Guest Post by Willis Eschenbach
Since at least the days of Da Vinci, people have been fascinated by perpetual motion machines. One such “perpetuum mobile” designed around the time of the civil war is shown below. It wasn’t until the development of the science of thermodynamics that it could be proven that all such mechanisms are impossible. For such machines to work, they’d have to create energy, and energy cannot be either created or destroyed, only transformed.
I bring this up for a curious reason. I was reading the Jelbring hypothesis this afternoon, which claims that greenhouse gases (GHGs) are not the cause of the warming of the earth above the theoretical temperature it would have without an atmosphere. Jelbring’s hypothesis is one of several “gravito-thermal” theories which say the heating of the planet comes from gravity rather than (or in some theories in addition to) the greenhouse effect. His thought experiment is a planet with an atmosphere. The planet is isolated from the universe by an impervious thermally insulating shell that completely surrounds it, and which prevents any energy exchange with the universe outside. Inside the shell, Jelbring says that gravity makes the upper atmosphere colder and the lower atmosphere warmer. Back around 2004, I had a long discussion on the “climateskeptics” mailing list with Hans Jelbring. I said then that his theory was nothing but a perpetual motion machine, but at the time I didn’t understand why his theory was wrong. Now I do.
Dr. Robert Brown has an fascinating post on WUWT called “Earth’s baseline black-body model – a damn hard problem“. On that thread, I had said that I thought that if there was air in a tall container in a gravity field, the temperature of the air would be highest at the bottom, and lowest at the top. I said that I thought it would follow the “dry adiabatic lapse rate”, the rate at which the temperature of dry air drops with altitude in the earth’s atmosphere.
Dr. Brown said no. He said that at equilibrium, a tall container of air in a gravity field would be the same temperature everywhere—in other words, isothermal.
I couldn’t understand why. I asked Dr. Brown the following question:
Thanks, Robert, With great trepidation, I must disagree with you.
Consider a gas in a kilometre-tall sealed container. You say it will have no lapse rate, so suppose (per your assumption) that it starts out at an even temperature top to bottom.
Now, consider a collision between two of the gas molecules that knocks one molecule straight upwards, and the other straight downwards. The molecule going downwards will accelerate due to gravity, while the one going upwards will slow due to gravity. So the upper one will have less kinetic energy, and the lower one will have more kinetic energy.
After a million such collisions, are you really claiming that the average kinetic energy of the molecules at the top and the bottom of the tall container are going to be the same?
I say no. I say after a million collisions the molecules will sort themselves so that the TOTAL energy at the top and bottom of the container will be the same. In other words, it is the action of gravity on the molecules themselves that creates the lapse rate.
Dr. Brown gave an answer that I couldn’t wrap my head around, and he recommended that I study the excellent paper of Caballero for further insight. Caballero discusses the question in Section 2.17. Thanks to Dr. Browns answer plus Caballero, I finally got the answer to my question. I wrote to Dr. Brown on his thread as follows:
Dr. Brown, thank you so much. After following your suggestion and after much beating of my head against Caballero, I finally got it.
At equilibrium, as you stated, the temperature is indeed uniform. I was totally wrong to state it followed the dry adiabatic lapse rate.
I had asked the following question:
Now, consider a collision between two of the gas molecules that knocks one molecule straight upwards, and the other straight downwards. The molecule going downwards will accelerate due to gravity, while the one going upwards will slow due to gravity. So the upper one will have less kinetic energy, and the lower one will have more kinetic energy.
After a million such collisions, are you really claiming that the average kinetic energy of the molecules at the top and the bottom of the tall container are going to be the same?
What I failed to consider is that there are fewer molecules at altitude because the pressure is lower. When the temperature is uniform from top to bottom, the individual molecules at the top have more total energy (KE + PE) than those at the bottom. I said that led to an uneven distribution in the total energy.
But by exactly the same measure, there are fewer molecules at the top than at the bottom. As a result, the isothermal situation does in fact have the energy evenly distributed. More total energy per molecules times fewer molecules at the top exactly equals less energy per molecule times more molecules at the bottom. Very neat.
Finally, before I posted my reply, Dr. Brown had answered a second time and I hadn’t seen it. His answer follows a very different (and interesting) logical argument to arrive at the same answer. He said in part:
Imagine a plane surface in the gas. In a thin slice of the gas right above the surface, the molecules have some temperature. Right below it, they have some other temperature. Let’s imagine the gas to be monoatomic (no loss of generality) and ideal (ditto). In each layer, the gravitational potential energy is constant. Bear in mind that only changes in potential energy are associated with changes in kinetic energy (work energy theorem), and that temperature only describes the average internal kinetic energy in the gas.
Here’s the tricky part. In equilibrium, the density of the upper and lower layers, while not equal, cannot vary. Right? Which means that however many molecules move from the lower slice to the upper slice, exactly the same number of molecules must move from the upper slice to the lower slice. They have to have exactly the same velocity distribution moving in either direction. If the molecules below had a higher temperature, they’d have a different MB [Maxwell-Boltzmann] distribution, with more molecules moving faster. Some of those faster moving molecules would have the right trajectory to rise to the interface (slowing, sure) and carry energy from the lower slice to the upper. The upper slice (lower temperature) has fewer molecules moving faster — the entire MB distribution is shifted to the left a bit. There are therefore fewer molecules that move the other way at the speeds that the molecules from the lower slice deliver (allowing for gravity). This increases the number of fast moving molecules in the upper slice and decreases it in the lower slice until the MB distributions are the same in the two slices and one accomplishes detailed balance across the interface. On average, just as many molecules move up, with exactly the same velocity/kinetic energy profile, as move down, with zero energy transport, zero mass transport, and zero alteration of the MB profiles above and below, only when the two slices have the same temperature. Otherwise heat will flow from the hotter (right-shifted MB distribution) to the colder (left-shifted MB distribution) slice until the temperatures are equal.
It’s an interesting argument. Here’s my elevator speech version.
• Suppose we have an isolated container of air which is warmer at the bottom and cooler at the top. Any random movement of air from above to below a horizontal slice through the container must be matched by an equal amount going the other way.
• On average, that exchange equalizes temperature, moving slightly warmer air up and slightly cooler air down.
• Eventually this gradual exchange must lead to an isothermal condition.
I encourage people to read the rest of his comment.
Now, I see where I went wrong. Following the logic of my question to Dr. Brown, I incorrectly thought the final equilibrium arrangement would be where the average energy per molecule was evenly spread out from top to bottom, with the molecules having the same average total energy everywhere. This leads to warmer temperature at the bottom and colder temperature at elevation. Instead, at thermal equilibrium, the average energy per volume is the same from top to bottom, with every cubic metre having the same total energy. To do that, the gas needs to be isothermal, with the same temperature in every part.
Yesterday, I read the Jelbring hypothesis again. As I was reading it, I wondered by what logic Jelbring had come to the conclusion that the atmosphere would not be isothermal. I noticed the following sentence in Section 2.2 C (emphasis mine):
The energy content in the model atmosphere is fixed and constant since no energy can enter or leave the closed space. Nature will redistribute the contained atmospheric energy (using both convective and radiative processes) until each molecule, in an average sense, will have the same total energy. In this situation the atmosphere has reached energetic equilibrium.
He goes on to describe the atmosphere in that situation as taking up the dry adiabatic lapse rate temperature profile, warm on the bottom, cold on top. I had to laugh. Jelbring made the exact same dang mistake I made. He thinks total energy evenly distributed per molecule is the final state of energetic equilibrium, whereas the equilibrium state is when the energy is evenly distributed per volume and not per molecule. This is the isothermal state. In Jelbrings thought experiment, contrary to what he claims, the entire atmosphere of the planet would end up at the same temperature.
In any case, there’s another way to show that the Jelbring hypothesis violates conservation of energy. Again it is a proof by contradiction, and it is the same argument that I presented to Jelbring years ago. At that time, I couldn’t say why his “gravito-thermal” hypothesis didn’t work … but I knew that it couldn’t work. Now, I can see why, for the reasons adduced above. In addition, in his thread Dr. Brown independently used the same argument in his discussion of the Jelbring hypothesis. The proof by contradiction goes like this:
Suppose Jelbring is right, and the temperature in the atmosphere inside the shell is warmer at the bottom and cooler at the top. Then the people living in the stygian darkness inside that impervious shell could use that temperature difference to drive a heat engine. Power from the heat engine could light up the dark, and provide electricity for cities and farms. The good news for perpetual motion fans is that as fast as the operation of the heat engine would warm the upper atmosphere and cool the lower atmosphere, gravity would re-arrange the molecules once again so the prior temperature profile would be restored, warm on the bottom and cold on the top, and the machine would produce light for the good citizens of Stygia … forever.
As this is a clear violation of conservation of energy, the proof by contradiction that the Jelbring hypothesis violates the conservation of energy is complete.
Let me close by giving my elevator speech about the Jelbring hypothesis. Hans vigorously argues that no such speech is possible, saying
There certainly are no “Elevator version” of my paper which is based on first principal physics. It means that what I have written is either true or false. There is nothing inbetween.
Another “gravito-thermal” theorist, Ned Nikolov, says the same thing:
About the ‘elevator speech’ – that was given in our first paper! However, you apparently did not get it. So, it will take far more explanation to convey the basic idea, which we will try to do in Part 2 of our reply.
I don’t have an elevator speech for the Nikolov & Zeller theory (here, rebuttal here) yet, because I can’t understand it. My elevator speech for the Jelbring hypothesis, however, goes like this:
• If left undisturbed in a gravity field, a tall container of air will stratify vertically, with the coolest air at the top and the warmest air at the bottom.
• This also is happening with the Earth’s atmosphere.
• Since the top of the atmosphere cannot be below a certain temperature, and the lower atmosphere must be a certain amount warmer than the upper, this warms the lower atmosphere and thus the planetary surface to a much higher temperature than it would be in the absence of the atmosphere.
• This is the cause of what we erroneously refer to as the “greenhouse effect”
Now, was that so hard? It may not be the best, I’m happy to have someone improve on it, but it covers all the main points. The claim that “gravito-thermal” theories are too complex for a simple “elevator speech” explanation doesn’t hold water.
But you can see why such an elevator speech is like garlic to a vampire, it is anathema to the “gravito-thermal” theorists—it makes spotting their mistakes far too easy.
w.
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dp says:
January 19, 2012 at 9:58 pm
dp, you should ask the author, Dr. Robert Brown, that question, as the description is his. I have only quoted part of it here. There is a link to his thread in the head post.
w.
Willis:
The problem is that the atmosphere Jelbring described cannot exist, and thus no meaningful statements can be made about it.
Suppose we have an atmosphere with a near surface temperature of 288K, and translucent in infrared frequencies. Then it must radiate to space at about the black body rate. Thus it will cool to the temperature of space, about 3K. In order to keep the atmosphere at a temperature above 3K, energy must be added. Indeed, to keep the atmosphere at 3K, energy must be added, via the cosmic background radiation. Once energy is added, whether at the top or the bottom of the atmosphere, the conditions for an isothermal atmosphere no longer apply.
It’s like dividing by zero. Once we let that slip in, whether by Jelbring or Eschenbach or Mann & Jones, we can get to any conclusion we want by following one chain of logic while ignoring another.
okay Willis, I take your point re gravity not affecting temperature. however I would argue that gravity is associated with temperature – I assume that temperature is highest in the core of astronomical bodies, as written.
yes temperature dissipates by convection and radiation.
but what is kinetic energy other than mainly a gravitational effect, a pull or a push ?
whilst this topic is ‘restricted’ to movements of air and temperature in a tube, and the subject of a planetary body minus an atmosphere, the real life said body of air operates in relation to a wider context – wind, height, temperature of surrounding air.
equilibrium is a temporary phenomenae. perhaps slowest rate of change is a near definition.
I have to go out. I’ll return to this later.
“It’s an interesting argument. Here’s my elevator speech version.
• Suppose we have an isolated container of air which is warmer at the bottom and cooler at the top. Any random movement of air from above to below a horizontal slice through the container must be matched by an equal amount going the other way.
• On average, that exchange equalizes temperature, moving slightly warmer air up and slightly cooler air down.
• Eventually this gradual exchange must lead to an isothermal condition.
I encourage people to read the rest of his comment.
Now, I see where I went wrong. Following the logic of my question to Dr. Brown, I incorrectly thought the final equilibrium arrangement would be where the average energy per molecule was evenly spread out from top to bottom, with the molecules having the same average total energy everywhere. This leads to warmer temperature at the bottom and colder temperature at elevation. Instead, at thermal equilibrium, the average energy per volume is the same from top to bottom, with every cubic metre having the same total energy. To do that, the gas needs to be isothermal, with the same temperature in every part.”
[-Sounds wrong.-]
“Yesterday, I read the Jelbring hypothesis again. As I was reading it, I wondered by what logic Jelbring had come to the conclusion that the atmosphere would not be isothermal. I noticed the following sentence in Section 2.2 C (emphasis mine):
The energy content in the model atmosphere is fixed and constant since no energy can enter or leave the closed space. Nature will redistribute the contained atmospheric energy (using both convective and radiative processes) until each molecule, in an average sense, will have the same total energy. In this situation the atmosphere has reached energetic equilibrium.”
So, it’s the “average molecule” which has same energy [velocity] and in less density the amount energy per cubic meter is less [and is therefore is colder- doesn’t warm as much something like thermometer].
But no molecule stays at same velocity. Or make it simple, maybe, molecules don’t go [normally] in any vector for more nanosecond.
In convection one can have a group “averaged molecules” going in one direction or vector or have something one calls velocity- but none of these actual molecules is actually going in one direction- it’s more like sound wave. it’s possible for a gas molecule travel say a meter or more in one direction, but chances are low of this happening. With zillion and zillion them, one probably goes mile fairly commonly:)
Example of the theory as I understand it.
Have two flat areas of real estate. Have them at different elevation. On at sea level and one at 10 km in elevation. And they are on same planet. They will have different density of air 1 meter above their surfaces.
On the land at 10 km in elevation should receive more sunlight. But let’s say the both receive same amount of sunlight. These areas of real estate are hundreds of square km and flat.
The one at higher elevation will not warm the air to as high a temperature as the land at sea level. The higher elevation land will increase the gas molecules speed to same velocity as the sea level land, but there is low density, so it’s cooler.
And even with more sunlight in higher elevation it would make the gas molecules travel faster, but it still would be cooler air temperature.
If this is true, than lower elevation has a “greenhouse effect”.
And the land at higher elevation should have it’s dirt at higher temperature [if has more sunlight]- and less greenhouse effect in this sense.
jorgekafkazar says:
January 19, 2012 at 10:14 pm
Not a clue, my friend. Your pseudo-algebra is meaningless. Try putting units on it to see what I mean. Unless you think that is symbolic logic, but it’s not any kind that I recognize.
jorge, i seriously have no clue what you are on about. It seems you think I’ve made a logical mistake. Let me recap the logic.
I stick a thermocouple in an isothermal atmosphere. I cannot extract any work.
I stick a thermocouple in an atmosphere that is cold at the top and warm at the bottom. I can extract work.
Which one is the lower energy state?
w.
Gravitational heating comes from contraction of a gas. Potential energy is converted to kinetic energy.
The sun when it originally formed would have had enough kinetic energy to shine for about 18,000,000 years. However – after about 18,000,000 years it would be cold and dead without nuclear fusion.
The atmosphere of earth may have contracted at some point but that energy dissipated several billion years ago.
The gravito-thermallers are kind of right – it may have happened, once. But that train has left the station and doesn’t impact current atmospheric conditions. The potential energy was converted and radiated away a long time ago.
Willis,
I am beginning to think that there may be another effect at work to raise the temp due to gravity, but not for the reasons suggested. Because of the small angle subtended by the earth vis a vis the sun, the energy comes in a collimated beam. Essentially, most of the photons, whatever their energy, have a pretty good shot at the surface of the earth. However, once the surface or the gas
near (first couple miles let’s say) the surface reradiate this energy, it goes off it all directions. Due to density differences in the atmosphere caused by gravity, I would expect some of this energy to be reflected back at lower angles of incidence to the horizon due to refractive effects in the atmosphere, similar to the mirror like effect seen on a hot highway at low look angles. The compressed gas of the atmosphere creates a diode effect for inbound radiant energy. No clouds required.
Thomas L says:
January 19, 2012 at 10:22 pm
First, this is a thought experiment. Einstein famously explained relativity using the thought experiment of an elevator in space. Of course, something like that cannot exist … but it can still be very useful in understanding the real world.
Second. suppose we find a planet with an atmosphere with no nearby sun. We put a foam shell around it, so it is almost entirely energetically isolated from the universe.
That is the atmosphere Jelbring describes … so why can’t that atmosphere exist?
w.
George Turner says:
January 19, 2012 at 7:34 pm
Nice try. Before you rotate the cylinder, the air at the top of the cylinder has lower pressure than the air at the bottom of the cylinder, and therefore the density at the top is less than the density at the bottom. So when you turn the cylinder over, you are doing work, as moving the bottom up takes more energy in a gravity field than you get from moving the top down. And the air in what was the top (now the bottom) gets compressed and heated, while the air in what was the bottom (now the top) expands and cools. But you knew this.
Bryan says:
January 19, 2012 at 4:30 pm
Perhaps Willis will suggest a real experiment with real materials to test the alternative conjectures.
Say with a thermoelectric device to make use of the temperature difference.
A computer simulation program would not be any kind of proof
I think he will find with real materials that this is beyond him
Perhaps this is why there has never been an experiment to settle the matter!
Yes, the distance from textbook to rigorously formulated physical hypotheses is huge. In fact, it is so large that it contains all the science, except for checking the predictions. To the Warmists it is a vast desert that they dare not enter.
JeffT says: Jan 19, 5:00 pm If there is a temperature gradient between two parts of a system, net heat flows from the warmer part to the cooler part.
/headslap! Thank you, Jeff, it is that simple. Total energy, chemical Potential energy, mechanical potential energy do nothing to prevent temperature equilibrium.
Take a pressure cylinder. Fill it with iron springs and Freon. Put in the piston. Place it is a bath or air, water, or any other fluid. Press the piston to its max. The contents of the cylinder heats, the iron springs compress, the Freon compresses then liquefies releasing more heat. Wait a bit. The Cylinder system contains a lot of potential energy, but it and its bath will come to the same temperature in time.
Release the piston and the temperature of the cylinder will drop and then be warmed by the bath back to a lower temperature. This example does not employ gravitational potential energy, but I’m convinced. Total energy is irrelevant.
So on a uniformly heated, dead planet at equilibrium, the atmosphere will have a zero lapse rate, isothermal atmosphere. But bring on a day and night cycle, then differential heating will force a lapse rate to become established by adiabatic processes regardless of the atmospheric composition. All we need to know is the specific heat of the atmosphere and the surface gravity: Cp/g. http://wiki.answers.com/Q/What_is_the_dry_adiabatic_lapse_rate_on_Venus.
Could one do the experiment using water as the fluid?
A long insulated tube (thermos like?) with temperature sensors along and a small tube inside for air to get out. Pour water at fixed temperature. Measure along tube. It would not have to be very long with good temperature sensors, a tower would do.
1. The elevator soliloquy:
The mass of the atmosphere, solar insolation, and the temperature of the upper atmosphere determine the temperature at the surface of a planet of sufficiently thick atmosphere. The exact makeup of the gasses of the atmosphere is not important, as it can be assumed that essentially all outgoing long wave radiation is absorbed and re-radiated in the lower atmosphere, finally being radiated to space from the upper atmosphere. Surface temperature is simply a function of the temperature of the effective black body temperature of the upper atmosphere required to radiate heat equal to the net insolation of the total planet, the mass of the atmosphere, and the gravity of the planet.
Climate change and excursions from average can be attributed mainly to changes in the mass of the atmosphere, albedo, and insolation, and are largely independent of minor changes in LW absorption in the atmosphere.
2. I must disagree with Dr. Brown. In a tall column of gas acted on by gravity, and at equilibrium, in a perfectly insulated container, the top of the column will be colder than the bottom of the column. All gas will be at an equal energy state (by mass, not volume), and since gas at the top has more potential energy, it must be at a lower thermal energy state in order to have the same entropy value. In other words, the adiabatic lapse rate still applies. This is essentially the situation that occurs in the atmosphere, on a very large scale.
Stephen Rasey says: 4:54 pm: Assuming that Index of refraction (Ir(r=ground) is higher near the ground than at high altitude Ir(r=large), then there is a focusing of energy.
Nope! Effects cancel out. Index of refraction might make it possible for someone on the surface of a planet to see more than 180 degrees of the outer shell. But each point on the outer shell is illuminating an equally larger area of the planet, thus diluting its contribution at any one point. Therefore, there is no net focusing, no net increase in energy, and an isothermal atmosphere even with index of refraction that increases with pressure.
Willis Eschenbach says:
January 19, 2012 at 10:32 pm
You have a planet not near a star and heat the atmosphere to 288K. You construct an insulating sphere to hold the heat in. If the atmosphere is isothermal, the TOA is at 288K. This will heat the insulation to 288K. The insulation will then radiate to space at 288K. Of course the radiation rate could be very slow, but we are talking about equilibrium here. If we make the foam slightly radioactive, then we get near equilibrium, but only until enough half-lives have passed.
If there is a material that is 100% reflective at microwave through visible frequencies, that would keep the atmosphere either isothermal or iso-energetic. Iso-energetic implies a lapse rate related to the gravity gradient. My memory of thermodynamics is not enough to tell me which way the energy gets partitioned, but I think it’s iso-energetic.
Or we could surround the planet with insulation in the form of turtles. Then it’s turtles, all the way up.
Thus spake Willis (re thermonuclear heating): “I ran the numbers in the past. I don’t have them in front of me, but it is very small, less than a tenth of a watt per square metre from memory. There are a variety of geothermal regions, and hot vents under the sea, and hot springs and pools on land. But you have to consider—for every hot springs you know of, there are thousands of square miles of land with no hot springs, where if you go to sleep in the morning, you wake up very cold. So yes, you are right, there is a heat source down under. But it is very small, even if we have greatly underestimated it.”
Right, Willis. I did the calculation a couple of years ago, and the conducted core heat is minuscule. If we look at mass transfer of hot material (hot springs, volcanoes, etc.), the numbers may be more significant. It is estimated that there are on the order of three million undersea volcanic vents. Most of these are small, but we don’t really have a handle on their total heat load, nor do we know exactly where it goes or how fast. (These seeps also emit CO2, but that’s another story,)
There is no equilibrium of a column of air in the gravitational field of a spherical planet unless there is a heat sink drawing energy away from the atmosphere. The heat equation does not allow a spherically distributed atmosphere to attain thermal equilibrium.
Without radiating gasses to draw the heat energy away, the atmosphere of a planet is like an ideal electrical capacitor hooked up to a constant voltage source, or an ideal inductor hooked up to a constant voltage source. It has nowhere to dissipate the energy, so the energy just keeps accumulating.
Eventually, the capacitor pops, or the inductor melts. In the real world, stray conductance or resistance, respectively, if they are large enough, prevent destruction. In the same way, a planet with insufficient radiation boils away its atmosphere, or achieves an equilibrium when sufficient radiation takes place from unaccounted emitting particles.
I have fleshed this all out in the previous thread starting about here.
So-called greenhouse gasses do not heat the surface – they prevent it from overheating. They are like a resistance put in parallel with that capacitor, or in series with the inductor.
In steady state, there is no difference between my description and the standard greenhouse hypothesis. On Earth, IR emitting gasses arrest the runaway heat accumulation that would otherwise exist. Backradiation equilibrates with the net overage in steady state Stefan-Boltzmann emissions from the planet’s surface.
So, there is no discriminator in the steady state, which is why people have been led astray for so long. They have been using Stefan-Boltzmann in ways it was not intended to be used – Stefan-Boltzmann only holds in conditions of equilibrium or, at least, quasi-equilibrium. The atmosphere of a planet without significant emissions is never in equilibrium, or even quasi-equilibrium.
The solution of the heat equation in spherical coordinates tips the balance in determining the real process going on. Without atmospheric emissions, there is always a temperature gradient leading into the atmosphere, and the atmosphere therefore continually accumulates energy. There is no adiabatic lapse rate possible without a heat sink in the atmosphere above. But, there is a lapse rate.
Look at the earlier thread. I’ve got it nailed. There is no doubt about it.
Erratum:
Without radiating gasses to draw the heat energy away, the atmosphere of a planet is like an ideal electrical capacitor hooked up to a constant current source…
“On average, that exchange equalizes temperature, moving slightly warmer air up and slightly cooler air down.”
Willis, I’m not sure whether that summarizes what Dr Brown said, but it’s wrong, and for a fundamental reason.
When the lapse rate is below the dry adiabat (toward isothermal) it is referred to as convectively stable. Above the adiabat, it is unstable. At the adiabat, it is neutrally stable.
At the adiabat, rising air cools by expansion, at exactly the rate at which the nearby air is becoming cooler (by lapse rate). And falling air warms. There is no buoyancy issue created. Moving air retains the same density as the environment. And no heat is transferred.
But in the stable regime, falling air warms faster than the change in ambient. It becomes less dense, so there is a force opposing its rise. That is why the air is stable. This motion both takes kinetic energy from the air and moves heat downwards (contra your statement). It is a heat pump which works to maintain the lapse rate.
Rising air does the same. It cools faster, and so rises against a buoyancy force. It takes KE from the air and moves “coolness upwards”, ie heat down. It pumps heat just as falling air does.
That is why air in motion tends to the adiabat lapse rate. A heat pump requires energy. Where from?
The atmosphere is famously a heat engine, driven by temperature differences, most notably from equator to pole, but also innumerable local differences, eg land/sea. This provides the kinetic energy that maintains the lapse rate, and it is hard to imagine any planetary atmosphere where the energy would not be available.
The effectiveness of the heat pump tapers as the adiabat lapse rate is approached. Beyond, in the unstable region, everything is reversed. The pump becomes an engine, with heat moving upward creating KE. This of course quickly diminishes the temperature gradient.
I blogged about this here.
It’s true that with no motion at all, conduction will render the air isothermal.
“a force opposing its rise”
I meant, opposing its fall.
A few posts from the other thread to help the discussion:
Solution of the Heat Equation in Spherical Coordinates
Bart says:
January 18, 2012 at 12:11 pm
In an atmosphere with no convection (the thought experiment world), the PDE governing the temperature is
dT/dt = alpha*del^2(T)
where alpha is the conductivity parameter and “d” is actually a partial differential operator and del^2 is the Laplacian. To solve this equation, we set T = T1(t)*T2(r), where T1 is wholly a function of t, and T2 is wholly a function of r. This leads to
(dT1/dt)/T1 = alpha*del^2(T2)/T2
where the “d’s” are now total differential operators. Since the left side is wholly a function of time, and the right wholly a function of r, they must both equal a constant, call it lambda.
Then, the solution of dT1/dt = lambda*T1 is elementary, T1 = T1(0)*exp(lambda*t). The solution of
del^2(T2) = (lambda/alpha)*T2
is a modified and adjusted zeroth order Bessel function of the second kind, which is qualitatively similar to a constant divided by radius.
Thus… the full solution is
T = T(0,0)*exp(lambda*t)*F(r)
There is always a gradient downhill in the radial direction. Thus, there is always heat flow into the higher altitudes. This heat flow will continue until it stops, either by high energy radiation if the atmosphere allows, or boiling away of the atmosphere.
Solution is a little more complicated than that, but same general morphology
Bart says:
January 18, 2012 at 1:56 pm
The alpha parameter is actually called “thermal diffusivity”, and it is inversely proportional to density. So, as the altitude increases, alpha will grow larger. So, even the Bessel function solution is not precise, and will only hold approximately in the lower atmosphere and before significant thermal expansion has taken place. A precise global solution would have to take all of these factors into account.
But, there is no chance of steady state because the steady state solution is T = B/r for a constant B, and since there is a gradient, there cannot be a steady state, and that creates a contradiction. So, the conclusion remains: there is always a thermal gradient pushing heat continuously into the atmosphere, and it will not stop until either there is some kind of radiative release, or the atmosphere flees.
Why the solution cannot be a constant, part I
Bart says:
January 19, 2012 at 1:52 pm
…Starting from some temperature at the base, in any finite time, the temperature is going to be less out farther than it is nearer, and it is going to go to zero at infinity. So, there must be a gradient over any finite time.
What is the form of that gradient? Well, through separation of variables, the temperature function is the product of a time varying part, and a spatial varying part. The spatial varying part is a function of spatial coordinates only, so it does not change with time. Therefore, since for any finite time, the spatial solution is of the form A + f(r), where A is a constant and f(r) is a monotonically decreasing function of r, and A must be zero at that time, then it must always be zero, and f(r) must go to zero as r approaches infinity.
QED.
Why the solution cannot be a constant, part II
Bart says:
January 19, 2012 at 8:14 pm
…So, if the “mode shape” of this thing is a constant independent of radius, that means the entire spherical shell has to heat uniformly. It starts at zero everywhere. It is at 1K everywhere at the same time. It is at 100K everywhere at the same time. This is physically impossible based solely on the fact that the rate of heat conductance is finite.
Willis,
If I understand what you are asking in this post, here is my elevator speech. If I have misunderstod, my apologies.
Imagine a 10km perfectlyl insulated column of gas.
Outside the tube connect one end of a suitably sensitive, perfectly insulated thermocouple to the top and bottom of the tube.
1. If a continual current flow is detected from the thermocouple, then the temperatures at the top and bottom of the column are different and one has a perpetual motion machine. *
* Series-parallel enough of these structures and our energy problems are solved – until gravity runs out, anyway.
2. If, after a suitable period of time no current flow is detected then the top and bottom of the column are at the same temperature.
I’m betting on #2.
I disagree with idea that the temperature on each layer is the same or that the energy per volume is the same, both are IMO wrong.
Let’s imagine an ideal gas container in gravitational field. In our gas particles don’t interact at all. As long as the gas is in equilibrium, each of its particles has the same total energy at every moment (but the important point is that total energy is sum of kinetic and potential energy). These particles travel on parabollic trajectory, bouncing off the bottom. This gives us overview of the energy distribution over the container volume – at the very top, the speed of particles is zero and the temperature is zero as well. At the very bottom the particle temperature is proportional to square of the height of the trajectory. The equilibrium condition is held too because on each horizontal layer boundary you get exactly the same number of particles traveling up and down at exactly the same speed.
This model I believe gives us very good idea about temperature, pressure and energy distribution over the container even for normal gases with interacting particles.
Another important thing is, your thermal perpetuum mobile cannot work in this temperature gradient as the medium you use in it follows the same rules the gas does. To raise your medium to the height where you intend to let it cool down you use more energy than you gain by letting it cool down there.
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Now, I may be completely wrong somewhere in the above (transition from ideal non-interacting gas to normal interacting one feels to be the weakest point) so my main points are:
– potential energy is completely omitted in the article arguments
– the argument with thermal perpetuum mobile is invalid because the medium must follow the same rules the atmosphere does. If gravity created temperature gradient, it cannot be used by thermal engine.
One needs to ask, why is our air at different temperatures on earth at different altitudes? For one thing, we do not have an impervious membrane around us, and we have this thing called the sun shining at us. Result, the ground and especially water is heated or energized, and conducts heat to the lower air by direct contact, evaporation, etc. Result, the air at the bottom is warmer than air higher up. However, if you go very high, you reach ozone, which can and does absorb UV radiation, making it hot (and heating the non ozone around it) and causing it to expand way out into space. This will have some effect heating from the top down, although since this air is very thin the total amount of energy it can add to air lower down isn’t much. There is also some heating caused by the various GHG’s (especially water vapor) being heated by radiation and heating the air around them. Conclusion, there are reasons why our air is at different temperatures at different altitudes and it has nothing to do with any gravito thermal effect.
Second, if earth were as described, with an energy impervious barrier around it (and it had no fissionables), it would be at absolute zero. Even if gravito thermal worked, any air that contacted the earth would contact a very cold surface. The much more dense earth would quickly absorb all the heat from the air, which would freeze out rapidly, goodby gravito thermal effect.
However, let us assume, for arguments sake, that the surface of the earth and the air start out at the same temperatures as our surface and air. If gravito thermal works, and the people make huge heat engines, they will have to produce a lot of energy to make up for the fact that the cold ground will absorb and keep on absorbing the energy. They could, I suppose, surround the entire earth with an insulating material to slow this heat loss. If they did, they could be comfortable assuming the heat engines can create enough heat and light for them and to make up for the slight loss to ground. An impervious insulator below them would result in the energy and heat from their engine building up rapidly in the space between the insulator above and below them, which would continue until it melted the engines and stopped their operation. A not quite perfect insulator below them could keep them comfortable until the ground below the insulator had warmed up all the way through, at which point the people would find it slowly getting warmer, continuing until once again it gets warm enough that the heat engines melt and no more heat is, apparently, being produced from nothing.
Third, let us assume that the earth, as ours does, has fissionables in it’s core, and that is why the surface is warm. However, since there is a barrier to energy above them, this heat will slowly build up, since it cannot escape, until all the fissionables are used up. If it has a lot of them, as our earth seems to, it will get very hot before that happens, and once all the fissionables are gone, it will stay at that hot uniform temperature forever. However, lets us say, for arguments sake, that the last fissionables are used up just as it reaches a nice comfortable temperature (one where air does not freeze out). Now let us say that the people there, somehow, in the dark, invent heat engines to take advantage of this gravito thermal effect to make light and power. This will slowly add heat to the earth (the solid planet will absorb most of it) and very slowly heat things up. Eventually, it will get too hot, all the people will die, the engines will stop, and it will stay at this new, somehow elevated temperature forever. If you could somehow create indestructible heat engines that never melted and didn’t need tending, the heat would slowly build up forever, eventually the inside of the membrane would be hotter than any sun (which, I suppose, since it melted the earth, would stop the gravito thermal effect). As we can see, even under ideal starting conditions, if we assume that this membrane surrounds the earth, and we assume ideal starting temperatures for gravito thermal to work, it always results in energy being created from nothing, and the space inside the membrane getting hotter and hotter.
I think all this shows that under any circumstance we start with, even absolutely ideal (and impossible) starting conditions, gravito thermal results in an impossible outcome, with energy inside the membrane building up from nowhere. It should also be noted that the whole thought experiment is ridiculous anyway, such and energy impervious barrier cannot exist in our universe with it’s laws anyway. Thus we see that the only way gravito thermal can work is in a universe that is entirely imaginary, and has different natural laws that ours does. In other words, the authors of this idea are living in a world all their own, literally. They would realize that if they thought through all the implications of all this, as I have above.
Bart says:
January 19, 2012 at 11:06 pm
So in the Jelbring thought experiment you are saying the atmosphere never, ever achieves thermal equilibrium? If not … why can’t we pull work out of it with a thermocouple?
Regards,
w.