Introduction The energy balance of the Earth and its importance in global warming considerations has been studied extensively for many years. Unfortunately, this has been based upon values of energy averaged over the whole surface of the Earth, and so cannot give accurate results for specific locations on the land surface. This must be of significance in many applications, such as meteorology.
The present paper deals instead with areas of one square metre at specific latitudes. Whereas NASA takes the energy incoming from the Sun, suitably modified by transmission through the atmosphere, and simply divides by 2 to obtain an average for the illuminated part of the Earth’s surface, and then divides by 2 again for the diurnal effect of day and night, the present paper concentrates on an individual square metre, but at each principal latitude.
The calculated temperatures apply to the land surface itself, NOT to the atmosphere close to the surface, and are mid-day values.
It is required to find the temperature of the surface for which the power it emits is equal to the total power which is absorbed by the surface. This is the condition of energy balance and the “energy balance temperature”.
Theory In line with NASA, the proportion of the energy from the Sun which is transmitted through the atmosphere and reaches the Earth’s surface is taken to be 0.48. This is the atmosphere transmission factor, atf.
With the usual value of 1366 Watts per square metre for the Solar Constant, this gives a gross incoming power of 656 Wats per square metre. NASA maintains that 30% of the power absorbed by the surface is given up to the atmosphere in non-radiative form by convection and evaporation. Therefore, only 70% is absorbed by the surface and re-emitted as electromagnetic radiation, which is 458.98 Wm^-2.
At higher latitudes, however, the effective power is reduced by the cosine of the latitude.
A further modification must be made because of the effect of the changing tilt of the Earth’s axis as the Earth revolves in its orbit around the Sun.
The net incoming power takes into account these three items.
Power radiated from the surface At the longer wavelengths, the radiated energy is absorbed by the greenhouse gases in the atmosphere, in line with the Greenhouse Gas Theory, but for the shorter wavelengths only some is absorbed, and the remainder travels freely out to space. This is known as the Atmospheric Window, and operates at wavelengths below 14 microns.
The energy absorbed by the greenhouse gases is subsequently re-emitted, and the overall effect is that 50% of the emission travels upwards to space, and 50% travels down to the Earth’s surface.
Let P be the total power emitted from the surface. w = the total emitted power at short wavelengths, 14 microns and less f = the proportion of the shortwave radiation w which escapes directly to space
Hence, the atmospheric window is fw, and so this power escapes directly to space………… fw……….Space (1) This leaves a power (P – fw) in the atmosphere. By the Greenhouse Gas Theory, this is absorbed by the atmosphere and is re-emitted, half going upwards to space, and half going downwards to be absorbed by the Earth.
That is power to space = 0.5( P – fw)…………………..Space(2)
And power to Earth = 0.5( P – fw)…………………..Earth(3)
So, from (1) and (2), the total power to space = fw + 0.5(P – fw) = 0.5(P + fw) Also, the power to Earth’s surface = 0.5( P – fw) ….Earth Return This Earth Return is of vital importance, because it is needed to augment the power from the Sun absorbed into the surface, and thus provide sufficient power for Planck’s Law to be viable.
The total input power to the surface is equal to the net incoming power from the Sun, PLUS the Earth Return, and thus depends upon the term fw.
The total input power to the surface is compared with power P radiated by the land surface, in accordance with Planck’s Law, which varies with the fourth power of the Absolute Temperature. This also provides the value of fw, and so the energy balance temperature can be established.
Results The net incoming power is taken to be 459 Wm^-2; the atmosphere transmission factor is 0.48, and the window transmission factor f is 0.50, throughout.
A computer model based on this theory and Planck’s Law has provided data for monthly, mid-day land surface temperatures, for the range of latitudes from 0 to 90 degN.
This is for the land surface itself, NOT the atmosphere close to the surface.
The data is shown in Fig 1.
In the winter months, at latitudes of 60 degN and more, very little of the Sun’s energy is absorbed into the surface, and so energy balance can only be attained at very low temperatures, tending to Absolute Zero. In such cases, the surface temperature for energy balance is indeterminate, and so no data is shown. (Also applies later in other figures.)
Such surface temperatures will, of course, be moderated by the usual atmospheric effects of circulating air currents.
It is seen by interpolation, that at a latitude of 35 degN, the mid-day surface temperature for June, month 6, is found to be between 62 and 59 deg C. This is almost, but not quite, hot enough to fry an egg on the sidewalk, as is tried in a competition each year in Oatman, Arizona. Success requires a temperature of 62 degC
These results are shown graphically in Fig 2.
Fig 3 shows the temperatures displayed against latitude for the three chosen months.
The outgoing longwave radiation (OLR) is shown graphically against latitude in Fig 4, and in table form in Fig 5.
Fig 6 shows both input power, Inp, and the associated outgoing radiation, OLR, for the range of months, and latitudes from 0 to 60 degN. The paired values are virtually identical.
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 wavelength400 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
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.
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)
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.
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.
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
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.
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.