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.
As a side note everyone should realize … no matter which side was right it would not change anything relative to the GHE. If there were no radiating gases in a world like this the only place the system could radiate energy is the surface. So, in one case the atmospheric temperature is constant and in the other it cools as you go up. In neither case does it change the surface temperature because the energy in MUST equal the energy out.
The only way to change this is to add radiating gases (and I don’t care if you think N2 and O2 are radiating), and you are left once again with a green house effect.
Robert Brown, your posts so far have served excellently (IMHO) to debunk “gravito-thermal” theories, in large part because your posts have scrupulously adhered to the main principle of The Debunking Handbook (page 5, available free from Richard Dawkins’ website Reason and Science) “Fill the gap with an alternative explanation” … that explanation being:
Nonskeptical Elevator Summary: “Solar heating of the Earth’s surface + GHG heat radiation sustains the nonisothermal / nonequilibrium profile of the Earth’s atmosphere.”
But I think you have to be ready for future skeptical articles that suggest an alternative, more sophisticated non-GHG theory:
Alternative (skeptical) Elevator Summary: “Solar heating of the Earth’s surface + day-night surface temperature cycling sustains the nonisothermal / nonequilibrium profile of the Earth’s atmosphere.”
To my mind this second, skeptical, non-GHG theory cannot be debunked primae facie, and so I will bet anyone a donut that such theories will appear as WUWT posts in the coming weeks and months.
Joe Born says:
January 25, 2012 at 5:34 am
“I would have thought that differentiating that expression for average molecular kinetic energy with respect to altitude would indeed yield a quantity that is proportional to lapse rate. And the result does indeed differ from zero.”
You would have thought erroneously – as has already been explained to you repeatedly. You are wilfully taking the extreme and irrelevant sub-thermodynamic case of a minuscule total number of isolated particles – in which regime the macroscopic temperature is increasingly ill-defined and no longer simply proportional to the kinetic energy per particle – and torturing it to produce something that looks a bit like a macroscopic lapse rate, but is really nothing more than a mathematical artefact of absolutely no significance. There is and can be no real lapse rate at all – if there were it would violate the second law.
Just to be clear – the adiabatic lapse rate which is caused by gravity does not make the surface any warmer than it would be otherwise. All that happens, as at least one other commenter noted, is that it makes upper layers colder than than they would be otherwise. Nothing happens except that kinetic energy becomes gravitational potential energy with increasing altitude. The kinetic energy of the surface atmosphere is the same regardless. Nikolov et al are still wrong. It just needs to be made clear they aren’t wrong about gravity creating a temperature gradient. They’re just wrong about gravity raising the temperature anywhere in the column. It doesn’t. All it does is change the way total energy at any given altitude is apportioned between kinetic and potential.
DavidB says:
January 25, 2012 at 6:45 am
Nice. Now consider two equally sized buckets of water. One on the top floor and one on the lower floor. Both buckets are the same temperature. Do they each have an equal amount of energy? No. The bucket on the upper floor has more gravitational potential energy. That energy had to come from somewhere. It came from whatever force was used to lift that water to the higher elevation. If you carried it then that extra energy came from the food you ate.
There’s no such thing as a free lunch. Follow the joules. One MUST account for the source of gravitational potential energy! In reality, despite pleadings to the contrary from sources who should know better, gravity creates the dry adiabatic lapse rate. The gravitational potential energy in the higher layers came from its store of kinetic energy. The Hamiltonian of all horizontal layers is constant but the kinetic energy apart from the potential energy is not.
What an interesting perpetual motion machine. We need a solid state physicist. Someone who knows the Debye theory of heat in solids. Someone who thinks that heat is transferred by phonons, and that phonons have a characteristic momentum (mass and velocity). So if a phonon moves uphill against gravity it must loose momentum – and energy because it is, in effect, an upward moving mass. I suggest that the wire too will have an ‘adiabatic lapse rate’ (temperature gradient in a gravity field) because of the nature of heat – and heat transfer – in solids.
Solid state physicists are sensitive creatures and would probably not post here for fear of getting a rhetorical custard pie in the face (even if they made a correct point).
Joules Verne didn’t answer my questions. Does he think the water at the bottom of the pipe is warmer than at the top? If not (and his ‘buckets’ example suggests not) when why does this not follow from Jelbring’s theory? What is the relevant difference between air and water?
DavidB says:
January 25, 2012 at 8:44 am Why on earth would you think that a 30ft column of incompressible water would react the same as a Miles high atmosphere of compressible GAS???
@Robert Brown
I’m not sure what you’re going on about in figure 2. The silver wire is a proxy for thermal conduction in the gas. Thermal conduction is accomplished via collisions. In the absence of gravity there is no preferential direction for collision energy. In a gravity field there is a preference. A molecule travelling upward loses thermal energy as it ascends and gains thermal energy as it descends. A molecule getting whacked from above gets hit harder than one getting whacked from below if everything else is equal.
Let’s take the situation of me throwing a rotten tomato at a Duke physics professor on his lecturn. I might be throwing it from the balcony or I might be throwing it from the orchestra pit. In either case the energy I can add to the tomato with my arm is the same. If I have a choice I’m going to choose to throw from the balcony of course for the obvious reason that gravity is an aid in one direction and a restriction in the other. Thermal conductivity in the gas works the same way. Kinetic energy is preferentially sequestered at the bottom of the gravity well and gravitational potential energy is sequestered at the top.
What part of that do you not understand?
“They’re just wrong about gravity raising the temperature anywhere in the column. It doesn’t. All it does is change the way total energy at any given altitude is apportioned between kinetic and potential.”
I thought N & Z said that pressure raises the temperature in the denser gases at the bottom of a column when an external energy source is added, not that gravity did it directly.
Gravity just places more energy at the bottom of the column by pulling molecules downward to creater greater density and pressure at the bottom. In the process it does apportion some of the available energy as you say.
I think Jelbring might be suggesting a separate purely gravitationally induced temperature gradient but I’m not convinced that it is significant as yet.
“Does he think the water at the bottom of the pipe is warmer than at the top? ”
As regards water in a column the difference is that water is incompressible and so upward convection dominates and the warm water rises to the top.
The compressibility of a gas results in the higher temperature being at the bottom at all times.
That accounts for the different temperature profiles in oceans and air despite both being affected by pressure.
“Joules Verne” says:
“What part of that do you not understand?”
Is “Joules Verne” another screen name for Dave Springer?
IanH says:
January 25, 2012 at 8:28 am
“What an interesting perpetual motion machine. We need a solid state physicist. Someone who knows the Debye theory of heat in solids. Someone who thinks that heat is transferred by phonons, and that phonons have a characteristic momentum (mass and velocity). So if a phonon moves uphill against gravity it must loose momentum – and energy because it is, in effect, an upward moving mass. I suggest that the wire too will have an ‘adiabatic lapse rate’ (temperature gradient in a gravity field) because of the nature of heat – and heat transfer – in solids.”
It wouldn’t matter if it did. So long as the adiabatic lapse rate is not fixed to the same value for every possible material (which we know it isn’t), there will be an exploitable temperature difference somewhere in the system.
Look, if phonons confuse you, forget the solid wire. Just divide the container with a vertical insulating partition. Put a light gas (say, helium) on one side, and a heavy gas (say, argon) on the other. The lapse rate will be less in the former than the latter (in this case by the ratio of molecular weights). Make a short horizontal connection at the top and the bottom, and presto there’s your perpetual motion machine.
DavidB says:
January 25, 2012 at 8:44 am
“Joules Verne didn’t answer my questions. Does he think the water at the bottom of the pipe is warmer than at the top?”
It will be warmer at the bottom to the degree that water is compressible. In order for there to be a temperature gradient established there must also be a density gradient. In the water column we have no practically detectable density gradient and hence no practically detectable temperature gradient. If it were water vapor you betcha there’d be a temperature difference we could measure even at 30 feet. It would be about 0.1C warmer at the bottom of the pipe.
The whole point is that if you could achieve the totally impossible conditions that Dr Brown proposes for his thought experiment, you probably would have a perpetual motion machine. After all to get the conditions proposed would take magic in the first place.
It is an imaginary concept, so is Alice in Wonderland.
Stephen Goddard was expounding this ‘temperature is due to pressure’ theory some time ago on WUWT. Where is he, to defend his theory??
http://wattsupwiththat.com/2010/05/06/hyperventilating-on-venus/
Come on Stephen, what say you now?
DavidB says:
January 25, 2012 at 8:44 am
“What is the relevant difference between air and water?”
Air is compressible and water is not. Air is seen under natural circumstances to flow up hill water has not. Would you rather have a hot water bottle keeping you warm at night or the same volume of warm air trying to do the same thing? There are lots of relevant differences between air and water?
Joules Verne says:
January 25, 2012 at 4:24 am
This is exasperating Professor Brown. Gravity CAN NOT maintain an energy gradient. That would be a violation of 2LoT. We KNOW for a fact that gravity creates a potential energy gradient in an atmosphere. Molecules at higher altitudes have more gravitational potential energy than those at lower altitudes. Therefore, to satisfy 2LoT, there must exist an equal and opposite energy gradient to make up for the gravitational energy gradient.
Well, I still haven’t worked out what any of you are talking about.., but isn’t the opposite and equal energy gradient to make up for the gravitational energy gradient, pressure? That wot gives buoyancy. More noticeable in the ocean though, but applied to the fluid gas volume of air above us.
http://www.usatoday.com/tech/columnist/aprilholladay/2005-02-18-wonderquest_x.htm
“Archimedes’ principle applies to air as well as water: a force equal to the weight of the air displaced buoys up an object surrounded by air.”
“We scarcely think of air at all; it’s just so nebulous and pervasive. But our atmosphere has considerable mass because it towers at least 50 miles (80 km) above Earth’s surface into space. Air provides a buoyant push just as water does. A column of air that extends from sea level to space with a tiny postage-stamp size cross sectional area — one square inch — weighs almost 15 pounds and, consequently, exerts a pressure of 15 pounds per square inch on the bottom of the column.”
“Imagine an air parcel immersed in the ocean of air that is our atmosphere, as shown in the figure. The surrounding air presses in on the air parcel from all directions but the pressure along the sides of the parcel are equal and opposite and thus cancel.
The pressure on the top of the parcel is less than the pressure at the bottom (since pressure decreases with altitude). That pressure difference is the buoyant force — the force that pushes up on the air parcel.
The air parcel, however, has mass and therefore weight. Gravity pulls it down. If gravity’s pull is less than the buoyant upward push, the parcel rises. If gravity’s pull is greater than the buoyant push, it falls.
If the parcel contains light hot air from a flame, then gravity’s pull is less than the buoyant push. That’s why fire goes up.”
Smokey says:
January 25, 2012 at 9:10 am
“Is “Joules Verne” another screen name for Dave Springer?”
Presumably the laws of physics would remain the same either way so I’m going to plead the fifth on that question.
Wayne,
The Venus atmosphere was discussed at length in a couple of threads at Science of Doom. I did find a reference to the solar radiative flux at the surface when the sun is above the horizon on Venus and the peak is about 36 W/m². That was sufficient for Venera 9 (I think) to be able to make visible light photographs of the surface and transmit them back to Earth. I also ran the radiative transfer calculations for the atmosphere and the lapse rate at Spectralcalc and that flux is indeed sufficient to maintain the lapse rate against radiation and conduction. If the atmosphere of Venus became truly opaque to incoming solar radiation at some altitude above the surface, the atmosphere below that point would be isothermal assuming no heat input to the surface from the core of the planet.
Stephen Wilde and mkelly,
Nope. Water can be compressed, just much less than air. From Wikipedia:
But not no decrease in volume. That means a water column has a potential temperature just like air.
Paul Birch says
“Look, if phonons confuse you, forget the solid wire. Just divide the container with a vertical insulating partition. Put a light gas (say, helium) on one side, and a heavy gas (say, argon) on the other. The lapse rate will be less in the former than the latter (in this case by the ratio of molecular weights). Make a short horizontal connection at the top and the bottom, and presto there’s your perpetual motion machine.”
Does it not give you pause to think why none of these seemingly very simple methods have been tried?
Robert Brown towards the end of the previous thread said that he wished he could demonstrate his conjecture by an experiment.
He is thinking along the lines of a centrifuge.
I wonder about something…..
Everyone assumes an ideal gas as a starting point in these treatments , based I presume, on the correct assumption that nitrogen, oxygen, and even CO2 closely approximate the definition of an ideal gas, duh. Here’s the thing though. This is an argument about the gas, in an of itself, i.e. it ignores the gas’ context.
Here’s my dilemma. One of the premises made regarding an ideal gas is that there are no inter-molecular collisions in an ideal gas and this is done so as not to introduce a ‘wall bias’ where molecules near a wall experience (inter-molecular) vector forces that tend to pull them away from the wall. The problem is that a gas constrained by gravity is unavoidably subjected to a ‘gravity bias’, irrespective of how ideal it is, as a function of its container.
With gravity as your container, any gas no matter how ideal, is being subjected to a force (gravity) that biases collisions and (i would think) completely invalidates any assumption of ideal behavior, doesn’t it?
The centrifuged cylinder of gas is a great example that someone brought up.
As the centrifuge spins up a pressure gradient is set up. What does the ideal gas law demand will happen to temperature as the pressure increases at one end of the cylinder and decreases at the other?
As the centrifuge spins down the gas at one end of the cylinder expands and at the other end it contracts. What does the ideal gas law demand must happen in this case?
It’s a given that no one here is willing to cast aside the ideal gas law we should all agree that spinning up and spinning down the centrifuge will create at least a temporary temperature/pressure gradient predicted by the ideal gas law.
So the question boils down to what happens when the centrifuge is left running indefinitely. Does the column temperature equalize? I say no because for conduction to equalize the temperature there has to be equal freedom of motion in any direction and this clearly isn’t the case. Freedom of motion is restricted going against the centrifugal force and is aided going in the direction of the force. The temperature will not equalize if we discount conduction through the solid walls of the cylinder. If we stuck a silver wire down the center of our cylinder as in Brown’s figure 2 the temperature would equalize because the thermal conductivity of the silver wire has no preferential direction due to centrifugal force.
I’m trying as many ways as I can think of to explain what I know must happen and how. So now I’m trying it via illustrating that thermal conductivity coefficient changes in a compressible fluid with greater conductivity going with gravity and lesser conductivity going against gravity. This does not happen with incompressible fluids.
Does that help? It should certainly help to explain to Brown why his silver wire is not a valid proxy for thermal conduction in a compressible gas in a gravity field.
Robert Brown,
The way to think about the difference between the stratosphere and the troposphere is to use the slab gray atmosphere toy model ( see 7.3.2 here, for example. Petty goes into more depth). In the troposphere, the atmosphere is more transparent to SW radiation and less transparent to LW radiation so the surface temperature is warmer than the slab. The opposite is true in the stratosphere. The stratosphere is less transparent to SW radiation because of absorption of UV by oxygen and ozone and more transparent to LW radiation. That makes the ‘surface’, i.e. the tropopause, colder than the slab.
Paul Birch says
So long as the adiabatic lapse rate is not fixed to the same value for every possible material (which we know it isn’t), there will be an exploitable temperature difference somewhere in the system.
AMEN, BROTHER!