Arguments
The Physical Chemistry of Carbon Dioxide Absorption
Posted on 23 December 2010 by hfranzen
Guest post by Hugo Franzen
Perhaps because I have been a Physical Chemist for more years than I care to mention, I have the idea that Physical Chemists have something important to contribute to just about any discussion about physical phenomena. I hope that I can convince you that this is in fact true in the case of global climate change. One reason I feel it important to be a spokesman for Physical Chemistry in this arena is because, for the most part, we P. Chemists feel it important to develop math based arguments that catch the essence of what is occurring. Of course we then leave the hard part of dealing with the ramifications to someone else. What I mean by this in the current discussion is that the problem of global warming can be broken down into two parts – the “forcing” part that deals with the difference between the energy input and output at the earth’s surface and the consequences of that forcing. The latter is the huge problem of the feedbacks and their consequences on the distribution of that energy over the globe.
The second part is the tough, ongoing job of the Climatological community while the first part is basically P. Chem. (even though it was in many cases done by folks in other disciplines). In this essay it is my purpose to discuss the easier, forcing part. When it comes to communication, the considerations in this realm have a distinct advantage in that the results follow directly from the solution of an elementary differential equation using specroscopic data that have been known since the 1960's. I have benefited greatly from others in coming to an understanding of the P. Chem. of Global Warming (GW). I feel myself qualified, on the basis of what I have learned, to say, from the point of view of Physical Chemistry, i.e. rigorous science of the if A then B type, that global warming as the result of carbon dioxide in the atmosphere is totally undeniable and that the extent of the forcing is beyond doubt close to what the climatologists are saying it is. These conclusions are not based on the earth’s temperature history or upon complex computer programs (which I certainly believe to be of great importance – it’s just that they are difficult to communicate about) but upon the type of calculation that is done in P. Chem. courses around the world. The calculation on which this essay is based can be found in the presentation (GWPPT6) linked at the bottom of this post.
Interactions between molecules and electromagnetic radiation have been an important part of Physical Chemistry since its inception. The first scientist to attempt a calculation of the GW (a term he introduced) was Svante Arrhenius, the great Swedish Physical Chemist. He did his climate change work in the mid 1890’s. The understanding of the interactions of molecules with radiation progressed enormously with the advent of Quantum Mechanics, and can easily be called a very mature science at this time. The science involves, among many other things, the observation and interpretation of spectra.
Carbon dioxide is a molecule that has been extensively studied in this way and there is available today an incredible depth of knowledge about the interaction of carbon dioxide with electromagnetic radiation. Among a number of interactions about which a great deal is known there are those involved in taking a carbon dioxide molecule (basically linear oxygen to carbon to oxygen) from its ground bending vibrational state to its first excited bending vibrational state. However in both the initial and the final vibrational states the molecule can be in any one of a very large number of rotational states which are separated by energies very much smaller than the energy difference between the vibrational states. Thus there are many transitions between the various rotational states associated with the ground and first excited vibrational states. Transitions between many pairs of these states can be brought about by absorption of infrared radiation of the correct wave length for each pair, and thus such radiation is absorbed over a range of energies (Fig. 1)
These transitions were studied by both theory[i] and experiment[ii] in the 1960’s and the results are highly relevant to global warming, for they provide experimental and calculated data for the linear transmittance of carbon dioxide gas in the infrared region. Transmittance is the fraction of the intensity of a beam that makes it through an absorbing sample. Absorbance is one minus transmittance. Although the data of Fig.1 are for a particular product of path length and carbon dioxide concentration, i.e. a given NL/V, where N/V is the concentration and L is the path length, it should be mentioned that Stull et. al. calculated results for a wide variety of NL/V values and wave lengths. The fit obtained between the calculated and observed spectra for one of the NL/V values (Fig. 1) provides assurance that the absorption coefficients, i.e. the proportionality constants relating the logarithms of the transmittance at the various absorbing wave lengths to the concentration of carbon dioxide in the gas phase, form a reliable basis for calculating the transmittance (or absorbance since they sum to 1) of carbon dioxide in the atmosphere.
The absorption coefficients reported by Stull, et. al. are linear absorption coefficients appropriate to the absorption that results in a decrease in intensity when the radiation is traveling in a single straight line. But the radiation, when the source is the earth, travels in all directions away from the earth. When the radiation is in a single direction, the relevant transmittance is the linear transmittance and the absorption coefficients for linear transmittances were reported by Stull, et. al. The transmittances required when the source is the earth are called the diffuse transmittances and these are calculated by integrating (summing) the intensity equation over all the angles. What results is a diffuse transmittance equation for flux rather than the corresponding equation for decrease in intensity. But this is just what we want to determine the energy audit for the earth, because the flux is the rate at which energy is radiated through a unit area of a surface.
The first task in applying Physical Chemistry to the Global Warming problem is probably to determine the flux of radiation at the earth’s surface. Fortunately in 1900, in his celebrated determination that radiation is quantized, Max Planck solved this problem for equilibrium radiation. It turns out that at equilibrium all matter emits radiation the distribution of which is determined only by its temperature. This distribution, which describes the rate at which energy is emitted at a given wavelength, is given by the Planck equation. But is the earth’s surface at thermal equilibrium? The answer to a very good approximation is yes, provided you restrict your attention to a small enough area and a short enough time. This can be seen immediately when you realize that to say that something is “at thermal equilibrium” means that it has a temperature and vice versa. So the very fact that we can report a temperature for a given place at a given day, and we routinely do that at any place on any day, means that the earth at that time and place is close enough to being in thermal equilibrium that we are justified in talking about its temperature. It then follows that the Planck distribution is a very good approximation to the distribution of the infrared energy radiated by the earth at that place and time.
There remains the problem that the earth then has many different temperatures. In the Earth Sciences it is common practice to use average temperatures as though they were ‘the temperature”. What we believe, and it has been borne out by many studies, is that in general we can do two different things: we can make a number of measurements, reach conclusions based upon those measurements and then average the conclusions, or we can average the measurements and reach conclusions based upon that average measurement. For example we could measure the temperatures at a very large number of places on the earth and 1. Use Planck’s law to calculate the energy radiated at each point and then average or, 2. Average the temperature and use that temperature with Planck’s law to calculate the radiated energy. What has been found is that the final results are essentially the same. In fact temperatures have been measured at a wide range of spots over the earth’s surface and Physicists have looked at the earth’s radiation using satellites. The observed distribution of radiant energy is nicely given by Planck’s Law and the earth’s average temperature. The earth’s spatially averaged temperature, when averaged over a year, comes out to be 288K. The balance: energy in = energy out for the earth for a year results because if the energy in from the sun (corrected for albedo) during a year were greater than the energy radiating out from the earth then the earth’s average temperature would rise and, according to Planck’s Law, the earth would radiate more energy and reestablish the balance. A similar argument holds for antithesis.
So we know the quantity and distribution of the average energy emitted from the earth’s surface from measurement, spectral observation and Planck’s Law. We also know, this time from the data of Stull. et. al. and generalization to the diffuse case, the diffuse transmittances of carbon dioxide for wave length intervals in the energy range of the earth’s Planck radiation. In order to get a “broadband” transmittance, i.e. the transmittance for the energy range over which the carbon dioxide absorbs, the diffuse absorbance for each wave length interval is multiplied by the fraction of the Planck radiation that is emitted in that interval and the product is summed to yield the Planck averaged, broad-band, diffuse transmittances (PABBDT’s) at the various NL/V values for which the linear transmittances were reported by Stull, et. al. These were fit to a curve (PABBDT vs. NL/V) in order to find the dependence of PABBDT for carbon dioxide on NL/V from which we can find the PABBDT of the atmosphere if we know NL/V for the atmosphere.
Thus the one piece of information that is lacking at this point is NL/V of carbon dioxide in the atmosphere. As I am sure everyone reading this essay knows, in 1958, a scientist, Charles David Keeling at the Mauna Loa observatory, initiated a program of measurement of carbon dioxide in the earth’s atmosphere and it has been in operation ever since. The current concentration is somewhere between 385 and 390 ppm. This concentration has been shown to be essentially the same everywhere in the atmosphere. Hence we know, in each case to a very good approximation, the average flux of infrared radiation emitted by the earth as a function of wave length, the PABBDT of carbon dioxide as a function of NL/V and NL/V for the atmosphere.
Therefore we can determine the amount of energy absorbed and reemitted by carbon dioxide in the atmosphere. Because one half of the reemitted radiation comes back to the earth (is the carbon dioxide greenhouse gas flux), this flux is equal to one half the Planck flux in the absorbing interval multiplied by one minus the diffuse, broadband transmittance. Knowing the earth’s average temperature at some initial time and the expected increase in atmospheric carbon dioxide (the Keeling curve) we can calculate the earth’s average temperature difference for these two times as follows. The energy leaving at the final time equals the energy entering at that time (for the reason discussed previously) and, because we know by how much that energy is increased by the carbon dioxide greenhouse effect, we know by how much the earth’s temperature is increased by that effect. When the calculation is done, as in GWPPT6, the conclusion is that the earth’s temperature is currently rising by 0.014 degrees per year because of the greenhouse gas effect of the additional carbon dioxide entering the earth’s atmosphere
In this essay I have restricted myself to words. What I concluded above was accomplished by the solution of equations, and these equations and their solutions are presented in my website. As I see it there are four possibilities 1. Someone could understand the methodology of GWPPT6 on hfranzen.org and agree with the conclusions, or, 2 they might understand and not agree, or, 3 it could be that they do not understand but agree for other reasons,or, 4. they might not understand and not agree. The folks in category 1 need to get the message out. I hope that those in category 2 will contact me with their criticisms. Those in category 3 deserve credit for sound intuitive thinking. Those in the last category are most troublesome. In my view they should either do the hard work of learning the basic science needed for understanding or find someone they trust who understands it to interpret the power point for them. The people of the world need to move on to some very serious changes in our consumption of fossil fuels and there is absolutely no place for obstruction by people who do not understand the nature of the problem.
[i] Stull,Wyatt, and Plass, Applied Optics, V.3,No.2, p.250 (1964)
[ii] Burch, Gryvnak, and Williams, Applied Optics, V.9, p750 (1962)
Your mathematical work is very detailed. Thanks.
Very nice work. That's something I've been trying to assemble for some time, and could only achieve some fragments. It will definetly be a resource I'll use for reference.
It's a set of equations the interested layperson with a minimum background (say an engineer) can understand and verify. This kind of informed person can be a valuable means to spread knowledge and hence influence public opinion.
Thanks.
I really nice piece of writing.
But I see the original on the web site had paragraph breaks in reasonable places, so yes, it is impressive and well written in many ways.
But (you knew there was a 'but' coming, right?) since it is targeted for "the interested layperson", it needs more figures breaking up the text at intervals (not necessarily regular). And it really didn't need that much repetition of "trasmittance + absorbance = 1", especially not once that is illustrated (hint, hint).
Finally, the "interested layperson" is not going to know what a "wave number" is, yet it it referred to w/o comment in the sole figure in the article. Normal people think in wavelength or frequency, only spectroscopists think of "wave numbers";)
Very impressive.
I'm hoping someone can explin the questions I have below:
Shouldn't the energy from the core of the earth somehow need be taken into account into the temperature increase calculations caused by higher levels of CO2?
Is there an assumption made that for every CO2 molecule put in the atmosphere, is a Nitrogen or Oxygen or Water molecule removed? If so, shouldn't the change in radiative effect be used istead of just the addition of the CO2 effect? I've been told that solar radiation is constantly sloughing off part of the atmosphere (more during solar flares or bursts). How does that fit into the puzzle?
A real puzzling question I have: On the moon (virtually no atmosphere, no heated planet core), how quickly does the temperature change from hot to cold as the planet surface goes sunlit to shaded? On the earth, what is the time period of sunlight to cooling in the atmosphere? Is the real reason the air temperature does not hit the extremes due to the radiated heat given off in the evenings from the oceans, planet core and land mass? I feel all of these effects and temperature fluctuations must be taken into account and modeled to better understand the overall effect of CO2 in the atmosphere.
Thanks so much for explaining this to me.
As you may have learned, the moon doesn't have any air around it. The air that surrounds our earth acts as a nice blanket to keep us warm and comfy! But the moon, since it doesn't have this blanket, gets much colder than the earth and much hotter than the earth. On the side of the moon that the sun is shining on, the temperature reaches 260 Fahrenheit! That is hotter than boiling. On the dark side of the moon, it gets very cold, -280 Fahrenheit.
The moon has a tenuous atmosphere comprised by argon, polonium, radon, helium, oxygen, methane, nitrogen, carbon monoxide and carbon dioxide. There is a greenhouse effect on the moon due to the surface and subsurface lunar regolith (soil) , which has the property of absorbing and storing the incident solar radiation. This is the reason by which scientists are considering that the greenhouse effect on Earth depends more on the surface and subsurface regolith (soil) than on the Earth's atmosphere.
http://science.nasa.gov/science-news/science-at-nasa/2009/23oct_ladee/
Read more: http://wiki.answers.com/Q/What_is_the_temperature_on_the_moon's_surface#ixzz18xpvUOer
The addition of CO2 in itself is a complex business, with some being removed from the atmosphere by sinks and some being added by sources.
The 'addition' is the result of these processes.
On the Moon the surface temperature swings to extremes of hot at day to cold at night.
Our planet's core radiates about 0.01% of the energy in comparison to what we receive from the Sun. I apologize for not providing references as this is being tapped out from my phone.
This isn't a greenhouse effect at all. The surfaces of the earth and moon both have the ability to absorb solar radiation, if they didn't they would appear transparent. In fact, the key to the greenhouse effect is that CO2 doesn't readily absorb solar radiation, but does absorb infrared. You should read up on the greenhouse effect and thermal radiation.
http://www.mantleplumes.org/Energetics.html
As Bibliovermis notes, the amount of energy (~40TW) is too small to make difference globally.
Also, I'm not sure about the "sloughing off" question. Are you suggesting that atmosphere loss plays a role in global energy balance?
On the other hand comes the question, what is the source of the increases of CO2 in the atmosphere?. I say in GWPPT6 that I see no reaonable alternative but to ascribe the CO2 increase to human consumption of fossil fuels, but that could a limitation on my part. In a way I am saying if you have a better idea please let me know, However in my experience the usual deniers claim is that it comes from the oceans and the second power point on my web site,
CB with Buffering (charge balance with buffering) points out some very serious restraints on the reactions of CO2, bicarbonate, and carbonate in the ocean.
Those who wish to have the oceans supplying CO2 to the atmosphere must explain the driving force causing the CO2 to evolve. The only reasonable possibility that I can see is temperature, and if this is claimed to be the cause then those who claim it must find the equilibrium constants at the elevated temperature and then calculate the amount of CO2 that evolves. Not a totally arduous task but one the results of which,from my calculation, they will find fall far short of what they are claiming.
Finally, the equations of CB with Buffering are for the average temperature of the earth so if the partial presssure of CO2 increases in one place it will decrease in another and there will be no net change. As a long time professor of chemistry it seems to me the deniers have simply not done their homework.
There is a greenhouse effect on the moon due to the surface and subsurface lunar regolith (soil) , which has the property of absorbing and storing the incident solar radiation. This is the reason by which scientists are considering that the greenhouse effect on Earth depends more on the surface and subsurface regolith (soil) than on the Earth's atmosphere.
Assuming this line of reasoning is based on Hertzberg and Siddons, you may want to read the discussion here.
Generally speaking, linking to anonymous content on WikiAnswers is not helpful.
Thanks for the response. I would think that when adding gases to the atmosphere, there may be some type of "balaning effect" where part of the atmosphere may recombine to form particulates that may go back to the ground.
Response to rocco #15
I will have to read these for I would think that the raising of the ocean temps would be much more than the link you posted.
Response to hfranzen #16
If the global temp increases 1deg Celsius, decomposition of matter in the oceans and land mass will increase. Any idea how much of an effect that would be? Also, based on huamn activites, can we measure increased decomposition (which increases CO2) accurately outside of calculating how much fossil fuel emissions are made.
Thanks for the healthy discussion! I studied Aero/Astro in college but now I am in biotech (past 20 years - much more complicated!) and feel we are missing some key pieces in studying the earth as a control volume part of the solar system (also a control volume and not a conrol mass, since energy from the sun constantly leaves the solar system).
Decomposition rates do not have a significant effect on CO2 levels in the atmosphere, as any CO2 lost via decomposition was taken from the atmosphere to begin with. The carbon from fossil fuels on the other hand has been sequestered from the atmosphere for millions of years. The general topic is discussed here.
Also, there is no need to guess about balancing effects, the amount of CO2 in the atmosphere is directly measurable and it is increasing rapidly. You can find the discussion here.
nice calculations, I have no doubts about their correctness. There is no doubt about the forcing due to the steady increasing levels of CO2, it is well established physics. I just have, as a really skeptical scientist, not at home in the climate science, some heretic questions to the statement "that the earth’s temperature is currently rising by 0.014 degrees per year". Does this rise refer to the whole mass of our planet, to the crust, to the sum of liquid and frozen water or just to the atmosphere or a part if it? Surely not to the surface, since it is two dimensional. Probably not to the whole mass of our planet, its thermal inertia is much too high. If the troposphere or the whole atmosphere is meant, than the heat exchange with the solid or liquid masses beneath should be taken into calculation. Could you please, specify, what is meant by "earth’s temperature"?
"Earth's temperature" refers to the air, water & terrestrial surface; not a 2-d idealized surface of a sphere and not the sum total of the planet's mass.
I would think that when adding gases to the atmosphere, there may be some type of "balaning effect" where part of the atmosphere may recombine to form particulates that may go back to the ground.
Can you give an example, please? Or failing that, a more detailed rationale for why you would make this assumption?
If such "particulates" were being created and going "back to the ground," it seems like it wouldn't be that hard to detect them, especially since this process would presumably be increasing along with CO2 emissions. Where should we look?
If I'm misunderstanding your comment, I apologize.
Please refer to Are surface temperature records reliable in which you will find details of the measurements that provide independent verification of hfranzen's detailed PChem calculations.
You will note in Figure 8 on that page that the ocean temperatures are, on average, rising at 0.14-0.15 degC/decade. Land-based temperatures are rising at a faster rate, 0.2-0.3 degC/decade. This article reconciles 30 years of satellite data and 130 years of surface measurements to a high level of consistency. If you look in detail at other data collections, you may find even higher rates in the last 30-40 years.
These data are overwhelming evidence of global warming, which no objective scientist should fail to recognize.
I assume that when averaged over a year and over the whole surface the flux into the eath's surface will equal that exiting from the surface. The rationale for this assumption is that if there is an imbalance the earth, if it is at a higher temperature than this steady-state temperature, will have a higher out-going than in-coming flux and the opposite is true if it is at a lower temperature. Imbalances will occur at various places over the surface and at different times but the average over a year and the earth's surface will yield the balance energy in = energy out.
Actually there is a slight imbalance due to the increasing greenhouse gas effect, but that can be calculated with even greater precision than has been achieved in GWPPT6. The fact that the increase in the earth's temperature is essentially what is calculated for the increasing GHG effect of CO2 means that the mentioned energy balance is a reality for the earth in the absence of astronomical effects such as the Milankovich cycles (which result from perturbations of the earth's orbital eccentricity and the angle of its axis to the orbital plane and are very well understood to occur on time scales very much greater than those concerning us in the current increase in the GHG effectI.
http://wattsupwiththat.com/2010/03/08/the-logarithmic-effect-of-carbon-dioxide/
Obviously this David Archibald guy is too big for his britches (I don't need help dissecting his rubbish). However, is this Modtrans graph showing actual forcing effect CO2 has on our planet (besides feed backs), or is there something missing? To me it looks like it is showing the 3.7 watts per square metre that the IPCC suggests in AR4 section 2.3.1. Thanks in advance.
I find that the assertion in your 12 slide (the anthropogenic origin of atmospheric CO2 excess) would benefit of a mention of the Suess effect. The evolution of the isotope ratios rejects other hypotheses with more certainty than the coincidence in the increase of concentrations with anthropogenic emissions.
http://en.wikipedia.org/wiki/Suess_effect
See also Figure 2.3 in IPCC-AR4
http://www.ipcc.ch/publications_and_data/ar4/wg1/en/ch2s2-3.html#2-3-1
This is just a suggestion for future versions. Thanks for your work.
jon
The article, when calculating the final result, switched from energy to temperature.
Certainly some of the (extra) energy trapped increases temperature but not all. Energy can also go into melting ice or vaporising water (no change on temp at the phase transition), wind, waves etc.
I feel that to quote the change energy balance is enough for a p. chemist; and leave details of where the energy goes to the geophysicists..
The 1.25 is the middle of 1.2 and 1.3 and is the estimate of the no-feedback climate sensitivity I got from the paper of Hansen in 1995:
"For example, for doubled CO2 the no-feedback climate sensitivity of 1.2-1.3 °C is increased to about 2-5 °C in the GCM simulations"
I don't think it changed much since then.
According to the Keeling curve approximation, the concentration CO2 in 2110 will be 688, currently we have 390, this is a (688/390)-1 = 76% increase. (must have used different number above, since I wrote 78%) Because you arrived at 1.4 C for not even a doubling of CO2 I wondered why you got different results from what Hansen stated. Are you calculating the same thing?
I calculated based on the 1.25 of Hansen what such an CO2 increase would do: 1.25*ln(1.76)/ln(2) = 1 degree C. (1.76 is for the 76% increase, 2 is for the doubling, and the ln's are because of the logarithmic relationship)
If I calculate the climate sensitivity based on your numbers I get 1.4*ln(2)/ln(1.76) = 1.7 C and this is about 35% higher than the number Hansen stated.
I agree that both 1.25 and 1.4 are serious, but the feedbacks are the ones to wreak havoc, and turning it into 2 - 4.5 C
The calculation is shown step-by-step on my web site (hfranzen.org) Anyone in the world should feel free to examine the argument for errors. I do not know where Jim Hansen got his numbers so it is impssible for me to say why our numbers differ. I can say that my calculation is primitive compared to those done by climate scientists with large computers. There are two points at which my pared down approach might introduce some error, namely in the neglect of water vapor absorption competing with carbon dioxide absorption (line overlap) and in the use of 50 reciprocal centimeter intervals (as opposed to 5 recip. cm intervals or line-by-line calculation).
It is impossible for me to say by how much these simplificaions alter the results, but, in the years I have been working on GWPPT6, all the feedback i have received has indicated to me that the errors are not significant. But, even if my number is off by some tenths of a degree, I feel that the calculation catches the essential features of the interaction of carbon dioxide in the atmosphere with the earth's infrared radiation and provides me with a basis to assert that global warming is a scientific fact, a statement which I am quite certain is accepted by most readers of these threads. It is my hope that the availability of the argument presented in GWPPT6 will provide for some a deeper understanding of the nature of the interaction.
Climate impact of Increasing Atmospheric Carbon Dioxide
I feel quite certain that this is not a major effect, but it could have some impact upon the calculations. I would suggest that an accurate estimate of the effect of carbon dioxide could be found by doing the calculation including the enormous number of absorption lines of both carbon dioxide and water vapor including allowance for line broadening (collision effects, and his would require partitioning the calculation into temperature intervals as a function of altitude). I not only lack the data and the experience to do this calculation, but it would also require a huge amount of serious computer time.
However I feel confident of the result because the number that I get is in the right ball-park. In fact until I am made aware of a more detailed calculation that yields a result that it significantly different from that of GWPPT6 I am inclined to believe that as long as the interest is in climate change, as opposed to local weather, the results of GWPPT6 are quite satisfactory.
I've taken a look at GWPPT6. I don't see that you have accounted for potential cloud and/or water vapor overlap in your calculations. These are major contributors to net effect of increasing CO2 in the atmosphere - both of which significantly diminish CO2.
Also, it's not clear to me how much additional absorbed power from increasing CO2 your inputing. Maybe I missed that? On page 44 you do apparently state that only half of the absorbed power affects the surface. Can you clarify?
1.I have not accounted for water vapor overlap. Since the absorptions are basically lines in the spectrum (a quantum mechanical effect -see the specrum from the astrophysics group at Ohio State in GWPPT6) it is only the occasional overlap that will have an effect and that to reduce the absorption by CO2 slightly. My guess is that this would result in less than a 10% reduction in the absorption and a far lesser effecct in the change in absorption with increasing CO2 ppm. Since my earth-year temperature increase result fits very well with what is actually observed (see #25 above) I am quite certain that overlap is not serious problem and what GWPPT6 shows is the basic thrust of what is occurring.
2.The increase that I am calculating comes directly from the increase in the broad-band diffuse transmissivity as a result of the increase in the ppm of CO2. The latter is measured by the Keeling curve. The former results directly from the physics of GWPPT6 generalizing Beer's Law from s linear absorption of intensity to a broad-band, diffuse abosrption of flux. I input nothing that is not calculated or observed.
3.Clarification: The absorption is a process in which carbon dioxide is excited from some rotational level in the ground vibrational state to some rotational level in the first excited vibrational state. The short explanation of the fact that half is returmed to the earth is: absorbed radiation is then reemitted through any of a number of processes and this emission is in all directions, i.e. half up and half down. Thus half the reemitted absorbed radiation returns to the earth as GHG flux. A slightly longer explanation of the reemission follows.
Once this excitation has occurred the molecule either relaxes to the ground state or, more frequently, gives up the energy to the translational motion of another molecule (e.g. nitrogen) through collision.
In the more probable collisional dectivation case this energy then becomes part of the thermal bath in which the molcules reside, in other words the atmosphere is locally heated above its steady state temperature. This excess bath energy is then lost through any of a myriad of collisional processes, say with the ubiquitous water molecules. This excitation is then lost through emission.
In either case - direct emission or collisional deactivation follwed by remission from some other infrared active molecule the remmission is isottropic, i.e. nondirectional, and thus occurs with equal probability up or down.
"I have not accounted for water vapor overlap. Since the absorptions are basically lines in the spectrum (a quantum mechanical effect -see the specrum from the astrophysics group at Ohio State in GWPPT6) it is only the occasional overlap that will have an effect and that to reduce the absorption by CO2 slightly. My guess is that this would result in less than a 10% reduction in the absorption and a far lesser effecct in the change in absorption with increasing CO2 ppm. Since my earth-year temperature increase result fits very well with what is actually observed (see #25 above) I am quite certain that overlap is not serious problem and what GWPPT6 shows is the basic thrust of what is occurring.
The increase that I am calculating comes directly from the increase in the broad-band diffuse transmissivity as a result of the increase in the ppm of CO2. The latter is measured by the Keeling curve. The former results directly from the physics of GWPPT6 generalizing Beer's Law from s linear absorption of intensity to a broad-band, diffuse abosrption of flux. I input nothing that is not calculated or observed."
With all due respect, you can't just make these assumptions. Water vapor absorbs a significant amount in the CO2 absorbing bands, especially on the high end above 15u but below too. Water vapor also exists in much higher concentrations than CO2. Then you have the issue of clouds as well. That is anywhere that is cloud covered, incrementally more power from additional CO2 would have little effect because the clouds would absorb the increased power anyway. Also, there is generally more water vapor in between the surface and the clouds, which would further reduce any effect from additional CO2 because of water vapor overlap.
On top of all this, in really dry and cloudless areas where there is the least amount of overlap - i.e. where more CO2 has the highest potential to increase the total surface power, is also where IR heat energy most easily and quickly escapes out to space. This is easily demonstrated by how cold it gets at night in very dry areas with little water vapor or clouds in the atmosphere, such as the desert. Even during the summer, the nights can be very cold - much colder than more moist and/or cloudy areas at the same latitude.
Unless you can properly account for and quantify all these things, your numbers really don't have much bearing on reality.
"Clarification: The absorption is a process in which carbon dioxide is excited from some rotational level in the ground vibrational state to some rotational level in the first excited vibrational state. The short explanation of the fact that half is returmed to the earth is: absorbed radiation is then reemitted through any of a number of processes and this emission is in all directions, i.e. half up and half down. Thus half the reemitted absorbed radiation returns to the earth as GHG flux. A slightly longer explanation of the reemission follows.
Once this excitation has occurred the molecule either relaxes to the ground state or, more frequently, gives up the energy to the translational motion of another molecule (e.g. nitrogen) through collision.
In the more probable collisional dectivation case this energy then becomes part of the thermal bath in which the molcules reside, in other words the atmosphere is locally heated above its steady state temperature. This excess bath energy is then lost through any of a myriad of collisional processes, say with the ubiquitous water molecules. This excitation is then lost through emission.
In either case - direct emission or collisional deactivation follwed by remission from some other infrared active molecule the remmission is isottropic, i.e. nondirectional, and thus occurs with equal probability up or down."
Thanks for the detailed reply, but I was looking for an actual number. The IPCC says a doubling of CO2 results in an intrinsic increase in surface power of 3.7 W/m^2. I've said before on here that if only half of the absorbed power affects the surface, the actual is only 1.85 W/m^2. What are you inputing in your formula? Since we both agree that because the re-radiation is isotropic, only half of the absorbed power can affect the surface and the other half is radiated in the same general direction it was already headed. The question then is specifically how much gross additional power is absorbed from a doubling of CO2?