiated to space by the Earth’s surface.  This absorbed power is then re-emitted, half resuming its way to space, and half going to Earth, which is called the Earth Return (ereturn) in this paper. So less power goes to space.  This is shown satisfactorily in the table in the column called “output”.

The system is now out of balance and so, heated by the power from the Sun, the surface temperature tends to rise to enable it to radiate more power, if available, and so to restore balance. 

But the Law of Conservation of Energy, means that the surface cannot MAKE any energy to radiate.    

A new supply is required if more is to be radiated.  


The program was run again, but for an assumed temperature of 1 degC higher;
namely 306.69 K, in order to determine the powers involved. 

ap         r        input         T             L            w           A        output ereturn total      
0.0    0.789    484.71    306.69    230.28    260.66    360.61    490.94        0.00    490.94
0.5    0.789    484.71    306.69    230.28    260.66    360.61    400.79      90.15    490.94
1.0    0.789    484.71    306.69    230.28    260.66    360.61    310.64    180.31    490.94

The table shows the data for the “output” radiated to space for absorbed proportions of 0.0, 0.5, and 1.0, together with the associated Earth Return, and the “total” of these two figures.

It is seen that the total is now out of balance with the input at this higher temperature, and requires an extra input 6.23 Wm^-2.  This is not possible because the initial condition was for the full input from the Sun.   The only other energy in the system is the Earth Return.  This is limited to half the amount absorbed from the initial balanced condition.  It is now in the state of sensible heat energy, maintaining the surface temperature, but might supply the surface for radiation.  But in spite of the higher temperature, the limit means that no more power can be radiated than before the supposed increase in greenhouse gas concentrations. In fact, it will be less, because some energy will still be required to maintain the supposedly higher surface temperature.

Therefore, the final effect will be that there is no increase in temperature, even though there has been an increase in greenhouse gas concentration.  

It is seen that power balance is maintained at the same temperature 305.69 K for all values of ap.


Summary

In the stable condition of power balance,
          output power to space is equal to input power from the Sun.

More greenhouse gases are emitted into the atmosphere
          (a) more power is absorbed from the output power.
          (b) half rejoins the output power to space
          (c) half returns to Earth, the Earth Return
          (d) the output power is no longer in balance with the Sun’s input power

The Greenhouse Gas theory claims that balance is restored by more power being added to the output, resulting from a rise in temperature.  So this means the output power to space is now equal to its value before the extra was added. 

BUT, there is also the Earth Return still to be taken into account. 

This means that there is now more power than before the extra GH gas was emitted, and this violates the Law of Conservation of Energy. 

The only solution to the problem is that the energy of the Earth Return was itself re-radiated as output power to space, and that this was the power that restored the output to its initial value before the emission of the extra gas.  The values are exactly correct. 

And, of course, this would happen without any rise in temperature. 

Therefore, the Law of Conservation of Energy means that the Enhanced Greenhouse Gas theory for global warming is invalid.
 
Increased emissions of carbon dioxide, methane and other such gases are not the cause of the global warming from which the Earth is suffering. 

A better idea is required, but which is clearly related to anthropogenic activities.

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