**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 effect 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 determines 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[1]. 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 [2] 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);

P[h]= P0*exp(-h/scale);

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.

**References**

1. http://www.cfa.harvard.edu/hitran/

2. http://home.pcisys.net/~bestwork.1/CalcAbs/CalcAbsHitran.html

Hi Clive,

Great article!

I have a question. Above figure 6 you state …

“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.”

This doesn’t make sense to me. In my mind, LTE would not be a 2d surface, but a 3d mass of air that will be larger as the thermosphere expands, much like a heat sink that can increase it’s surface much more than in a strictly vertical sense.

Thanks Eric,

Yes, you are quite right. LTE really covers a 3d shell of finite thickness. In my code I assume each layer to be 100m thick. So really the heights I am talking about should be at the centre of each shell and define the temperature of the surrounding atmosphere within 100 m of that height.

I agree that the wording should be changed .

The atmosphere would indeed expand a little if the local temperature increases. I think convection would maintain the lapse rate. However the lapse rate itself could change with an increased relative humidity. These are just the known unknowns.

I think the wording is fine, but that the IR surface expands in 3 dimensions and that the integration procedure would be more accurate if it accounted for that?

Looking at the model, it seems clear that the IR surface/manifold? is much rougher with 600ppm of CO2. This is clearly shown in 5b that the effective emission height is not only higher, but has much greater variance. This also makes sense intuitively to me. Sort of like a puffer fish whose spines would also extend along with his body. http://animals.nationalgeographic.com/animals/fish/pufferfish/

So while the average temperature goes down, the surface area increases greatly.

My mind starts aching when trying to follow the integration procedure, but would it not be possible to account for the increased IR surface area in the integration? Perhaps spreading including a probability with each shell and integrating over the vertical?

You are right – there is a small change in surface area with height but I don’t think that is why the 600 ppm has greater variance particularly in the troposphere. The surface area doesn’t really change that much because the centre of the earth is so far away. 4piR^2 and 4pi(R+dr)^2. I think it may be because the pressure (density) change is exponential.

I will put the code on-line when I have finished and you will see how simple it really is !

Clive

I’d like to look at some code, although I can assure you that it’s not simple (for me at least :)). Thank you for entertaining my questions and for publishing this online.

Clive,

Thanks for devoting the time to do this analysis. It gives me confidence that when we are talking about the “no feedbacks” case we know what we are talking about (no data fiddling involved). If you get some free time you might look at the difference between the “Dynamic Tropopause Potential Temperature” and surface or near-surface temperature. Any relevance for climate sensitivity?

http://climaterealists.com/?id=11197

Regards

Hi Clive,

Perhaps a simple way of looking at the effect of GHGs are by considering each molecule a small IR transmitter/receiver, with a small delay (and chance that energy gained in reception) is given up in a collision w/O2 or N2. Since the direction of transmission is random, the molecules are transmitting toward space approximately 50% of the time, while they are receiving (a briefly storing) energy from all directions (hence 100% of the time). The density of GHGs (not any particular one, because thermal equilibrium ‘shares the energy’ between all species present, not just the GHG active at band for a given event) thus determines the mean free path for IR moving through the atmosphere, hence the number of absorb/emit events, the total delay and hence the energy per unit volume (and temperature via the energy/temperature relation for the species).

In this view, all other things being equal, the temperature must decline with altitude because energy per unit volume declines with altitude (until it is so low that UV – O2/O3 starts to make a significant contribution), given that 50% of emission events in a layer result in ‘permanent’ loss of the energy to a higher altitude and hence ultimately to space.

I’m not sure that worrying about the frequency of the emission to space ultimately matters that much, unless you are trying to match the ‘last look’ NIMBUS spectrum data, for example, because presumably a given quantity of energy will have spent part of it’s trip up from the surface manifested in any number of frequencies, given the range of active frequencies and the thermal equilibrium at each absorb/emit point.

On the other hand, is it possible that a quantum of energy emitted upward at a wavenumber of 680 is trapped there and will be making it’s whole trip being absorbed/emitted at that wavenumber?

As a thought experiment, would it make a difference to the temperature profile of the atmosphere if all of the CO2 in the lower atmosphere were replaced by H2O and CO2 were somehow confined to only the top layers that are not 100% opaque to space (in the present proportions so the NIMBUS data looks the same)?

Thanks for you tremendous help in understanding how this works… I appreciate your insights.

Regards,

Randall

Randell,

Sorry for the delay- I only just saw your comment. Your general analysis is spot on. However I think the frequency does matter because the mean free path for an IR photon depends both on density and absorption cross-section. The cross-section varies strongly with frequency. So for example the quantum line for CO2 at exactly 15 microns has so large a cross-section that the atmosphere is opaque until way up into the stratosphere. Other lines are much weaker and radiate to space at low altitudes. So even if much of the CO2 is thermally excited rather than directly by photons, the energy loss upwards depends on density and frequency.

Regarding your thought experiment. In the lower atmosphere about 2/3 of heat flow upwards is by convection and evaporation, and only 1/3 is by radiation. This proportion changes as you go upwards until eventually at the tropopause ALL energy loss upwards is by radiation. In that sense changing the H2O/CO2 mix may change these proportions and both the height of the tropopause AND by implication the surface temperature will change. The dry adiabatic lapse rate would remain the same although the environmental lapse rate would fall as there is now more water vapor.

regards

Clive

Hi Clive,

I think the statement “energy loss upwards depends on density and frequency” might well be right at the heart of the matter. Consider 2 end-point scenarios, one where frequency does not matter and the other where density does not matter. Also, to make this simple to think about (at least for me), consider that each of these end-point scenarios has an analogous simple electrical circuit, where the voltage corresponds to energy difference, the current to heat flux, and the resistance to specific GHG influence (which is a function of density and frequency). Using this analogy, in the one case, the electrical configuration is like a bundle of vertically stacked resistors (say each 1 meter long, one for each frequency and GHG) that are all tied together so there is no difference in potential at the end of each resistor, and at each elevation, the resistor with the lowest resistance of course carries the most current. Call this the parallel configuration. The corresponding ‘series’ configuration is similar except the only potentials that are equal are at the bottom and top of the stacks. Everywhere else the potentials are determined by the relative sizes of the resistors. Summarizing, one looks like a ‘series of parallels’ (“SOP”) and other is a ‘parallel set of series’ (“POS”) configuration.

I put together a spreadsheet for a 2X2 resistor array in SOP and POS, in order to compare the total resistance calculated and see under what conditions SOP vs. POS makes any difference. Let column 1 be H2O and column 2 be CO2. Let row 1 be the upper atmosphere and row 2 be the lower atmosphere. For this analogy, let’s make the resistances dimensionless by dividing all of them by the ‘resistance’ of H2O in the lower atmosphere, whatever that is, so we can set the ‘resistances’ of upper atm H2O, and CO2 in relation to this value.

I take the ‘conductance’ (the inverse of resistance) for lower atmosphere H2O to be relatively high, given that there is both a lot of water and a lot of IR bands where water is active. Let that have a value of 1. For the sake of this tiny model, set the upper atm conductance/resistance also to 1 (although it might be lower given nothing above to interfere with radiation away… but there is less present).

As for lower atm CO2, the resistance value (inversely proportional to the heat current that flows through this path) must be much higher, given relatively little CO2 and fewer active bands. I set this resistance to 99. (in effect saying that in the lower atm 99% of the heat flow is H2O related, and 1% in CO2 related). In the upper atm, let’s say that 1/3 of the heat flows via CO2, so the resistance is a value of 2? Twice the upper atm H2O value.

Now we can compare SOP and POS and also how sensitive the total resistance is to a changes in upper atm CO2 resistance. In this tiny model, a doubling of upper atm CO2 resistance in the SOP mode from 2 to 4 increases total resistance by 8%, a significant change given how water dominated the model as a whole is. In POS mode, the increase is significantly smaller… .03%. In fact, POS mode predicts virtually no sensitivity to changes in upper atm CO2 resistance, because once energy is ‘in the water channel’ it stays there all the way up.

I think this could be (and probably has been) resolved with lab bench experimentation… it would be along the lines of a test chamber illuminated with each CO2 band at the relevant range of atm pressures, temperatures and CO2 (and H2O) concentrations… looking orthogonally into the test chamber we should be able to see how much of the CO2 band is the result of a quick absorb/emit of this illumination vs. something that looks much more like the full thermal equilibrium (a function only of the temp). I would bet that at higher relative pressures it’s very much a thermal pattern and only at really low pressures does the CO2 illumination band show up strongly. Just guessing though. Anyone out there aware of such lab data?

If my relative ‘resistance’ numbers are nonsense, then the whole exercise is highly suspect. The sensitivity % could be much higher but probably is much lower (for SOP). Please comment if it looks too far off…

Best regards,

–Randall

Why do you choose 100 m as the critical attenuation length for photon emission? As atmosphere is clearly many km thick, photons with 100 m attenuation length would most likely not escape the 100-meter-environment.

It is probably a bit dubious to choose any value as clearly photons are emitted from a fair thickness of air with varying escape probability. However, for comparison, how would the results change if you take, say, 1 km or 3 km as the critical attenuation length? This probably would move more of the emission to the stratosphere.

Second point, I think it is a bad idea to do any averaging of the spectra. At large altitudes, the emission/absorption lines are very slender and the peak-valley variation may have large impact on the emission altitude.

Something in the ballpark of 3 km might actually be fairly good guess for the critical attenuation length as such a change in altitude will cause an appreciable change in pressure and the associated narrowing of lines enhances the escape probability for radiation moving upwards.

Marko, I am choosing a vertical grid with arbitrary 100m intervals. So I don’t think I am assuming anything about the attenuation length, which anyway varies dramatically with atmospheric pressure. The advantage of calculating from TOA downwards is that we know the attenuation length is many times larger than 100m high up. I am calculating the height for each wavelength where more photons are absorbed than are transmitted through. This is the height where for that wavelength the atmosphere becomes opaque for IR photons. This is what I define as the effective emission height.

The trick is to imaging a flux of IR photons going down from space rather than going up from the surface. The result for the effective emission height will be the same.

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