Guest post by Robert G. Brown
Duke University Physics Department
The Problem
In 2003 a paper was published in Energy & Environment by Hans Jelbring that asserted that a gravitationally bound, adiabatically isolated shell of ideal gas would exhibit a thermodynamically stable adiabatic lapse rate. No plausible explanation was offered for this state being thermodynamically stable – indeed, the explanation involved a moving air parcel:
An adiabatically moving air parcel has no energy loss or gain to the surroundings. For example, when an air parcel ascends the temperature has to decrease because of internal energy exchange due to the work against the gravity field.
This argument was not unique to Jelbring (in spite of his assertion otherwise):
The theoretically deducible influence of gravity on GE has rarely been acknowledged by climate change scientists for unknown reasons.
The adiabatic lapse rate was and is a standard feature in nearly every textbook on physical climatology. It is equally well known there that it is a dynamical consequence of the atmosphere being an open system. Those same textbooks carefully demonstrate that there is no lapse rate in an ideal gas in a gravitational field in thermal equilibrium because, as is well known, thermal equilibrium is an isothermal state; nothing as simple as gravity can function like a “Maxwell’s Demon” to cause the spontaneous stable equilibrium separation of gas molecules into hotter and colder reservoirs.
Spontaneous separation of a reservoir of gas into stable sub-reservoirs at different temperatures violates the second law of thermodynamics. It is a direct, literal violation of the refrigerator statement of the second law of thermodynamics as it causes and maintains such a separation without the input of external work. As is usually the case, violation of the refrigeration statement allows heat engines to be constructed that do nothing but convert heat into work – violating the “no perfectly efficient heat engine” statement as well.
The proposed adiabatic thermal lapse rate in EEJ is:
where g is the gravitational acceleration (presumed approximately constant throughout the spherical shell) and cp is the heat capacity per kilogram of the particular “ideal” gas at constant pressure. The details of the arguments for an adiabatic lapse rate in open systems is unimportant, nor does it matter what cp is as long as it is not zero or infinity.
What matters is that EEJ asserts that 
in stable thermodynamic equilibrium.
The purpose of this short paper is to demonstrate that such a system is not, in fact, in thermal equilibrium and that the correct static equilibrium distribution of gas in the system is the usual isothermal distribution.
The Failure of Equilibrium
In figure 1 above, an adiabatically isolated column of an ideal gas is illustrated. According to EEJ, this gas spontaneously equilibrates into a state where the temperature at the bottom of the column Tb is strictly greater than the temperature Tt at the top of the column. The magnitude of the difference, and the mechanism proposed for this separation are irrelevant, save to note that the internal conductivity of the ideal gas is completely neglected. It is assumed that the only mechanism for achieving equilibrium is physical (adiabatic) mixing of the air, mixing that in some fundamental sense does not allow for the fact that even an ideal gas conducts heat.
Note well the implication of stability. If additional heat is added to or removed from this container, it will always distribute itself in such a way as to maintain the lapse rate, which is a constant independent of absolute temperature. If the distribution of energy in the container is changed, then gravity will cause a flow of heat that will return the distribution of energy to one with Tb > Tt . For an ideal gas in an adiabatic container in a gravitational field, one will always observe the gas in this state once equilibrium is established, and while the time required to achieve equilibrium is not given in EEJ, it is presumably commensurate with convective mixing times of ordinary gases within the container and hence not terribly long.
Now imagine that the bottom of the container and top of the container are connected with a solid conductive material, e.g. a silver wire (adiabatically insulated except where it is in good thermal contact with the gas at the top and bottom of the container) of length L . Such a wire admits the thermally driven conduction of heat according to Fourier’s Law:
where λ is the thermal conductivity of silver, A is the cross-sectional area of the wire, and ΔT=Tb–Tt . This is an empirical law, and in no way depends on whether or not the wire is oriented horizontally or vertically (although there is a small correction for the bends in the wire above if one actually solves the heat equation for the particular geometry – this correction is completely irrelevant to the argument, however).
As one can see in figure 2, there can be no question that heat will flow in this silver wire. Its two ends are maintained at different temperatures. It will therefore systematically transfer heat energy from the bottom of the air column to the top via thermal conduction through the silver as long as the temperature difference is maintained.
One now has a choice:
- If EEJ is correct, the heat added to the top will redistribute itself to maintain the adiabatic lapse rate. How rapidly it does so compared to the rate of heat flow through the silver is irrelevant. The inescapable point is that in order to do so, there has to be net heat transfer from the top of the gas column to the bottom whenever the temperature of the top and bottom deviate from the adiabatic lapse rate if it is indeed a thermal equilibrium state.
- Otherwise, heat will flow from the bottom to the top until they are at the same temperature. At this point the top and the bottom are indeed in thermal equilibrium.
It is hopefully clear that the first of these statements is impossible. Heat will flow in this system forever; it will never reach thermal equilibrium. Thermal equilibrium for the silver no longer means the same thing as thermal equilibrium for the gas – heat only fails to flow in the silver when it is isothermal, but heat only fails to flow in the gas when it exhibits an adiabatic lapse in temperature that leaves it explicitly not isothermal. The combined system can literally never reach thermal equilibrium.
Of course this is nonsense. Any such system would quickly reach thermal equilibrium – one where the top and bottom of the gas are at an equal temperature. Nor does one require a silver wire to accomplish this. The gas is perfectly capable of conducting heat from the bottom of the container to the top all by itself!
One is then left with an uncomfortable picture of the gas moving constantly – heat must be adiabatically convected downward to the bottom of the container in figure 1 in ongoing opposition to the upward directed flow of heat due to the fact that Fourier’s Law applies to the ideal gas in such a way that equilibrium is never reached!
Of course, this will not happen. The gas in the container will quickly reach equilibrium. What will that equilibrium look like? The answer is contained in almost any introductory physics textbook. Take an ideal gas in thermal equilibrium:
where N is the number of molecules in the volume V, k is Boltzmann’s constant, and T is the temperature in degrees Kelvin. n is the number of moles of gas in question and R is the ideal gas constant. If we assume a constant temperature in the adiabatically isolated container, one gets the following formula for the density of an ideal gas:
where M is the molar mass, the number of kilograms of the gas per mole.
The formula for that describes the static equilibrium of a fluid is unchanged by the compressibility (or lack thereof) of the fluid – for the fluid to be in force balance the variation of the pressure must be:
(so that the pressure decreases with height, assuming a non-negative density). If we multiply both sides by dz and integrate, now we get:
Exponentiating both sides of this expression, we get the usual exponential isothermal lapse in the pressure, and by extension the density:
where P0 is the pressure at z=0 (the bottom of the container).
This describes a gas that is manifestly:
- In static force equilibrium. There is no bulk transport of the gas as buoyancy and gravity are in perfect balance throughout.
- In thermal equilibrium. There is no thermal gradient in the gas to drive the conduction of heat.
If this system is perturbed away from equilibrium, it will quickly return to this combination of static and thermal equilibrium, as both are stable. Even in the case of a gas with an adiabatic lapse rate (e.g. the atmosphere) remarkably small deviations are observed from the predicted P(z) one gets treating the atmosphere as an ideal gas. An adiabatically isolated gas initially prepared in a state with an adiabatic lapse rate will thermally equilibrate due to the internal conduction of heat within the gas by all mechanisms and relax to precisely this state.
Conclusion
As we can see, it is an introductory physics textbook exercise to demonstrate that an adiabatically isolated column of gas in a gravitational field cannot have a thermal gradient maintained by gravity. The same can readily be demonstrated by correctly using thermodynamics at a higher level or by using statistical mechanics, but it is not really necessary. The elementary argument already suffices to show violation of both the zeroth and second laws of thermodynamics by the assertion itself.
In nature, the dry adiabatic lapse rate of air in the atmosphere is maintained because the system is differentially heated from below causing parcels of air to constantly move up and down. Reverse that to a cooling, like those observed during the winter in the air above Antarctica, and the lapse rate readily inverts. Follow the air column up above the troposphere and the lapse rate fails to be observed in the stratosphere, precisely where vertical convection stops dominating heat transport. The EEJ assertion, that the dry adiabatic lapse rate alone explains the bulk of so-called “greenhouse warming” of the atmosphere as a stable feature of a bulk equilibrium gas, is incorrect.
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Do people really have to have stuff like this demonstrated? I saw, but didn’t read, the previous post on this topic. I thought it was a weird thing to have on a climate blog, though since it did deal with a gas and our climate is a gas, I didn’t think too much of it.
Now, I’ve gone back and read that previous post and glanced through the LONG bunch of comments, a disturbing number of which actually supported the impossible idea. This is basic physics, Second Law of Thermodynamics sort of stuff – you can’t get perpetual-anything.
I thought everyone realized you can’t magically use gravity or magnets to generate perpetual energy machines. It blows my mind that there are actually people who think a thermally graded column would result. It’s nothing but a variation on a perpetual energy machine.
Kudos to WUWT for spending some time debunking this sort of nonsense. It’s sad that there are apparently so many people who swallow this sort of nonsense in the first place.
There seem to be a lot of people who think that a lapse rate is a proprty of the matter itself, rather than a convenient description of the behaviour of the gas.
I’m not sure that Dr Brown will convince them otherwise.
But maybe Hans Jelbring can convince us all that the heat in figure 2 cannot rise up the silver bar because graivity is holding it down!
Sometimes something like this comes along that refutes a common idea so well and simply that it just makes a guy go ‘holy crap!’. Thanks for this excellent post as for me it was a revelation.
‘what goes up, has to come down,spinning wheel has to go round’ blood sweat and tears. it’s a bit warm !
“If this system is perturbed away from equilibrium, it will quickly return to this combination of static and thermal equilibrium, as both are stable.
_______________________
This is also perfectly expressed by Le Chatelier’s Principle.
What would complement this theoretical explanation is if an experiment backed it up.
So far as I know no experiment has ever been carried out.
All suggested proposals seem to run into problems when real components and physically accurate numbers are used.
“Spontaneous separation of a reservoir of gas into stable sub-reservoirs at different temperatures violates the second law of thermodynamics. It is a direct, literal violation of the refrigerator statement of the second law of thermodynamics as it causes and maintains such a separation without the input of external work.”
No, Robert. The second law requires that no energy gradient can be maintained without input of work. It requires the reservoir of gas to be isoenergetic not isothermal. A horizontal layer may be a different temperature than another if the cooler layer has its lesser kinetic energy balanced by greater gravitational energy. This is obviously the case since it goes without a shadow of a doubt that a molecule of air in a higher layer has more gravitational energy than a molecule in a lower layer. Thus the second law actually demands a temperature difference of equal and opposite polarity to compensate for the difference in gravitational energy. An isothermal atmosphere in a gravity field is the one that violates the second law.
My name is Joules because I know how to find and count them no matter how they try to hide.
Thanks for playing.
Hmmm. Temperature is the integral of the number and energy of particles seen at the measuring surface. Less particles equal lower temperature given equal velocity profiles of the particles. Less particles also equals lower pressure. Temperature and pressure are thus directly linked. Hence the dry lapse rate. Gravity and pressure are also directly linked. Hence the dry lapse rate.
Well thanks for taking the time to post Doc.
I gave up trying to convince Tallbloke in the other thread as he is clearly in ‘La La La’ mode.
His reply as to what was going to stop heat flow in a system with a temperature gradient was that at the interface between hotter and colder the actual temperature was the same and therefore no heat would flow!
Lets see, take a rod with a temperature gradient and cut it into the thinnest possible slices allowed in physics and apparently adjoining slices are the same temperature!
However, that would mean the rod was actually the same temperature all the way through as not only are those two touching slices (call them a & b) the same temperature, then obviously, the two slices c & d touching them are also at the same temperature and the two slices e & f touching c & d are the same as c & d and a & b also. Well you get the picture.
Anyway I am sure he will be along soon to show that gravity has some magical quality that allows work to be done indefinitly in a closed system.
Alan
The dry adiabatic lapse rate is derived several ways, each of which assumes an isentropic atmosphere.
Example 1. A packet of dry air rises. Let’s first assert that it is adiabatic (hopefully not controversial), as it rises it expands and does work on it’s environemnt. But that work is reversible. Thus the process is adiabatic and reversible, hence isentropic. By definition.
Example 2. We know from calculus that dx/dy(w)*dy/dw(x)*dw/dx(y) = -1. Let us apply this identity to dT/dp(s), the temperature pressure relationship at constant entropy. We will find that dT/dp=-dS/dP * dT/ds. the second RHS term is Cp/T by the definitoin of entropy. The first RHS term is, by a Maxwell relationship, -dV/dT, which for an ideal gas is R/P (you can google Maxwell relationship). Making the substitutions: dT/dp = RT/PCp = V/Cp for an ideal gas. Lastly we recognize that the gravity imposed pressure gradient is dP/dz = rho*g, but rho is 1/V, so when we multiply: dT/dp * dp/dz = V/Cp * g/V = g/Cp.
Example 3. The state of an ideal gas is fully specified when two variables are specified. We shall choose T and p. The total entropy differential is: dS = dS/dT * dT + dS/dp * dp. at constant entropy, dS=0, then dT/dp = dS/dp * dT/dS and we proceed as above.
So, it seems quite clear to me that the near surface atmosphere is isentropic.
The moist adiabatic lapse rate MALR differs from the DALR because it must account for the latent heat release as water condenses. This extra heat slows the rate of cooling of a rising air packet, so the MALR is less than the DALR – but it is still isentropic as it satisfies all the condition of my example 1, but the math is kinda nasty.
The adiabatic lapse rate need not be an equilibrium critereia, but instead a steady state condition if heat transfer in the near surface atmosphere is dominated by convection.
Let’s set moisture aside for the moment and set up a simple dry ideal gas atmosphere. Incoming radiation heats the planet surface. Air at the surface is heated and begins to rise, thus convection begins. Convection is attempting to return the atmospheric temperature gradient to zero, but as it rises it experiences an isentropic expansion, which causes it to cool (the reverse also occurs). If convection is the dominate mode of heat transfer (conduction negligible), it cannot drive the temperature gradient to zero, but only to the DALR.
Thus, I think the DALR is a steady state condition that arises because, in the near surface atmosphere, convection is the dominate heat transfer mode and it is a compressible fluid with a gravity imposed pressure gradient, hence convection is constrained by the DALR.
Lastly, if I assume the entire atmosphere is isentropic – all the way to the tippy top – then T1/T2 = P1/P2^0.4. If I solve for T2 letting T1 and P1 be the conditions at the tippy top of the atmosphere, I calculate an enormous value for T2, the surface temperature. Clearly the atmosphere cannot be isentropic all the way up. At some point it becomes non-isentropic.
I think the isentropic condition breaks down when convection ceases to be the dominant mode of heat transfer. As you rise, the atmosphere becomes less dense, convection less effective, until eventually heat transfer is dominated by radiative heat transfer. Radiative heat transfer is not constrained by the DALR and can drive the temperature gradient to zero. Hence the planet surface temperature is not enormous.
So, the non-radiative atmospheric thermal effect becomes an exercise in identifying at what point in the atmosphere does convection cease to dominate, which is also the point where the isentropic assumption breaks down. If that point is known (yes I know it won’t be a sharpt break between convection and radiation, but let’s keep it simple), then the equation above can be used to find the increase in surface temperature resulting from an isentropic near surface atmosphere.
Now, I could very well be wrong, or have made mistakes along the way (I do that sometimes – maybe even more than sometimes). But please don’t wave your hands at me and tell me to “check the meaning of entropy” – I find that frustrating. I think I’ve shown sufficient work…
Show me my mistake. Anybody. I won’t be offended.
T = temperature
p = pressure
V = olume
R = ideal gas constant
Cp = ideal gas heat capacity = 5/2R
g = gravitational constant
rho = density = 1/V
z = vertical spatical coordinate.
Very interesting but really irrelevant given that our atmosphere is not an ideal gas in a cylinder. Gas, ideal or otherwise, is a very poor heat conductor compared to Silver so perhaps your example above is not very good and enrtopy will still increase. It can be demonstrated by observation that convecting gas does rise, due to the density difference between the rising gas and that of the surroundings. If we consider real air it can be saturated with water vapour and rise to form clouds. The rising air will cool adiabatically at the SALR (4c/1000m rise) but will descend, as it must as dried air having the water vapour removed in the cloud formation, and warm at the DALR (9.8C/1000m) and end up with more heat than it started with. Rather like a Foehne Wind in a vertical loop. Katabatic winds are warmer at the bottom of their descent than the cold mountain start. The atmosphere is never in equilibrium because the planet rotates, there is a non-uniform surface and moving clouds which alter the solar energy falling on the surface. It is the lack of atmospheric equilibrium that gives us weather.
I am not being obtuse here but there is so much wrong with the GHG theory, cooling planet and rising GHG’s, lack of the predicted tropospheric heat, and the violation of 2nd law with this heat transfer from a cool troposphere to the warmer surface. If you could do that you really would have a PMM. So an alternative mechanism must be found to explain the BB heat anomaly, assuming that this is correct.
“In nature, the dry adiabatic lapse rate of air in the atmosphere is maintained because the system is differentially heated from below causing parcels of air to constantly move up and down.”
I live half-way up a mountain at 6100 feet. The valley below is at 4500 feet. The temperature difference is nearly always the dry lapse rate 8 degrees F, whether it is calm or windy. Only if it is raining or snowing will it be different. It then goes to the moist lapse rate. Most of the time it is sunny, heating the ground equally, both in my back yard and in the valley below. What maintains the lapse rate temperature difference?
In the real atmosphere the Earth surface is heated and radiation cools at top of atmosphere.
Convection is the major method of heat transfer.
Is convection always present?
The answer according to textbooks is no.
We can have the interesting situation where there is little or no convection, still air in other words.
This condition is called the Neutral Atmosphere.
This atmospheric condition is known as the neutral atmosphere and can be stable particularly at night.
See the near Neutral RESIDUAL LAYER page 31
What happens then?
Robert Brown says
“What maintains the adiabatic lapse rate is convection”
Nick Stokes and Joel Shore would agree.
So if convection is absent presumably the lapse rate disappears!
Well not in the real world.
If the air is dry, the lapse rate is at its maximum of g/Cp = – 9.8K/km
Climate Science define convection as an UNSTABLE vigorous vertical exchange of air. .
See bottom of page 13.
The stable condition (hydrostatic approximation) is used to derive the DALR. See page 12
This condition holds for still air and air parcels moving up and down at constant speed (no unbalanced force) will track the DALR.
These air parcels are assumed not to exchange heat with their surroundings.
On going up expansion work PdV is stored by the surroundings(temperature dropping by 9.8K/km)
At TOA there will be a loss of heat by radiation to space causing the down phase
On going down the surroundings do work on the parcel (PdV) (temperature increasing by 9.8K/km)
Stationary parcels will not change temperature.
These two idealised adiabatic processes (like the adiabatic stages in the Carnot Cycle) will result in the parcel returning to Earth with nearly the same temperature as leaving (the slight drop being accounted for by radiation at TOA).
http://www-as.harvard.edu/education/brasseur_jacob/ch2_brasseurjacob_Jan11.pdf
@Joules Verne: “This is obviously the case since it goes without a shadow of a doubt that a molecule of air in a higher layer has more gravitational energy than a molecule in a lower layer.” Does the fact that the air in a higher layer also has fewer molecules matter? That is, are we talking individual molecules or rather volumes of molecules here?
Two points:
1) If the atmosphere were isothermal, then a unit mass of atmosphere would have greater total energy (thermal + potential) the higher it is located and the further it is away from the surface heating source. That is not a stable situation.
2) The silver wire will transport heat from warmer region to the cooler region, but in so doing it short circuits the transport of heat by convection. So with the wire present, convection will be less, but the net transport of heat will remain the same.
This head post now makes me want to more fully question all the textbooks to which I’ve ever been exposed.
Consider this simplest and no simpler demonstration:
Robert Brown’s wire is U shaped. This sudden U-turn enables the wire to enter a 2nd thermal energy reservoir (another control volume that happens to be a gray colored one). The wire, in Robert’s example in his words, is: “adiabatically insulated except where it is in good thermal contact with the gas at the top and bottom of the container”.
Thus the wire is adiabatically insulated from the temperature field of the gas in the white colored area. Heat will indeed flow until the gray reservoir is in thermal equilibrium with the white reservoir. This just shows why there are no perfect insulators – Perpetuum Mobiles could be constructed in gas in a gravity field. This IS textbook stuff.
Why did Robert Brown have to go to the trouble of constructing a second gray reservoir with the U-turn? Robert Brown needed a second thermal body.
Trick’s view is Robert Brown should run this analysis again with the wire not leaving thermal contact w/white colored control volume gas and report back with only one thermal body or one energy reservoir or one thermodynamic system. Call it what you will.
Meaning Robert Brown is allowed only one heat reservoir to demonstrate his proposed isothermal gas column where the wire stays in thermal contact with the white colored gas everywhere – no U-turns as here to a 2nd thermal reservoir. Trick’s view is Robert will be unable to do so – the gas column will not be isothermal – there will be a temperature lapse rate.
Trick’s view remains that Robert Brown’s proper application 0th, 1st,2nd Thermo Laws will enable Robert Brown to eventually see the one thermodynamic system GHG-free gas column w/gravity is not isothermal in theory since Robert Brown is smart and the thermo grand masters are right.
NB: I am posting here b/c I have had a miserable head cold last few days and was looking for a way to pass the time. It has been interesting & fun to re-learn about thermo. I have to thank Robert Brown (and Willis) for a more enjoyable few days than I would have had otherwise .
steveta_uk says:
January 24, 2012 at 6:42 am
“I’m not sure that Dr Brown will convince them otherwise.”
Brown won’t convince him this way.
“But maybe Hans Jelbring can convince us all that the heat in figure 2 cannot rise up the silver bar because graivity is holding it down!”
The device in figure 2 doesn’t work because it’s a closed system and the work extracted will reduce the total energy of the column until eventually there’s no more energy to extract at which point the gas reaches a temperature of absolute zero and has presumably vanished from this universe being totally converted to kinetic energy in the extracted useful work. In the real world the gas will collapse to the surface as a liquid before it gets to absolute zero and this will shut off further extraction of energy because the cold side of the thermocouple no longer has any cold gas to cool it.
When doing thought experiments it is wise to think about what might have been assumed.
I see two assumptions above:
1. It does not matter what the density of the gas is. It will equally conduct heat at the bottom into silver wire, as the wire will be able to conduct its heat into the gas at the top, even though the density at the bottom and top is different, due to the gravitational effect on the gas.
2. The cross-section of the wire will stay the same, which means the ability of the wire to conduct the heat, which depends on its cross-section, is the same at the bottom and top.
The gravitational field will actually pull down a considerable part of the mass to the bottom, making it far wider at the bottom then the top (depending on the length of the wire and its tensile strength), deforming it more into a tear drop shape.
With your setup you may be able to change the lapse rate, but I doubt that you achieve an isothermal state in this way.
I find the analysis quite reasonable – but it is so idealized as to be useless. A more interesting thought experiment has a spherical planet heated by a remote star, rotating on an axis roughly normal to the line to the star, with an atmosphere of non-greenhouse gases. The equator would be warmer than the poles, so there would be Hadley-type circulation that would cool the equator and warm the poles. Would there then be a vertical thermal gradient? I think there would be, but I’m sure someone would like to argue to the contrary.
I see a lot of people talking about heat and temperature as if they’re the same thing here, then basing their arguments on that false premise.
Isn’t that, for the most part. why we have variable wind? And the reason there is not a pocket of ‘missing’ heat 800 meters below the oceans surface, as is commonly conjectured by Hansen, et al.
It clearly shows that without an already present differential of temperatures, gravity cannot create one. But the atmosphere is a different problem. It is heated at the bottom and it loses its heat in altitude. So the question is, what can impede the flow of heat from the warmer ground to the cooler “layer of emissions”. It seems to me that both greenhouse gases and gravity would enhance the lapse rate or impede the flow of heat between those 2 layers.
I often hear people say that nights would not be as warm if IRs were not coming from the atmosphere, but what about gas particles falling and hitting them all of the time?
This is funny : ) Almost everyone is right.
Take a single gas molecule and put it at the top of the tube. It has zero kinetic energy and zero temperature. Let it fall and just before it hits the bottom it will have a lot of kinetic energy and heat. That is the lapse rate and it isn’t in equilibrium by definition.
Now place the atom at the bottom of the tube, it is now in isothermic equilibrium, its kinetic energy and temperature is zero.
If we let the atom bounce up and down in the tube, and don’t allow any energy to be extracted, it will stay in perpetual motion (and gravity will be continuously accelerating it) and will have a lapse rate (as long as the lapse rate isn’t measured).
Wayne2 says:
January 24, 2012 at 7:39 am
“@Joules Verne: “This is obviously the case since it goes without a shadow of a doubt that a molecule of air in a higher layer has more gravitational energy than a molecule in a lower layer.” Does the fact that the air in a higher layer also has fewer molecules matter? That is, are we talking individual molecules or rather volumes of molecules here?”
No. The fewer molecules must have sufficient gravitational energy that, if it were converted to kinetic energy, would be able to raise the temperature of a larger number of molecules in the lower layer the temperature in that lower layer. It MUST be isoenergetic to satisfy the second law. It need not be isothermal. In politics they say to follow the money to arrive at the truth. In physics you want to follow the
jewelsjoules.“Follow the air column up above the troposphere and the lapse rate fails to be observed in the stratosphere, precisely where vertical convection stops dominating heat transport. The EEJ assertion, that the dry adiabatic lapse rate alone explains the bulk of so-called “greenhouse warming” of the atmosphere as a stable feature of a bulk equilibrium gas, is incorrect”.
So your wonderful assertion, is that the radiative forcing of Co2, occur after its entry into the thermostats of the tropopause, and that extra radiative forcing, causes that missing hot spot, increasing the temperature back through the stratosphere and down again through the thermostat of the tropopause.
Been there, done that.