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## New IPCC report

Roger has been told that his post with his own figures are not acceptable. He can cite data from mainstream scientific sources, such as temperature recordings, greenhouse gasses verified in reputable scientific journals, etc.. and then discuss them.

If he wants to discuss anything, he'll need to cite sources. So his post has been closed, the entire thread does not need to be closed.

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This thread was started and finished back in February 2007. The answer to the question of the thread is that the IPCC 4th assessment report is available at IPCC Fourth Assessment Report: Climate Change 2007 (AR4).

The thread has been reopened in Sept 2009 by Roger Taguchi, with a post containing claims about the science of climate that conflict with conventional physics applied by scientists working on climate. I have also seen the unpublished article where Roger has written up this in more detail.

I did want to defer making a detailed response until it was clear whether or not the new post would remain in the thread. I'm happy now to go ahead with an explanation of why it is incorrect, properly sourced to appropriate references.

Climate sensitivity

 Quote by Roger Taguchi The accurate value for climate sensitivity is 0.277 K/(W/m^2), which is 3 times smaller than the generally accepted value of 0.8 K/(W/m^2).
Roger has confused the notion of climate sensitivity with the notion of Plank response. Neither value is known with three figures of accuracy.

The concept of Planck response is explained, with references, in a number of recent discussions. See, for example, msg #69 of thread "Physics of Global Warming", and following; msg #47 of "Ocean Heat Storage", and msg #147 of "Need Help: Can You Model CO2 as a Greenhouse Gas (Or is This Just Wishful Thinking?)" and following.

In brief, the Planck response is the temperature change you should expect per unit forcing, if nothing at all changes about the Earth other than temperatures. The climate sensitivity is the temperature change you should expect as other aspects of the Earth come in to equilibrium with the new temperature, such as humidity, surface cover, lapse rate, cloud cover, and so on.

The calculation Roger uses is for Planck response. Furthermore, it is a bit too low. Here's a very useful and widely cited reference on climate feedbacks, which includes some helpful definitions and explanations of the terms being used (link to 3.2Mb pdf):

Here's an extract from Appendix A, on definitions (page 3475)
The Planck feedback parameter λP is negative (an increase in temperature enhances the LW emission to space and thus reduces R) and its typical value for the earth’s atmosphere, estimated from GCM calculationsA1 (Colman 2003; Soden and Held 2006), is about -3.2 W m-2K-1 (a value of -3.8 W m-2K-1 is obtained by defining λP simply as -4σT3, by equating the global mean OLR to σT4 and by assuming an emission temperature of 255 K).
This reference is using the inverted value (forcing per unit temperature), but it corresponds to 0.31 K/(W/m2). The simplistic method Bony et al describe gives 0.26 K/(W/m2), close to Roger's value.

Roger speaks of 0.8 as the "generally accepted value", and this is a reference to climate sensitivity. There is no "generally accepted value"; climate sensitivity is one of the great unknowns in climate science and there are many different values proposed, with large associated uncertainties. However, for the most part the empirical and theoretical estimates tend to fall in a range about 50% either side of 0.8; with a couple of odd exceptions. The IPCC summarizes the range of credible values as being from about 0.5 to about 1.2. The paper by Bony et al (2006) cited above is a review article, which is focused upon the feedbacks that amplify Planck response to give sensitivity. The paper does not give sensitivity explicitly, though it can be inferred from the feedback numbers provided, to be in the range about 0.61 to 1.14. (See also the reference in that paper to Soden and Held, 2006).

The thread A low likelihood for high climate sensitivity also considers the range of likely values for sensitivity, and how they are estimated. Note that sensitivity is sometimes given in terms of 2xCO2 rather than W/m2 for the forcing unit. There are references in that thread to recent papers on estimation and constraints for climate sensitivity.

Beer's law and the effect of doubled concentrations

 Quote by Roger Taguchi Because the IPCC data show that doubling CO2 will not double absorption of infrared radiation, the Beer-Lambert law is not being followed, because of diminishing returns after more-than-50% absorption. Thus further doublings of CO2 to the point of suffocating levels can only result in a fraction of a degree increase.
The conventional physics of radiation transfer in the atmosphere is of course fully consistent with the Beer-Lambert law, as is the inference of substantial forcings for successive doublings of CO2. In thinking there is some conflict, Roger is, I suspect, mixing up the effect on a single frequency of light with the effect on the whole spectrum. In any case, his statements quoted above are incorrect.

The Beer-Lampert law, or Beer's Law, indicates that the proportion of light at a given frequency which is transmitted is a negative exponential function of the density of absorbing gas in the atmosphere. Let I be the intensity of incident light, and T be the intensity of transmitted light. Let l be the path length, N be the number density, and σ be the absorption cross section. Then Beer's law in a gas can be expressed:
$$\frac{T}{I} = e^{- \sigma l N}$$
This law does not consider re-radiation within the gas, and more importantly, it applies for a single frequency of light and the absorption cross section associated with that frequency. The simple consequence is that if the density of a gas is doubled, the proportion of light transmitted is squared (and hence reduced, since a proportion is in the range 0 to 1). However, this applies for each frequency individually; and does not carry across to the whole spectrum.

For example, suppose that a gas absorbs in three distinct bands. In the weak absorption band, there is 99% transmission. In the strong absorption band there is 1% transmission. And in the partial absorption band there is 50% transmission. When you double the density of the absorbing gas, these numbers become 98%, 0.01% and 25% respectively, by application of Beer's law. Now if the incoming light is equally intense in all bands, then the transmitted fraction is originally 50%, and after doubling it becomes 41%. All consistent with Beer's law. The great majority of the additional absorption occurs not in the weak or strong absorption bands, but in the partial band, where the derivative of absorption with density is greatest.

The actual absorption spectrum of a gas is much more complex than the simple three band case I've given as an example here, but the same principle holds in general. The major effect of doubling concentrations of a greenhouse gas (one that absorbs thermal radiation) is to increase absorption in the "shoulders", or "wings", of the main absorption band. This is a basic detail of the transmission of light in a gas which should be covered in any decent undergraduate text.

CO2 absorbs strongly at around 12 microns. Hence pretty much all of that frequency is absorbed. Further doublings of CO2 have negligible further effect on the absorption in the middle of that band. However, the width of the saturated band increases, as more and more frequencies move from being minimally absorbed to mostly absorbed; and the net effect of this is the basis for the logarithmic relation between density of a greenhouse gas, and temperature impact.

My reference for this effect is the online textbook Principles of Planetary Climate, by Raymond Pierrehumbert at the Uni of Chicago. Beer's Law is discussed on page 155. Here's an extract, including figure 4.12, from page 186, which explains the effect of increasing concentrations of a greenhouse gas, using CO2 as the example.

Figure 4.12: Lower panel: The absorption coefficient for CO2 at 1 bar and 300K, in the wavenumber range of interest for Earthlike and Marslike planets. The horizontal lines show the wavenumber range within which the optical thickness exceeds unity for CO2 paths of 1/10 , 1 and 1000 kg/m2. Upper panel: The corresponding OLR for the three path values, computed for the same temperature profiles as in Figure 4.5. The OLR has been averaged over bands of width 10 cm−1.
Figure 4.12 explains why the OLR reduction is approximately logarithmic in greenhouse gas concentration for CO2 and similar greenhouse gases. The key thing to note is that the absorption coefficient in the principal band centered on 675cm−1 decays exponentially with distance from the center. Hence, as the CO2 path is increased by a factor of 10, from 1/10 to 1 kg/m2, the width of the ditch within which the radiating temperature is reduced to cold stratospheric values increases only like the logarithm of the ratio of paths. This is true for paths as small as .01 kg/m2 and as large as 100kg/m2. However, when the path gets as large as 1000kg/m2, the weak absorption bands on the shoulder, near 950 and 1050 cm−1 start to become important, and enhance the optical thickness beyond what one would expect on the basis of the central absorption peak. 1000 kg/m2 corresponds to a partial pressure of CO2 of about 100mb for Earth’s gravity, or equivalently a molar mixing ratio of about 10 % for Earth’s current surface pressure. This is far in excess of any CO2 concentration on Earth likely to have been attained in the past 300 million years [...]. Many greenhouse gases also have a central absorption peak with exponential skirts, and these will also exhibit a nearly logarithmic dependence of OLR on the concentration of the corresponding greenhouse gas.
You can also see msg #7 of thread "Rising Carbon Dioxide Levels Don’t Increase Earth’s Temperature" for two spectra obtained from the MODTRAN calculator which shows clearly the effect of additional absorption in the wings of the main absorption band when concentrations are doubled.

Water vapour in the atmosphere

 Quote by Roger Taguchi The same IPCC data show that water vapour is 1.5 times as important as CO2 as a greenhouse gas, and it still seems to follow the Beer-Lambert law (doubling the concentration doubles the absorption). Thus climate changes are more sensitive to changes in water vapour than to CO2. Since water vapour is released on the combustion of alcohols (including methanol and ethanol) and alkanes (including methane, propane, gasoline and diesel fuels), but not coal, and by transpiration in plants in forests and crops, efforts to mitigate climate change by reducing human-produced water vapour would run in exactly the opposite direction to efforts to reduce climate change by controlling CO2 alone.
Water vapour is indeed the strongest greenhouse gas in the atmosphere. The interactions of gases is not simple, but in general I would say water is at least twice as important as carbon dioxide for the Earth's greenhouse effect. 1.5 times seems an underestimate of its importance.

The major point about water vapour is that it cycles through the atmosphere so quickly. Enormous volumes of water evaporate into the atmosphere and precipitate out again every day. The total water content is determined largely by temperature. As temperature rises, you get more water vapour (higher specific humidity), as relative humidity remains about the same or even reduces a bit. If you add a lot of water into the atmosphere, it rains out again in very short order, to restore the natural equilibrium for humidity.

Hence human emissions of water into the atmosphere don't have much effect on the total water vapour content. The best way to have a significant effect on water vapour is to somehow raise temperatures. This is why water vapour is treated as a "feedback", and carbon dioxide as a "forcing". Added carbon dioxide remains in the atmosphere a long time, and contributes to a stronger greenhouse effect and higher temperatures. This tends to raise specific humidity, which increases water in the atmosphere as well, and that makes the greenhouse effect stronger again. There are many complexities with the feedbacks, discussed in the reference Bony et al (2006) cited above. The point is that it is a feedback, because humidity is so strongly influenced by temperature, rather than by anthropogenic emissions.

Roger speaks above of "efforts to mitigate climate change by reducing human-produced water vapour". There are no such efforts. It would be a waste of time, because human produced water vapour has so little effect.

References. The various kinds of feedback that are involved with water vapour are discussed in Bony et al (2006). The relation of water vapour and temperature on Earth is discussed on page 101 of Principles of Planetary Climate, and also page 208 and following. Here is an extract from page 209:
On a planet without a substantial condensed water reservoir, water vapor could be a wellmixed noncondensing greenhouse gas much like CO2 on modern or early Earth. In most known cases of interest, though, atmospheric water vapor is in equilibrium with a reservoir -- an ocean or glacier -- which fills the atmosphere to the point that the atmospheric water vapor content is limited by the saturation vapor pressure.
Conclusion

There are plenty of open questions in climate science, and room for reasonable skepticism about some of the conclusions that are reported. However, the various sides of real scientific questions all work from a common underlying physical foundation. In my opinion, the post to which I am responding is not so much skepticism as basic error and misunderstanding of foundational material from which all the many genuinely open questions should be examined. I am not in any personal doubt about this myself; but I don't demand agreement from everyone. I have tried to give the explanation in some detail, with appropriate references, and am happy to let it stand as a response for readers to consider.

Cheers -- sylas

PS. Evo, thanks for your work on this. I hope this is okay. If there are still problems, let me know. I was not able to PM you; your box is full. Feel free to delete this postscript if you see it.

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Let's have a look at this:

 Quote by sylas Water vapour in the atmosphere Water vapour is indeed the strongest greenhouse gas in the atmosphere. The interactions of gases is not simple, but in general I would say water is at least twice as important as carbon dioxide for the Earth's greenhouse effect. 1.5 times seems an underestimate of its importance. The major point about water vapour is that it cycles through the atmosphere so quickly. Enormous volumes of water evaporate into the atmosphere and precipitate out again every day. The total water content is determined largely by temperature. As temperature rises, you get more water vapour (higher specific humidity), as relative humidity remains about the same or even reduces a bit. If you add a lot of water into the atmosphere, it rains out again in very short order, to restore the natural equilibrium for humidity. Hence human emissions of water into the atmosphere don't have much effect on the total water vapour content. The best way to have a significant effect on water vapour is to somehow raise temperatures. This is why water vapour is treated as a "feedback", and carbon dioxide as a "forcing". Added carbon dioxide remains in the atmosphere a long time, and contributes to a stronger greenhouse effect and higher temperatures. This tends to raise specific humidity, which increases water in the atmosphere as well, and that makes the greenhouse effect stronger again. There are many complexities with the feedbacks, discussed in the reference Bony et al (2006) cited above. The point is that it is a feedback, because humidity is so strongly influenced by temperature, rather than by anthropogenic emissions.
Now, repeating my remark,

 Quote by Andre It would also be interesting to calculate the energy required versus energy available to evoporate the additional water vapor required to keep that feedback going.
Did somebody do the paperwork here, how high is the energy bill? And is global warming paying for that? For getting all that extra water in the atmosphere?

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 Quote by Andre Did somebody do the paperwork here, how high is the energy bill? And is global warming paying for that? For getting all that extra water in the atmosphere?
All the energy that matters ultimately comes from the Sun.

I don't really understand the point of your question. Is the answer I give above what you want?

For getting water into the atmosphere; you are asking here about the water cycle, it seems. That's definitely driven by energy from the Sun.

As for doing "paper work", that sounds like a question about figuring out energy balances. Yes, of course scientists do the paper work in that sense.

Cheers -- sylas
 Blog Entries: 2 Recognitions: Gold Member I just would like to see how many W/m2 of energy is required to sustain the higher evaporation rate, associated with keeping relative humidities more or less constant with higher temperatures as proposed for the positive feedback. Obviously that additional energy has to come from the increased IR back radiation and it also is not available to warm the surface. I did not encounter calculations for that so far, so can you shown them? and transparantly of course, like independently reproduceable. Of course we could try some ball park figures for a very rough order of magnitude estimate.

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 Quote by Andre I just would like to see how many W/m2 of energy is required to sustain the higher evaporation rate, associated with keeping relative humidities more or less constant with higher temperatures as proposed for the positive feedback.
We've had this discussion long ago I vaguely remember.

There is no constant energy flux necessary to do this, as what is "lost" at evaporation is also "gained back" at condensation. In other words, you do not need a power flux to maintain a higher humidity. Consider a closed bottle with a liquid in, at a certain temperature. There's a certain partial pressure of vapor in equilibrium with the liquid. Now, heat the bottle somewhat. This will increase the temperature of the liquid and vapor en bottle. That costs some energy of course (heat capacity and latent heat). However, to maintain this bottle at this higher temperature, with the higher vapor pressure, doesn't need any energy anymore, once it is there. There is nevertheless constantly evaporation in the bottle, and constant condensation (dynamical equilibrium). The power input is 0 nevertheless.

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 Quote by Andre I just would like to see how many W/m2 of energy is required to sustain the higher evaporation rate, associated with keeping relative humidities more or less constant with higher temperatures as proposed for the positive feedback.
The total energy is the same, no matter what feedbacks are involved. It is the energy from the Sun. We get about 342 W/m2, of which about 30% is reflected, leading 240 W/m2 absorbed into the Earth. All the energy must be radiated again.

In the process of absorbing and re-radiating all the energy, we have what is effectively a massive engine, which provides all the energy for the water cycle, for weather, and all the other dynamic features of our active planet powered by this flood of solar energy.

The mean temperature of the Earth is determined by the need to shed this energy. The composition of the atmosphere bears upon this, by determining how easily the radiant energy from the surface gets out into space.

When the atmosphere absorbs more infrared radiation, there is a shift in distribution of energy up from the surface, with a bit more of the total upwards energy flux being from convention and latent heat, and a bit less being from radiant transfers. But the total flux of energy out the top of the atmosphere remains the same, by the first law. It has to balance the energy input from the Sun.

 Obviously that additional energy has to come from the increased IR back radiation and it also is not available to warm the surface. I did not encounter calculations for that so far, so can you shown them? and transparantly of course, like independently reproduceable. Of course we could try some ball park figures for a very rough order of magnitude estimate.
I am having a hard time sorting out what you want.

I've done all kinds of back of the envelope calculations for different energy fluxes here, and I can usually give references to where there are similar things in the literature. But what precisely you are asking doesn't seem to make sense to me yet. I think you may be mixing up different things, but I am not sure.

Do you mean the energy balance equations? If so, the best reference for this would be Trenberth et al (2009), cited in msg #1 of thead "Estimating the impact of CO2 on global mean temperature".

Do you mean what additional energy is required to bring up the Earth to a new temperature? The main heat sink is the ocean. You can estimate how much energy it takes to heat up the ocean by 1 degree, for example (since this is the largest capacity heat sink) but this is a one time cost, and until that cost is paid the energy is out of balance, with a small trickle of energy going into heating the ocean. That's roughly 0.5 W/m2 at present. See also the first part of msg #89 in thread "Estimating the impact of CO2 on global mean temperature".

Do you mean the additional energy to maintain a new rate of convenction and latent heat? There's no such thing. The total energy flux remains the same as ever, as required by the first law. Once you get to a new equilibrium, you maintain it with exactly the same amount of energy as before.

Cheers -- sylas

PS. Added in edit. vanesch seems to have stated much more concisely and clearly what is wrong with the assumptions behind the question.

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 Quote by vanesch We've had this discussion long ago I vaguely remember. There is no constant energy flux necessary to do this, as what is "lost" at evaporation is also "gained back" at condensation.
Exactly, but where is that condensation happening? and what happens to that energy after condensation, heating the earth surface? or is it lost in space for a big part?

Maybe this bottle is NOT closed.

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 Quote by sylas PS. Added in edit. vanesch seems to have stated much more concisely and clearly what is wrong with the assumptions behind the question.
What can be wrong with assumptions behind a question? But it seems to me that we may have hit the core of the matter here about the water vapor feedback. back later with some ballpark figures
 Blog Entries: 2 Recognitions: Gold Member okay, let's try again the atmosphere is not a closed bottle as proposed by Vanesch but it consists of several conveyer belts which move air, thermal and latent energy up and down. The most important one is the Hadley cel, comprising about half of the world surface, roughly between the tropic of cancer and the tropic Capricon. It creates monsoons and trade winds as part of a gigantic vertical conveyor belt. the winds towards the equator pick up humidity depending on the sea surface temperature, the mixture rate and wind speed and the surface temperature of the atmosphere. This process takes up energy and a lot, about 2500 Joules per gram (J/g) of water or 600 calories per gram (cal/g) of water aka latent heat. Energy that is taken away from the heating process at the place of the evaporation. Then the winds converge at the Intertropical Convergion Zone where it convects up, cools and the moisture starts to condensate, releasing the latent heat again, much higher in the atmospher. The resultant clouds have many different feedback effects, like reflecting direct solar sunlight, "reflecting" or outradiation long wave IR radiation both up and down. This causes much of the original surface warmth, via latent heat convection to escape to space. Now, again we have a conveyer belt here, so if there should be more water vapor in the atmosphere it;'s not enough to give only an initial boost like in the closed bottle, as the increasing cloud cover at higher temperatures will also increase out radiation from higher levels. Perhaps compare it with a leaky bucket, filled from the top but water leaking away from the bottom, which holds the water level in dynamic equilibrium . So it you open to tap more momentarily the water level will rise a bit but that will soon diminish if the tap is turned down again. You need to keep the tap more open to increase the water level. Likewise you need to evaporate more water constantly to have a higher atmospheric humidity. And the simple question is, if there is enough energy available to get that done, especially given the high evaporation energy required.

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 Quote by Andre Exactly, but where is that condensation happening? and what happens to that energy after condensation, heating the earth surface? or is it lost in space for a big part? Maybe this bottle is NOT closed.
That is correct. However, we are talking about a higher level of absolute humidity (or constant relative humidity). That, by itself, doesn't require any extra energy. Now, it is possible (but not necessary) that this higher humidity also means a FASTER cycle of evaporation at sea level and condensation at higher altitudes (although that is by no means necessary to have higher absolute humidity by itself). This would then simply mean that convection, through this latent heat, has a bigger heat transport from the lower layers to the higher layers. However, convection will adapt perfectly to what is missing in heat transport to keep the (half-wet) lapse rate. If less transport is done through radiative transfer, more transport will be done by convection and vice versa, because the adiabat has to be restored.
So if a certain "matter flux" of convection is now more efficient in transporting heat (because containing more absolute humidity, and hence more latent heat), if convection has to transport the same amount of heat, the matter flux will diminish, to compensate (the air will rise SLOWER).

If the adiabat has to be restored, convection will compensate *perfectly* the "missing part" of the heat flux by radiation. The humidity level can (will) influence the adiabat and the lapse rate. But that, by itself, doesn't tell you anything about the amount of transported heat. And the humidity level by itself doesn't require any "extra energy".

You could picture this still differently. If you consider that humid air has more "latent heat" than "dry air" (which is the case), then this doesn't mean that HAVING more humid air requires a constant heat flux in. However, *making humid air go up and condense*, will require some heat flux up. How much, will depend entirely on the matter flux of that humid air up. That can adapt to the heat flux to be transported. And the regulatory mechanism is the lapse rate. If the humid air goes up too fast, lower layers will cool too much as compared to higher layers (where the heat is deposited), the lapse rate decreases, and convection slows down and stops. On the other hand, if there's not enough convection, the lower layers will accumulate heat, the lapse rate will increase, and convection is stimulated. Stationarity will set in when the lapse rate is given by the adiabat of the humid air.

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 Quote by Andre And the simple question is, if there is enough energy available to get that done, especially given the high evaporation energy required.
And the simple answer is yes, of course there is.

The energy available comes from the Sun, and it is enormous. The guts of this question is about the energy NOW, to drive the massive circulations of air and moisture of our dynamic atmosphere. The energy comes from the Sun. The atmosphere IS an enclosed system, since we look at the entire Earth; and there's no other energy source that matters for circulating the air in the atmosphere.

If the capacity of the atmosphere to absorb thermal radiation is increased a little bit (and in absolute terms, we are speaking of small changes of a few W/m2) then there will be some changes to a new equilibrium condition. Another degree or two or temperature, and hence a higher specific humidity, and various changes in circulation patterns as the planet readjusts. But the total energy? It's all exactly the same. It's still 342 W/m2 in from the Sun, which is about 30% reflected and the result powering up from the surface as always, driving the water cycle and weather patterns and circulations, the inevitable result of the thermodynamic necessity that the Earth must shed that energy back out into space again.

Cheers -- sylas
 Blog Entries: 2 As I understand it, with rising CO2 levels, what changes is the atmospheres ability to hold water vapor. The elevation of the atmosphere grows into higher elevations. That is, its thickness is greater. As many of us understand, the atmosphere at sea level is saturated in the IR mostly by water vapor. However, at the edge of the troposphere, water vapor is nil and CO2 predominates. This is because CO2 is a well mixed greenhouse gas while water vapor is not. Air Temperature drops about 6.5C for every 1000 meters; this is the lapse rate. So, how high up in the atmosphere does the lapse rate apply? Obviously, it doesn't apply all the way to the moon since that would imply impossibly low temperatures. Instead, it only applies to the elevation at which there is no longer any significant water vapor and that elevation in turn is dictated by the level of CO2, CH4 and NOx in the atmosphere and the amount of heat being transferred.

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 Quote by Xnn So, how high up in the atmosphere does the lapse rate apply? Obviously, it doesn't apply all the way to the moon since that would imply impossibly low temperatures. Instead, it only applies to the elevation at which there is no longer any significant water vapor and that elevation in turn is dictated by the level of CO2, CH4 and NOx in the atmosphere and the amount of heat being transferred.
Actually, the lapse rate applies up to the tropopause; and that is determined not by the presence or absence of water vapour, but by whether there is net heating or cooling from the effects of radiation transfers. Water vapour actually works to reduce the lapse rate, because the moist adiabat is significantly weaker than the dry adiabat.

A dry atmosphere has a much stronger lapse rate. But whether dry or moist, the lapse rate applies up until the atmosphere is back in a radiative equilibrium, and this transition marks tropopause, the end of vertical convection, and the start of the stratosphere (stratified) in which the "lapse rate" is governed by completely different principles, of radiative equilibrium rather than of adiabatic convection and radiative-convective equilibrium.

The theory of lapse rate and tropopause height is explained in Principles of Planetary Climate, in section 4.8 "Tropopause height for real gas atmospheres" (page 255). The theory is general, and applies for all kinds of atmospheres and planets. Other texts on atmospheric physics should explain the same ideas; I am consistently referring to this text (PoPC) primarily because it is online and easily accessible as a common reference point for discussion. There is a progression of material; dipping into a section will give useful conclusions; and for deeper understanding it's well worth working through the previous chapters.

In brief, if works like this. In the absence of any convection or conduction of heat, where the only energy flux is from radiation, we expect a sharp discontinuity in temperature at the surface. The so called natural "skin temperature" of the atmosphere is 2-0.25 = 0.84 times the surface temperature. (See PoPC section 3.6 "Optically thin atmospheres: The skin temperature" page 141.) This is of course unstable; and there is a flow of heat up into the atmosphere from the surface by convection, and a natural temperature gradient is formed based on adiabatic energy transfers, up until the skin temperature is reached. In this region, the atmosphere is in "radiative-convective equilibrium", with net cooling of the atmosphere from radiation balanced exactly by net heating from convection. See also figure 3.14 from PoPC.

Above the tropopause the atmosphere is in a pure radiative equilibrium. I have also given some more discussion, with some extracts from PoPC, in other posts. See, for example msg #154 of thread " Need Help: Can You Model CO2 as a Greenhouse Gas (Or is This Just Wishful Thinking?)", and surrounding discussion.

Cheers -- sylas

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Thanks sylas;

At the tropopause, the atmosphere is dry and water vapor is no longer a significant constituent. However, the elevation of the tropopause is not constant. It varies about the earth and is also rising as the levels of greenhouse gases rise. Hence, global warming.

Don't know if it's elevation could be deried from fundamental constant and physical properties or not.

The World Metrological Organization has the following definition for tropopause:

 The lowest level at which the lapse rate decreases to 2 °C/km or less, provided that the average lapse rate between this level and all higher levels within 2 km does not exceed 2 °C/km.

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 Quote by Xnn Thanks sylas; At the tropopause, the atmosphere is dry and water vapor is no longer a significant constituent. However, the elevation of the tropopause is not constant. It varies about the earth and is also rising as the levels of greenhouse gases rise. Hence, global warming.
Yes; but the reduction in moisture is not the reason for the tropopause. You would still have a tropopause even if the atmosphere was completely dry all the way through.

The elevation of the tropospause is indeed not constant. It is highest at the equator... and the major reason for this is that the equator is more humid, and has a weaker lapse rate. Hence you go up much higher to get to skin temperature and radiative equilibrium. (PoPC page 258.)

The reason that global warming is leading to a rise in the tropopause is that the surface temperature increases, but the lapse rate remains about the same or perhaps a bit weaker. Hence there is a greater distance from the surface to the tropopause and radiative equilibrium.

Note also that a lapse rate is essential for greenhouse warming to work (PoPC Fig 3.6, section 3.3); but that it is not sufficient. An optically thin atmosphere provides negligible warming, but it still has an adiabatic lapse rate and a tropopause. (PoPC, section 3.6).

 Don't know if it's elevation could be deried from fundamental constant and physical properties or not.
You can get a good quantified estimate. I cited previously Principles of Planetary Climate section 4.8. The calculations are based on well defined physical principles, but they are an approximation.

I recommend very highly the exercise of reading though the first four chapters of Principles of Planetary Climate; or indeed any other similarly detailed undergraduate level text on atmospheric physics and energy balance. I found it hard work, but it is an excellent way to develop a deeper understanding of how the temperature structure of the atmosphere works.

 The World Metrological Organization has the following definition for tropopause:
This is a very Earth-centric definition. If you go through Principles of Planetary Climate a general physical theory is developed which can be applied in general to a whole range of different planets and different conditions. (See footnote 1 at the bottom of page 75, PoPC)

The major benefit of this is understanding better why there is a tropopause in the atmosphere, and why its altitude tends to increase with with greenhouse driven warming, and why the tropical tropopause is so much higher. When you can actually calculate an estimate for the height of the tropospause for given conditions, the level of understanding is greatly enhanced. I'd have to refer to notes to carry through such a calculation, at present.

Cheers -- sylas
 I haven't worked through the details, but I think Ray Pierrehumbert loosely defines the tropopause as the height of convection.