**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 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.

Pingback: weltklima - Seite 288

Clive,

Better late than never coming across this excellent post and the next one, where I have another question as well.

Fig. 8 shows a spectrum from NIMBUS which corresponds nicely to your calculated spectrum in Fig. 7. The y-axis is labeled radiance which suggests greater emission occurs with 300ppm vs. 600pm. In fact, Fig. 9 defines the change in radiance as 300ppm – 600ppm and the values are mostly positive. This seems counter intuitive unless fewer molecules at lower warmer altitude emit the greater amount of energy.

Can you straighten me out on this? Also, does this analysis account for the average absorption and emission which would occur during a normal 24 hour time frame?

Sorry about the confusion. The difference in figure 9 is a sign change. In fact the radiance is reduced in going from 300 to 600 ppm. What I have plotted is the “radiative forcing” which is the negative of that value !

The whole AGW argument is based on increasing levels of CO2 reducing OLR. This creates a global energy imbalance causing the surface to warm to rebalance the reduction. This “forcing” is equal to the fall in OLR caused initially by increased CO2.

Forcing = – reduction in OLR from top of atmosphere.

This is of course the consensus view of climate scientists, but is it also that of physicists? I read the responses of Jack Barrett and Peter Dietze to criticisms of the Heinz Hug’s “Climate Catastrophe – A Spectroscopic Artifact?” and wonder if the matter is settled. To your point, Barrett would argue “Certainly, at lower temperature [from higher altitude], the collision rate is reduced, but by no means as much as to offset the larger number of radiative molecules.”

I also interested in how transient changes in the lapse rate over 24 hour period is figured into line-by-line calculations converting emission temperature to radiance that appear to be based on a spectral snap-shot. If this is done, I missed it or haven’t got there yet along my path to understanding climate physics. Grateful for any advice on this.

Of course none of the climate models take into account the diurnal changes in the lapse rate between night and day. They just average over that as far as I know. However night time radiative cooling in clear skies must be important and is especially effected by clouds. Clouds are the elephant in the room. They cool the earth during the day and warm the earth at night. The net average effect though on earth of clouds is cooling -22 W/m2. A reduction in global cloud cover of 1-2% would offset all of AGW.

Regarding Barret. As far as I see it CO2 molecules are in thermal equilibrium with N2 and O2 molecules at any given height. Unlike N2/O2 they can also radiate “heat” energy. However they radiate due to collisions with other gases governed by the local temperature.

Hi Clive,

Many thanks for this explanation of the GHE. The “black body surrounded by a shell of greenhouse gas” model was never any better, for me, than a plausibility argument for how the GHE works. So it’s good to find something that is both understandable and seems realistic.

There is a small point I am not sure of. You say:

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.This clearly is near reality since your calculations reproduce the spectrum measured from space.

However, the “100m thick slice of the atmosphere” seems arbitrary (eg why not a 176m thick slice?). And although such a slice may itself be just about transparent, you might have a significant number of such slices above you, giving something not really all that transparent, through which photons don’t have a high probability of escaping to space.

Would it be appropriate instead to work down to the level at which 50% of incoming photons have been absorbed? So that, if my understanding is right, you’ll have calculated the height above which a majority of outgoing photons escape for good?

I hope my question makes sense.

Dr Best; Thank you for bringing this immense step toward sanity to the discussion. You say:

“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.”

My naive question is why is there such a large change in emission altitude for the side bands? I would expect the free path to be the same for all of the frequencies in this narrow band. The reason that their amplitudes are small being emitted to space is not because of the thick atmosphere above their emission altitude but their emitted amplitudes are small to begin with.

What am I missing? Or can you point me to something which explains your calculation of fig. 5a.

Thank you so much for your post.

Ron Chappell

I don’t know if this will be of any help to you, but here are some absorption coefficient figures I’ve gathered.

http://bartonlevenson.com/AbsorptionCoefficients3.html

All lines and bands are in microns, all coefficients in m^2 kg^-1. For notes on translation (you probably know this stuff already):

http://bartonlevenson.com/ConvertingAbsorptionCoefficients.html

Thanks so much for the very thorough and thoughtful reply. I’m overwhelmed with confusion. Some seem impossibly large.

arationofreason

Just background:

http://journals.aps.org/pr/abstract/10.1103/PhysRev.41.291

http://www.rmets.org/sites/default/files/qjcallender38.pdf

Callender’s 1938 paper also appears in

The Warming Papers, edited by David Archer and Ray Pierrehumbert.Pingback: Any doubts about Climate Change? - Page 183 - TeakDoor.com - The Thailand Forum

Clive,

What do you mean by tracking IR photons downward from space? The graphs that I’ve looked at show incoming from space as mostly visible light range and outgoing (upward) as IR.

It is just a thought experiment that allows us to get the same result as doing a complicated radiative transport calculation up from the surface. There are no real IR photons coming from space except perhaps those from the Big Bang microwave background !

Thanks Clive – I’m new to this specific field of study and find your discussions helpful. Do you know why CO2 is (has been) considered nearly saturated when the satellite IR spectroscopy graphs don’t show the near 667 wavenumber (or near 15 microns) as bottoming out? That means the IR detectors are still picking up a reduced amount but an amount nonetheless (40 mW for example in Fig 8). Is the consensus that it is all from upward retransmission from energized CO2? If so, intuitively that would seem a bit much. I understand why the peaks get wider so that’s not my issue. It’s the remaining 40 mW that still reaches the detectors that puzzles me.

That is a very interesting question. The central central lines are already saturated way up into the stratosphere where temperature actually increases with altitude. This means that IR emissions there are actually higher than those lines emitting from near the top of troposphere – hence the spike. As CO2 increases so does the emission height increasing emissions from the central lines producing a GH ‘cooling’ contribution. However the net effect when integrated over all lines is a reduction in outgoing IR and an induced warming effect.

The NIMBUS II data was from early 70s and there is AIRS data from after 2002. The altitudes for both satellites seemed similar (over 400 miles) but AIRS measured tropospheric temps so I’m not sure what that means regarding its altitude.

https://airs.jpl.nasa.gov/data/products

Anyway the AIRS spectrum provided by Pierrehumbert looks almost identical to the earlier NIMBUS II:

https://geosci.uchicago.edu/~rtp1/papers/PhysTodayRT2011.pdf

I confess I still don’t grasp why the 667 dip stops at that level. To me, I interpret this as more CO2 tomorrow could trap even more IR at 667. But on the other hand – why are the NIMBUS II and AIRS graphs so similar when more than 30 years had passed and CO2 levels increased from 325 ppm to 380 ppm?

Without understand this I won’t be able to appreciate what going to 600 ppm will actually do.

I assume you mean the 667 central spike. This will strengthen as CO2 increases but it releases more energy to space, the opposite of trapping more energy!

Basically the atmosphere radiates at different heights and wavelength, but always roughly as a black body with temperature Tlocal. The central spike already radiates from CO2 molecules way up in the stratosphere. The stratosphere temperature

increaseswith height and radiant energy goes as T^4 so more heat is radiated to space from this wavelength as CO2 levels rise. This increasescooling !You can already see that in the spectra when compared to BB spectra at various temperatures. The spike radiates at 245C whereas the side bands radiate at 220K.

A Black Body!!!! Hardly. Schoolboy error. Is the rest of your nalysis worth reading?

Clive,

I’ve done a little more research. I think the biggest reason the central spike at exactly 667 is a spike is because there is much less CO2 moving perfectly laterally compared to the amount of CO2 encountered with vertical velocity toward or away from the approaching IR (Doppler broadening). That would explain more absorption between 640 -665 and 669-700 than the amount absorbed at the actual 667 wavenumber itself. So the result is more IR at exactly 667 will make it all the way to the satellite detectors.

I reread chapter 4 of David Archer’s Global Warming book. He discusses the Band Saturation effect. Even though he appears to be in the “CO2 is ruining the planet” camp, this section (if I understand correctly) is basically an admission that the levels of CO2 have hit their limit of absorbing any more at 667. He does warn that more CO2 will contribute to additional warming due to available CO2 to absorb near (+ or -) 667 and “broaden” the band even wider. I get that and in a sense don’t technically disagree with it, but the farther you get from 667 in either direction, the harder it is for IR to find a CO2 traveling at that velocity. So if that’s the reason for concern, I can’t see the world collapsing as CO2 moves toward 450 ppm some time in the future.

However, I think a more likely reason for additional warming is something you mentioned in your post: DOUBLING CO2 and BASIC PHYSICS from Feb 4, 2010. There you mention how increased CO2 concentrations could cause the saturation at lower altitudes in the atmosphere. So as CO2 concentrations continue to rise, a higher percentage absorbs increasingly closer to the surface. There may be something to that especially if lower altitude CO2 has a better chance of transferring energy to O2 and N2 while higher altitude CO2 presumably has more time between collisions and has increased chance to re-radiate the energy away as IR before it transfers to other gas molecules to increase kinetic energy. Has someone done the math on this? Could additional warming be simply from continually increasing the ratio of: transfer to kinetic vs radiate to space?

Can you recommend any articles or books that explore this issue?