The Atmospheric Window

The Atmospheric Window

A E Banner     Oct, 2020 

Introduction

The Earth receives energy from the Sun, and must emit an equal amount to space in order to achieve the requisite energy balance. The energy is in the form of electromagnetic radiation which can readily pass through space.  The energy from the Sun is at very short wavelengths and some can pass through the atmosphere, although a significant amount is reflected into space.  The amount of power reaching the Earth’s surface is generally accepted to be 239.7 Watts per square metre of the surface, for an input beam incident normal to the surface, as would be the case if the Sun were directly overhead.  This amount of power decreases as the latitude, and therefore the incident angle, increases. 


The necessary energy (and power) balance is achieved by radiative emission from the Earth’s surface, in line with Planck’s Law.  Much of this power is absorbed and then re-emitted in accordance with the Greenhouse Gas theory, but a significant amount escapes directly to space.  This is called the “atmospheric window”, and plays an important part in the explanation of global warming.

The Atmospheric Window

The electromagnetic radiation emitted from the Earth’s surface comprises a wide range of wavelengths from nearly 10,000 microns down to 2 microns.  In order to deal with energy (and power), it is necessary to work with wavenumbers, so the corresponding range is from 1 to 4500 per cm, ie cm^-1.  A typical Planck curve is shown in Fig 1.  It should be noted that the energy is proportional to the fourth power of the Absolute temperature.



The units of the Planck power calculations are given here as milliwatts per square metre per cm wavenumber; that is milliW.m^-2.((cm^-1)^-1).  This applies for output per steradian, but in order to align with Stefan-Boltzmann results which are in Wm^-2 for total angular output, the Planck figures must be multiplied by π.

The energy of the range of wavenumbers up to about 714 cm^-1 is absorbed by the greenhouse gases in the atmosphere, in accordance with the Greenhouse Gas theory, and 50% is radiated to space and 50% is returned to the Earth’s surface. 

At shorter wavelengths down to about 2 microns and, therefore, greater wavenumbers, there is little absorption because of the very small absorption cross-section of carbon dioxide in this region, and so some of the energy in this range can escape directly to space.  This is known as “the atmospheric window”.  The remaining energy in this range is absorbed and re-emitted as usual.  The atmospheric window is from about 14 microns down to 2 microns overall, but there is some discontinuity because of the presence of water vapour.  

The proportion of energy escaping directly to space is the transmission factor  f  which will be referred to later, in the appendix. It is clear from a consideration of latitude, and the change of the tilt angle of the Earth’s axis with respect to the Sun as the Earth travels in its orbit, that the amount of power received per square metre of the Earth’s surface is dependent on the incident angle of the energy and the Earth’s position in its orbit.  Appendix A derives the relationship below.

Atmospheric Window = 2*Gcos(lat – tilt)  – P  

where G is the amount of power for normal incidence, (zero angle),  and  P is the power radiated from the surface, depending on the fourth power of the Kelvin temperature.

Therefore, the atmospheric window does not have a fixed value, but it changes from one latitude location to another, and it changes also from month to month throughout the year.

It will be seen from the first term on the rhs of the equation that the input power from the Sun received at the Earth’s surface will be greatest at low latitudes, but this is where the temperature is also at its greatest and so is the radiated surface emission power P due to the fourth power dependency of the Planck Law.  Therefore, the window will not have a maximum at low latitudes. 

Again, the power received at the surface (first term) reduces towards zero as the latitude increases, and so the atmospheric window also tends to zero at higher latitudes.

Therefore, it follows that the atmospheric window will have a maximum at some mid-range latitude.  This is supported by the results below.

Results

In order to verify this statement, it was necessary to obtain data for typical temperatures at various locations in latitude.  To this end, certain cities were chosen at the required latitudes, and the appropriate recorded temperature history was obtained in each case covering a series of 12 months.  A list of the cities and latitudes is given in Appendix B. 

It was necessary to correct for the Urban Heat Island effect in each case.  There is considerable discussion on the internet about this effect, but although little hard figures are available, there seems to be a general idea that 3 deg C may be appropriate for large cities, and this figure has been used generally in this work. 

However, the city of Houston in Texas is an exception.  An excellent paper by David Streutker was published in 2002 which gave the UHI basically as 3.0 degC then, and it has been assumed to be 6.0 degC by now for this work.  Baghdad has also been the subject of extensive investigations; its heat island is thought to be sufficiently bad to warrant the term Super Urban Heat Island, SUHI, and 10.0 degC has been used for Baghdad in this work.   

Therefore, the UHI temperature was subtracted from the observed temperatures for each city in order to obtain a reasonable estimate of the true temperatures at that latitude in the absence of human interference.    

The Atmospheric Window in each case was calculated from the formula in Appendix A and the results are provided for 16 locations in the Northern Hemisphere.
 
Some idea of the importance of the UHI can be seen in the first set of results, March to August, where two values are shown for Houston (at 29.76 degN) by means of  + , for 3.0 degC and 6.0 degC, which is the upper one.

It is clear that the atmospheric window in the northern hemisphere is most significant in April and May, reaching a power of 80 Watts per square metre at latitudes between 30 and 40 degN. 

The results presented here are simply approximations because of the lack of true experimental measurements of the temperatures at specific latitudes, but such measurements are themselves subject to temporary meteorological effects.  It could be that the best approach for further work might be to pursue the method here used, but with much better information about the local UHI values.

Appendix A

Earth’s Surface Emission 

The Earth’s surface emits electromagnetic energy as photons, very small quanta of energy, with a wide range of wavelengths, in accordance with Planck’s Law, in two broad regions; 400 to 14, and 14 to 2 microns.   

The emitted Planck power for long wave region 400 to 14 microns is denoted as u, and the power for the shorter wavelengths 14 to 2 microns, which is the atmospheric window region, as w.

For example, at a surface temperature of 287.6 K, (14.5 degC), the Planck Law provides the values
u  = 195.52 Wm^-2,
 and w  = 184.10 Wm^-2, so that the complete power emission is 379.62 Wm^-2.   

The following work applies for a surface area of the Earth of 1 square metre.

Long wavelength   400 to 14 microns                                                                                                                                    Power emitted from surface = u
Power absorbed in atmosphere = u

 In accordance with natural Greenhouse Gas theory, half of this absorbed power is re-emitted upwards to space, and half is re-emitted downwards to the Earth’s surface.
So,   power to space  =  0.5u   ———————–space

and   power to Earth  =  0.5u    ———————-Earth return

 
Short wavelength   14 to 2 microns
For this wavelength range, the power emitted from 1 square metre of the surface into the atmosphere is w.
A fraction f of this w is transmitted directly to space, and so the term “atmospheric window” has come to be applied to the amount of power, fw, escaping without absorption, per square metre.  So f must be within the range 0 to 1.

The remaining power, (1 – f )w, is absorbed and re-emitted, 50% to space and 50% to Earth.

Power to space  =  fw     ——————————space
Power to space  =  0.5(1 – f)w    ———————space

(Power to Earth  =  0.5(1 – f)w  ———————–Earth return The power returning to the Earth’s surface, the “Earth Return”, is of vital importance because it is needed, together with the power received by the surface from the Sun, to supply the power emitted from the surface in accordance with the Planck Law for the given temperature.)

Summing the three terms to space,

Total power to space  = 0.5u + fw  +  0.5(1 – f)w   ———————–(1)

For energy balance, this must be G Watts per square metre, assuming normal incidence.
Therefore            0.5u  +  fw  +  0.5(1 – f)w  =  G
                            0.5u  +  fw  +  0.5w  –  0.5fw  =  G
                            0.5(u + w)  +  0.5fw  =  G
                            fw = 2G – (u + w)

Now, (u + w) = P      where P is the total power emitted per square metre surface

Therefore,            fw = 2G – P    Watts per square metre

This is the Atmospheric Window  if power from the Sun is incident normal to the Earth’s surface, as may happen at the Equator. 

However, the angle of incidence increases as the latitude increases, and so a cosine factor must be applied to the first term, which becomes 2Gcos(incident latitude angle). 

Again, as the Earth moves in its orbit around the Sun, the tilt angle of its axis with respect to the Sun also changes, and so the effective angle must also be modified.

Starting at the Northern winter solstice in December, the cosine term becomes
cos(latitude – tilt), where

tilt =  – 23.5cos(n*30.417)   where n is the month number starting with 0 for December.
Hence,     Atmospheric Window = 2*Gcos(lat – tilt)  – P

Therefore, the atmospheric window is not an invariable constant, but depends on the latitude and the orbital position of the Earth with respect to the Sun.








Physics proves no more global warming

 


Introduction
The Greenhouse Gas Theory is 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.  However, after 30 years with little progress in satisfactorily establishing the theory, it seemed to be worth a complete re-evaluation of the problem, from a different starting point.

Hitherto, the general approach has been to consider the workings within the atmosphere, whereas the present paper deals instead with the simple Physics and Maths of the energy from the Sun which reaches the Earth’s surface, and the energy emitted to space by the Earth.

There is a fundamental requirement for energy balance, which means that the power emitted to space by radiative emission from the Earth must be equal to the power received by the Earth’s surface from the Sun.  This is generally taken to be 239.7 Wm^-2.  

It is clearly shown that at current temperatures all the energy is totally accounted for, so that an increase in concentration of carbon dioxide cannot produce an increase in temperature because there is no more energy left to be absorbed.   

The graph on the website of the GWPF (1) shows the Earth surface temperatures from 2001 to 2019.  The values of the ordinates have been manually measured from the base temperature of 14.0 deg C, and have been found to be in excellent agreement with the temperature anomalies on the HadCRUT4/Met Office data (2),(3).  This means that the ordinate values are themselves the temperature anomalies referred to a base temperature of 14.0 deg C.  This temperature is also, therefore, the base temperature for the anomalies in the Met Office data. 

The average value of the Met Office anomalies from 2001 to 2014 is 0.49 deg C, and so this gives 14.49 deg C as the average surface temperature for this period, in line with the GWPF graph.  This is 287.6 K.

The surface temperature is seen to have been substantially constant at this value in spite of the fact that the atmospheric concentration of carbon dioxide has been increasing steadily throughout this period.  Since extra CO2 has had no effect on temperature, this means that ALL of the emission power from the Earth’s surface was already being absorbed even in 2001, and has been continually totally absorbed since then, and must proceed so in the future.  Therefore, no further temperature increase above 287.6 K can occur as a result of increased carbon dioxide, or other greenhouse gases.

 Earth’s Surface Emission

The Earth’s surface emits electromagnetic energy as photons, very small quanta of energy, with a wide range of wavelengths, in accordance with Planck’s Law, in two broad regions; 400 to 14, and 14 to 2 microns.    

The emitted Planck power for long wave region 400 to 14 microns is denoted as u, and the power for the shorter wavelengths 14 to 2 microns, which is the atmospheric window region, as w.

For a surface temperature of 287.6 K, the Planck Law provides the values
u = 195.52 Wm^-2
w  = 184.10 Wm^-2
The complete power emission is, therefore, 379.62 Wm^-2.  It is this total which is used in the following calculations. 

Long wavelength 400 to 14 microns
Power emitted from surface = u
Power absorbed in atmosphere = u

 In accordance with natural Greenhouse Gas theory, half of this absorbed power is re-emitted upwards to space, and half is re-emitted downwards to the Earth’s surface.
So,   power to space  =  0.5u   ———————–space

and  power to Earth  =  0.5u    ———————-Earth return

 
Short wavelength   14 to 2 microns
For this wavelength range, the power emitted from the surface into the atmosphere is w.
A fraction f of this w is transmitted directly to space, and so the term “atmospheric window” has come to be applied to the amount of power, fw, escaping without absorption. 

The remaining power, (1 – f )w, is absorbed and re-emitted, 50% to space and 50% to Earth.

Power to space  =  fw     ——————————space
Power to space  =  0.5(1 – f)w    ———————space
Power to Earth  =  0.5(1 – f)w  ———————Earth return

The power returning to the Earth’s surface, the “Earth Return”, is of vital importance because it is needed, together with the power received by the surface from the Sun, to supply the power emitted from the surface in accordance with the Planck Law for the given temperature.

Summing the terms above:-
Total power to space  = 0.5u + fw  +  0.5(1 – f)w   ———————–(1)

For energy balance, this must be 239.7
Therefore            0.5u  +  fw  +  0.5(1 – f)w  = 239.7
                            0.5u  +  fw  +  0.5w  –  0.5fw  =  239.7
                            0.5(u + w)  +  0.5fw  =  239.7
                           0.5fw  =  239.7  –  0.5(195.52 + 184.1)
                                      =  49.89
                                fw  =  99.78 This is the Atmospheric Window
f = 0.541988
                                f  =  0.542 

Total Earth return  =  0.5u + 0.5(1 – f)w   ———————————(2)

For re-supply, this must be equal to total emission from surface, less the energy from the Sun at the surface, ie   u +w – 239.7   ie   139.92 Wm^-2

Therefore    Earth return  =  0.5u + 0.5(1 – f)w 
                                         =  0.5(195.52) + 0.5(0.45801)(184.1)
                                         =  97.76  +  42.16
                                         =  139.92 Wm^-2,    as required.

The calculated value for f has been confirmed within about 10% by visual estimations of the window data available in the link below.
https://en.wikipedia.org/wiki/Infrared_window#/media/File:Atmosfaerisk_spredning.png


Conclusion

At a surface temperature of 287.6 K, the total power emission from the Earth’s surface has been shown to be completely absorbed / directly transmitted to space through the atmospheric window. Therefore, increased concentration of carbon dioxide cannot cause an increase in surface temperature because there is no more energy to absorb.

(1)https://www.thegwpf.org/

(2)https://www.metoffice.gov.uk/hadobs/hadcrut4/data/current/download.html

(3)https://www.metoffice.gov.uk/hadobs/hadcrut4/data/current/time_series/HadCRUT.4.6.0.0.annual_ns_avg.txt

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

Appendix A  
Suppose, contrary to this hypothesis, that not all the power emitted from the Earth’s surface is absorbed.  What would the surface temperature be if this were the case?

Let the power NOT absorbed be h.

Therefore, this power h is transmitted directly to space.

Long wavelength region
Power from surface  =  u
Therefore, power absorbed in atmosphere  =  u – h
Then         power to space  =  h   ——————————space
                 power to space  =  0.5(u – h )  ——————–space
                 power to Earth  =  0.5(u – h )  ——————–Earth

Short wavelength region
Power from surface  =  w
Then         power to space  =  fw            ———————space
                 power to space  =  0.5(1 – f)w  ——————space
                 power to Earth  =  0.5(1 – f)w   ——————Earth

Summing, 
Total power to space  =  h  +  0.5(u – h )  +  fw  +  0.5(1 – f)w    ———(3)

Total Earth return  =  0.5(u – h)  +  0.5(1 – f)w    ——————-(4)

In order to re-supply the (u + w) Planck emission from the surface,

Earth return  +  239.7  =  u  +  w

Therefore,  0.5(u – h)  +  0.5(1 – f)w  +  239.7  =  u  +  w
Whence,    0.5u  +  0.5h  +  [1 – 0.5(1 – f)]w  =  239.7
Substituting for f  =  0.542,
                   0.5u  +  0.5h  +  0.771w  =  239.7
                   u  +  h  +  1.542w   =  479.4
                    h  =  479.4  –  u  –  1.542w   ————————-(5)

So we need to know the relationship between the power which is not absorbed, h, and the stable surface temperature supplying the powers u and w.  From the Planck Law we find:- 
Temp K       u          w        h Wm^-2
287.6      195.52   184.1    0.00
287.0      194.46   182.0    4.30
286.0      192.71   178.5  11.38
285.0      190.96   175.1  16.65

This clearly shows that if not all the Earth’s emitted power is absorbed, the surface temperature would be significantly less than the present 287.6 K.

Appendix B

Suppose an increase in surface temperature due to El Nino

The Met Office data show a very small temperature increase around 2016.  This was only about 0.2 K, and could well have been caused by El Nino effects.  But for the sake of argument, let us consider a value of 1 K greater than that used above, more in line with NASA’s claims.  So, take a figure of 288.6 K. 

For 288.6 K, the Planck data are:-
u = 197.29 Wm^-2
w = 187.64 Wm^-2
Therefore, total power emission from surface = (u + w)  =  384.93 Wm^-2

Substituting these figures into equations (1) and (2), and with f = 0.542, we find
Total Power to space  =  243.32 Wm^-2
Total Earth Return      =  141.62 Wm^-2

Therefore, the available power for emission from the surface is qual to the 239.7 from the Sun PLUS the Earth Return of 141.62, which makes 381.32 Wm^-2.  So, the available power is short by 3.61 Wm^-2, and this means that the 1 K rise considered is NOT sustainable. 

Therefore, any temperature increase such as might be caused by El Nino effects would return to the stable 287.6 K as shown.   


Appendix C

 Variation of surface temperature with transmission factor f                                       
The power emitted from the Earth’s surface is (u + w).
For energy balance conditions, this must be re-supplied at the surface.  The Sun supplies a power of 239.7 Wm^-2.

More power is returned to the surface by downward emission from the absorbing greenhouse gases in the atmosphere.  This total Earth return is shown in equation 2.

Therefore,        0.5u + 0.5(1 – f )w + 239.7  =  u + w
                             (1 – f)w  =  u + 2w – 479.4
Whence                            f  =  [(479.4 – u) / w] – 1.0  ———————(6)

The surface temperature determines the powers u and w, and so for energy balance there is clearly a relationship given by this equation between temperature and factor f.  This is shown in the graph below, where the limits of f are 0 and 1.0




The importance of the transmission factor is clearly demonstrated.  
The presence of HFCs and HCFCs in the atmosphere is a great potential threat because they have very large absorption coefficients.  Also, sulfur hexafluoride should be included.  Although this does not affect absorption of energy at wavelengths outside the window as already explained, these compounds do, nevertheless, have absorbing wavelengths within
the window.  Therefore, less energy can be directly transmitted to space.  That is, the value of the transmission factor f is reduced, and so the surface temperature increases as shown in the graph












 

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