Abstract: This post describes a new approach to calculating the CO2 greenhouse effect. Instead of calculating radiative transfer from the surface up through the atmosphere to space, exactly the opposite is done. IR photons originating from space are tracked downwards to Earth in order to derive for each wavelength the height at which more than half of them get absorbed within a 100 meter path length. This identifies the height where the atmosphere becomes opaque at a given wavelength. This also coincides with the “effective emission height” for photons to escape from the atmosphere to space. A program has been written using a standard atmospheric model to perform a line by line calculation for CO2 with data from the HITRAN spectroscopy database. The result for CO2 is surprising as it shows that OLR from the central peak of the 15 micron band originates from high in the stratosphere. It is mostly the lines at the edges of the band that lie in the troposphere. The calculation can then show how changes in CO2 concentrations affect the emission height and thereby reduce net outgoing radiation(OLR). The net reduction in OLR is found to be in agreement with far more complex radiative transfer models. This demonstrates how the greenhouse effect on Earth is determined by greenhouse gases in the upper atmosphere and not at the surface.
This post looks in detail at the emissions to space by CO2 molecules from the atmosphere. The main CO2 absorption band lies at 15 microns. It is composed of hundreds of quantum transitions between vibrational states of the molecule. The reference database for the strengths of these lines is called HITRAN and is maintained by Harvard University. I requested a copy of this database and have been studying it. Fig 1. shows in detail the transition lines within this band and Fig 2. shows the fine detail within the central spike. The line strengths are recorded at 296K in units of cm-1/(molecules cm-2) corresponding to the absorption cross section for one molecule in vacuum.
In the real atmosphere these lines are broadened due mainly to motion of molecules. This is a rather complex subject but luckily I found a Fortran program  which takes as input the line strengths from HITRAN and then integrates them over pressure to derive a net absorption cross section per Mole of CO2. This result is shown in Figure 3. Notice how strong the central peak now becomes with 2 clear side fans of absorption with fine structure.
To make progress to locate from exactly where IR is emitted to space we need a model of the atmosphere. For this we assume a standard lapse rate of 6.5C/km up to the tropopause at 11 km, then stationary temperatures through to 20 km followed by a linear increase of 1.9C/km in the stratosphere until 48 km above the surface (see fig 4).
The barometric pressure profile is taken to be
scale = RT/($molar*$g);
The objective now of the calculation is to take each CO2 transition line in turn and then descend from space to find at which altitude the absorption of photons of that wave length within a 100m thick slice of the atmosphere becomes greater than the transmission of photons. We define this height as the transition between opaque and transparency. This is the height at which thermal photons within the CO2 absorption bands are free to escape to space. – the effective radiation height. The absorption rate is simply the molar cross section times the numer of moles of CO2 contained in a 100 meter long cylinder of cross-section 1m^2. A graph of emission heights versus wavenumber is shown in figure 5a for a CO2 concentration of 300ppm in black and 600ppm in red. Fig 5b is a smoothed average over 20 adjacent lines.
Note how it is mainly emission heights from the side lines which lie in the troposphere. The emission height of the central peak actually lies in the stratosphere with the central spike reaching up to 25000 meters where the temperatures are actually increasing with altitude. As expected doubling CO2 concentrations rises the emission height significantly but the effect on radiation loss depends on the temperature difference beween the old emission height and the new emission height. Below the emission height, radiation in CO2 bands is in thermal equilibrium with the surrounding atmosphere. This is usually called Local Thermodynamic Equilibrium (LTE). The lapse rate of the atmosphere is driven by convective and evaporative heat loss from the surface, but energy loss to space can only occur through radiation. So the local temperature from where IR photons escape to space determines the radiation flux for that wavelength. The temperatures at the emission heights for CO2 are shown in figure 6.
The effective temperature of the emission height now allows us to calculate the planck spectrum for the CO2 lines. The result is shown in Figure 7.
So how does this compare with a real spectrum as measured by satellite ? Figure 9 shows a spectrum taken from NIMBUS. There is an overlap with the water vapour contiuum lines below 550 cm-1, which reduces the left shoulder. But apart from that the agreement is really rather good, and in particular note the upward spike at the centre of the line corresponding to emission from the warmer stratosphere. Similarly the flat bottom corresponds to the tropopause at around 216K.
Finally now we can make an estimate for the radiative forcing due to a doubling of CO2. To do this we first derive the net change in outgoing IR from an increase in CO2 from 300ppm to 600ppm as shown in Figure 9. Note how for the central peak the radiation actually increases for a doubling of CO2 as they emission height lies high up in the stratosphere. This is because temperatures are actually increasing with height.
Next we integrate the change in the radiative flux over all lines in the CO2 band going from 300ppm and 600ppm concentration. The result of this integration works out to be 1.17 watts/m2/sr.
However, to derive the net change in OLR we need to integrate this over the outgoing solid angle for photons that reach space. Quoting from Wikipedia.
The integration over the solid angle should be the half sphere of out going radiation. Furthermore, because black bodies are Lambertian (i.e. they obey Lambert’s cosine law), the intensity observed along the sphere will be the actual intensity times the cosine of the zenith angle , and in spherical coordinates, .
This then adds a factor which when you evaluate the integral gives an extra factor .
So finally the reduction in outgoing IR radiation caused by a doubling of CO2 from 300ppm to 600ppm becomes 4.7 watts/m2. This is not far away from the value as calculated by climate models – 3.7 watts/m2 ! This is usually called “radiative forcing”. Note how in the stratosphere the energy loss increases with CO2 concentration. This predicts that the stratosphere should cool, as the troposphere warms. All predictions of warming/cooling are of course based on the assumption that all else remains constant – lapse rate, H2O, clouds etc. The real signature for a CO2 GHG effect would be to observe cooling in the stratosphere where these effects are much smaller.
In the next post I will examine in detail how “radiative forcing” depends on CO2 concentrations.