The Atmospheric Window
A E Banner Oct, 2020
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.
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.
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.