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.
A physicist says:
January 24, 2012 at 8:48 am
The answer is easy. By conduction.
Convection will shut down quite quickly as the heat transported upwards by convection reduces the environmental lapse rate. Once the lapse rate is below the moist adiabatic lapse rate even cloud formation cannot drive convection anymore. Oh, except you removed all the H2O. Therefore once the lapse rate is below the dry adiabatic lapse rate all convection will stop. Without GHG to cool the upper atmosphere there is no way to regain a lapse rate suitable to drive convection. All vertical motion will cease. Conduction will become the dominant mode of heat transport. That too will stop when the atmosphere becomes isothermal.
Joe Born says:
January 24, 2012 at 8:21 am
“…If an ideal monatomic gas subjected to gravity in a thermally isolated container consists of only a single molecule, its kinetic energy K–and thus the mean translational kinetic energy–at any altitude z is given by K = mg(z_max -z), where m is molecular mass, g is the acceleration of gravity, and mgz_max is the total (kinetic + potential) energy of the gas. This is true no matter how long you’ve allowed the gas to “equilibrate.” In other words, temperature depends on altitude at equilibrium: there’s a non-zero temperature lapse rate.”
As I pointed out in the other thread, this is not a thermal system at all – or even a gas! – unless the atom is allowed to thermalise with the walls of the container. In which case the temperature (as measured by its average kinetic energy as it passes through a given level) is independent of height.
I have now read the Velasco et al article, and it agrees with what I said: in either the microcanonic (totally isolated) ensemble (with a reasonable number of particles in the gas) or the canonic ensemble (in thermal equilibrium with the surface or walls, irrespective of the number of particles), the gas is isothermal.
You are trying to make a haystack out of the negligible point that, for a tiny number of isolated particles, the statistics aren’t precisely the same as for the usual smooth distribution – they’re “lumpy”. However, even in this extreme case, the temperature at equilibrium will still be the same throughout the entire height, in the crucial sense that no net work could be extracted from the gas by connecting different levels, by any means whatsoever. The “lapse rate” is still zero.
Note by the way that any thermometer capable of measuring the the temperature of two levels of any such system alternately is itself either extracting work from any measured temperature difference, or has to have work done on it to obviate this happening.
All these tree people making a joke of climate science. You have Mann with his cherry picked tree proxies and
Ned Nikolov, Ph.D.
Ph.D. in forestry making a joke of Thermodynamics.
Any way the 600mb temp is going cold right now.
http://discover.itsc.uah.edu/amsutemps/execute.csh?amsutemps
I’ll take a stab to see if I can answer it correctly.
The short answer is it will not maintain the same kinetic energy, the same temperature. This has never been postulated. What’s stated is that eventually the average kinetic energy (the temperature) of all molecules will be the same throughout the gas. That’s not necessarily the same kinetic energy as any individual molecule will have at any given time – it’s the average that’s the temperature. Individual molecules can be hotter or cooler than this average.
“Will heat flow in the silver wire forever?”
Will the molecules continue to jostle forever?
If we never take any energy away from the system, what is the difference between jostling in the gas and flow through the wire?
The questions that remain are: Is isothermal state the lowest energy state for a compressible gas in a gravity field? And on what measure is there a stable (mininum energy) outcome if we are asked to accept that molecules at higher altitude have more total energy than molecules at low altitude?
Just asking.
Ged says:
This is a very subtle point, but one worth understanding. “Temperature” is a measure of the average thermal energy at some location. As implausible as it might seem at first, the average kinetic energy of the molecules that make it 17 km will be the same as the average KE of the molecules at the bottom. You see, only a very few molecules will make it 17 km high. The rest are pulled back by gravity before they get that high, so they never get counted. The self-selected molecules that DO make it 17 km high are the ones that had LOTS of KE to start with. Sure they lose a bunch on the way up, but that ends up leaving this subset 17 km high with the same average KE as the ENTIRE set had at ground level. (The same is true at any other level up thru the atmosphere).
This has been discusses MANY times in related threads recently.
Willis Eschenbach wrote:
Just don’t expect your belief, that gravity can do continuous unending work forever and ever amen, to be widely shared in the scientific community …
So Willis, when will we start flying off the planet??? When will the pressure on my feet from standing in one place stop?? When will the oceans boil from lack of pressure? Oh yeah, gravity is apparently an unending source of energy that counteracts centrifugal force. If not unending, we haven’t yet measured its reduction.
Alan Millar,
“The Sun will die and become a cold white dwarf with no solar wind to blow any atmosphere away.”
This is an assumption without much empiraical proof.
MDR says:
January 24, 2012 at 9:32 am
“…Now, we just showed that internal energy decreases with height, as explained above. Since internal energy [of an ideal gas] is directly proportional to temperature, this must mean that temperature also decreases with height…”
The correct statement would be ” that ALL ELSE BEING EQUAL temperature also decreases with height”. But the internal energy of an ideal gas is also directly proportional to density, which decreases with height. So all else is not equal. The argument fails.
In other words the decrease of internal energy with height manifests itself as decreasing density, not as decreasing temperature.
(In case someone does not see why internal energy is proportional to density it is because if stuff has energy, and you have more of the same kind of energetic stuff, you must have more energy.)
loads of politics going on here. Sceptical AGW blogs still trying to prove that planets are at temperatures dictated by their atmospheric gas composition rather than their distance from the sun. The Greenhouse effect still rules concensus thinking be you an aye or a nay for AGW.
In 1964, Manabe and Strickler published “Thermal Equilibrium of the Atmosphere with a Convective Adjustment”:
In a nutshell, this adjustment eliminates the possibility that a greater potential gradient might compensate for increased transport resistance by GHGs. Instead, only flux changes are allowed which can be nullified by positive feedback. Et voilà, CAGW!
@Tim Folkerts,
Maybe that is the case. Maybe you are totally right. But I still find that hard to believe form observations.
As you move up in altitude, the temperature, the measurable, observable temperature drops steadily. In fact, form my calculation I was less than 3x off just looking at KE turned into PE from the actual temperature at that height.
So the problem is, the average thermal energy you are talking about includes -all- altitudes, and this is a value that cannot change unless absolute energy is given or take from the system. But it does not -follow- the temperature change that occurs as any molecules moves from low to high against gravity. And that’s what I’m talking about. That’s what this whole discussion is actually about.
Yes, the molecules that make it 17 km up had more KE when they started than those that don’t (well, to a degree, as it’s a random walk). But once they MAKE IT 17 km up, they’ve lost a great deal of kinetic energy, temperature, into potential energy, versus what they had at sea level, as energy most be conserved. Correct?
In that way, gravity does NOT DO WORK, but gravity TRANSFORMS ENERGY from one type (kinetic) to another (potential). This must drop the temperature of the molecules in question, since temperature is only a measure of the KINETIC ENERGY side of the TOTAL ENERGY equation.
So, if air rises, it must lose temperature as a consequence of moving against gravity. Average kinetic energy of the entire system means little, as that will not change unless energy is removed from the system or inputted to the system. Average kinetic energy of the cohort of air that moves from low to high, now that does mean something, and that is going to be changed into potential energy.
So then, gravity seems to be able to maintain a temperature gradient -in this way-, as it seems to me, and maybe I’m wrong and your way of looking at it is right. But, what predictions would we make for the real world from what I’m seeing? For starters, we’d expect that if the average energy of the entire system increased, so there was more kinetic energy at the surface, that the atmosphere would “puff up” from the Earth and it’s edge would get higher due to more KE being available for transforming into PE. And this is -exactly what we see- in the real world. And the converse is true. Which is what we see with Martian poles in the winter, or Pluto’s entire atmosphere whenever it gets far enough away from the Sun.
“Why is it that you want to fight over physics that you can actually see with IR eyes? Save your energy for useful things, like arguing about the magnitude of the GHE, the sensitivity of it to changes in CO_2 concentration, the sign and nature of climate feedback or albedo modulation or the complex effects of atmospheric convection on local heating or cooling rates, or the ocean’s effect. The IR spectra render arguing about GH warming per se moot.”
Thank you Robert.
I sometimes wonder why skeptics waste their time an energy fighting against working science.
The real question is sensitivity. Think of all the energy and time devoted here on WUWT to clearly false theories. Imagine if that effort were put to better purposes. Like the surface stations project expanded on a global basis.
sad to see so much human energy, curiousity and intelligence wasted on crap like this and N&Z
Actually Willis, heat flowing through the silver wire forever doesn’t mean it’s impossible, as heat always flows forever in any system above absolute zero. Take any object and an arbitrary plane that defines it. The two parts will never be in exact thermal equilibrium because atomic collisions are discrete, so half the time one side is hotter and half the time it is colder. Thus heat flows back and forth across the boundary – forever. That doesn’t mean the existance of an object above absolute zero is impossible.
Or, take the case of the Maxwell Demon I constructed in my first comment in the thread. It’s made of nothing but pine and mylar but should maintain an imbalance of temperatures between two objects forever, because photon transmission and reflection can be made to be asymmetrical or unidirectional by having mirrors moving to create a path that is only valid in one direction. The concept is like the way volleyball players set the ball to each other. Given the timing of their motions, a volleyball can’t travel the reverse path and find the right hands in the right places at the right times. You can do the same with light if your mirrors are moving very fast.
So if you can construct a system that never reaches the isothermal state, and attach another path for heat flow, the heat will travel from hot to cold forever. This changes the stable temperature difference between the two objects, turning the Demon into a less efficient Demon. You can’t extract anything from the permanent heat flow without having the energy dissipate.
As for thoughts on the problems of heat flow through a long silver wire, the equations we use have never needed to include any gravity component because it’s unimportant to most purposes. It’s only fairly recently that physicists got irked enough to tweak heat flow equations so the heat couldn’t travel at infinite velocity, exceeding the speed of light.
Equations for electricity, from Ohms law to Maxwell’s equations, likewise completely neglect gravity, implying that I can connect a copper wire to a black hole with a potential of -24 VDC and have electrons pour right up out of the gravity well. Or I could show that an atmosphere of electrons and protons (hydrogen) wouldn’t have more pressure at the bottom than the top because Ohm’s law or some other handy formula says the electrons would be evenly distributed, as gravity was never factored in to the completely accepted electrical formulas we use every day.
I shall repeat in different words the argument i provided at
son of mulder says:
January 24, 2012 at 9:59 am
which no one challenged (or read). If I’m wrong please challenge.
Heat cannot flow up the silver conductor unless more heat is pushed in at the base. Heat is energy and hence by Einstein mass.
If you consider all the heat in the silver conductor as a distribution of mass then that mass has a center of gravity in the gravity field. If heat were to flow from the base to the top (without input of new heat at the base) then there would be reducing heat (mass) a the bottom and increasing heat (mass) at the top. Overall the centre of gravity of that heat (mass) would rise. ie work would have to be done against gravity. Put another way a force against gravity would have to be applied. If no new energy is being pumped into the base then the law of conservation of energy means that heat cannot be conducted up in a gravity field and so the top will be colder than the base and remain so ie no force can be applied.
The same logic can be applied to the atmosphere. You just have to ensure no new heat is applied and no heat is removed. If no new heat is applied and energy is allowed to be radiated away then eventually all temeprature would be equalised only at absolute zero.
Willis,
Seems like folks here want to argue that the laws of thermo are not settled science.
same with the laws of radiative transfer
Let me toss my in my idea to show what is wrong with this thought experiment. Take the earth system and remove gravity. What happens to the GHG’s, the water, loose objects, the adiabatic lapse rate…
Take gravity away from the thought experiment and what happens. Nothing except a very slight redistribution of mass as the pressure equalizes through the column?
For a volume of gas to have a thermal gradient requires a heat source and heat sink at each end of the gradient, so that heat is transported along the gradient from source to sink. Remove the heat source and the heat sinks from the gas, and the thermal gradient will disappear.
In the case of Earth’s atmosphere, the heat source is the heating of the surface of the earth by the sun, and the heat sink is radiative to Deep Space. So it is not appropriate to dismiss Robert’s because it’s not the atmosphere – Robert is just showing the mechanism in the paper being criticised is not correct.
Peter Spear, (IMHO) your scenario is correct.
The resulting no-GHG Earth would have nearly isothermal, hence stratified atmosphere, for the physical reason that every thermal would carry heat into the atmosphere that could never be radiated away, making it ever-harder for thermals to rise on subsequent days.
The no-GHG weather would be freezing no-wind nights followed by still cold days. As (relatively) warmer tropical air slowly circulated (colder poles), first Earth’s polar oceans would freeze, then the mid-latitudes, then even the equatorial oceans.
Some folks we’ve seen this scenario even here on earth, way back when the sun was cooler and GHG’s were scarcer, a world of low GHG’s and frozen seas … This hypothesis is called Snowball Earth.
Elevator Summary: GHG’s prevent Snowball Earth.
kuhnkat says: “Oh yeah, gravity is apparently an unending source of energy that counteracts centrifugal force.”
Repeat after me: Force is not energy; energy is not force.
Gravity can and does provide a continuing force. That does not mean that it is providing any continuing energy. Gravity only does work when there is a net movement inward. Since the atmosphere is not continually falling, gravity is not doing work.
(NOTE, on the gas planets, there is no solid surface to stop contraction, so there the planets are indeed contracting and generating continued thermal energy of the sort many people seem to think exists on earth. This is also how protostars warm as they collapse inward.)
steven mosher says:
January 24, 2012 at 2:38 pm
100% correct Sir.
Sensativity is the question and there have been no answers with acceptable confidence from a model yet. The present parameters do not hindcast with enough accuracy to forcast with any confidence.
Physiscs, as we presently know it, is physics.
Unless a new law of thermodynamics is found, Dr. Robert Brown is 100% correct.
Nick Shaw says:
January 24, 2012 at 9:52 am
Space has no temperature. To have a temperature there must be kinetic energy which means that there must be mass.
Things in space get cold because they radiate heat away. Look at photos (or drawings) of the space station – look for the radiators. These are positioned so that the Sun does not shine on them.
In the case of an atmosphere without IR radiators, it will have no way to cool itself.
Do I have this correct : the temperature is dependent on the kinetic energy of individual molecules(?)
When I throw a single ball upwards it’s velocity decreases with height. Do gas particle’s velocities not decrease as they go higher and higher,?
Can someone explain why, if their velocities do not decrease, there fewer of them at height.
Sorry if I am being dim!
While I agree with the underlying point, I’m not sure why the wire would necessarily violate the laws of thermodynamics if it continuously transferred heat. Under normal circumstances, it would radiate some of this energy away and otherwise be an imperfect conductor. If, however, we’re assuming a closed system with a perfect conductor surrounded by a perfect insulator, why would any energy be lost?
To put another way, assume I have a wheel with a frictionless axle at rest in a vacuum. If I spin it, it will spin endlessly. The conclusion that it will have perpetual motion doesn’t violate thermodynamics; the assumption that there is no friction does. Likewise, the wire would not violate any physical laws by endlessly transferring heat; those laws were broken by the assumption of a closed system with a perfect conductor/insulator.
I know I’m disputing people far above my pay-grade, so I’m assuming that I’m wrong in this. I’m just curious as to why.
What the recent theories regarding gravity induced temperature gradient are really saying is that the ideal gas law as commonly stated is incomplete. A factor is left out because in most terrestrial situations it is irrelevantly small.
They are implying the ideal gas law should be stated as:
PV = NkT +(delta PEg)
where PEg = the change in gravitational potential energy.
If you radically increase the gravitational potential energy of a mass of gas, you have changed the total energy in the system unless you give up an equivalent amount of energy in the form of temperature.
Larry