Where The Temperature Rules The Sun

Although the conclusion of those “proofs” is essentially true, I must once again point out that they are based on faulty assumptions.

Dr. Brown’s proof was a thought experiment involving a silver wire whose ends are thermally coupled at different altitudes to a gas column. The argument was that, if we assume that the lapse rate of a gas at equilibrium is non-zero, we are forced to conclude that the temperature gradient imposed on the wire by the gas column’ s lapse rate would cause heat to flow—and thus enable work to be performed—indefinitely. Since that’s perpetual motion, the argument went, the non-zero-lapse-rate assumption that led to it must be false.

But that argument tacitly made two questionable assumptions. The first was that coupling the wire to the gas column would not affect the gas column’s equilibrium lapse rate. The second was that the thermal coupling would impose the gas column’s lapse rate on the silver wire.

As can readily be seen by simulating molecules colliding in a gravitational field, neither assumption stands up to scrutiny for a finite number of molecules, no matter how large that finite number is.

Again, the basic conclusion that the equilibrium atmosphere’s lapse rate would not differ detectably from zero is correct. But, since the assumptions on which those arguments were based are invalid, it’s incorrect to call them proofs.

The column of gas used in Brown’s scenario is constrained laterally and so leads only to a linear decline in density and pressure with height which does not properly reflect the real world scenario.

Actually, no. As Coombes and Laue demonstrated, a uniform temperature is entirely consistent with an exponential pressure reduction.

Sorry,

w.

Don’t be. Your post provided a lot of insight, which I appreciate. As to your failing to comprehend the shortcomings of your “proof,” I completely understand that not everyone is comfortable with starting from first principles to re-examine his beliefs; each of us has his respective limitations.

For the benefit of readers with a somewhat broader perspective, though, I’ll explain it thus:

If you simulate a monatomic gas comprising two constituents, one consisting of molecules of mass and another consisting of molecules of mass , all randomly traveling in one dimension for the sake of simplicity subject to a gravitational field and among them having total (kinetic + potential) energy , what you find after a long period of “thermalization” is that at altitude the first constituent’s average molecular kinetic energy is , while the second constituent’s is zero. (By “what you find” I mean what you find after averaging over a long time; variances are so great that over short time periods these averages not repeatably undetectable. I had attempted to get a post published here that would explain this in more detail, but apparently I’ve become persona non grata here since I disputed Christopher Monckton’s bizarre mathematics.)

In any event, those energies translate to respective temperatures at that altitude of and absolute zero for the different constituents, where is Boltzmann’s constant. You will also find that both constituents have the same average temperature, , at altitude zero and that both temperatures change linearly with altitude: both lapse rates are non-zero, but, since the constituents are both at equilibrium there is by definition no average heat flow.

In other words, the equilibrium temperatures at the higher altitude are different for the different constituents, the different constituents have different lapse rates, and these values’ averages over long periods persist even though the constituents are intimately mixed, but despite their temperature differences and their intimate mixing no net heat flows on average between them.

Moreover, you’d find that each constituent by itself would exhibit a lapse rate twice the value it exhibits when the two are mixed.

Now, a silver wire is not the same as a gas, and mixing two gas components together is not the same as coupling a silver wire to a gas column. But adding the second constituent to the first reduces the original first’s lapse rate by removing the constraint that the first constituent’s total energy remain fixed. Since coupling the gas column to the silver wire removes a similar constraint from the gas column, we are entitled to question the silver-wire proof’s assumption that coupling the silver wire to the gas column would leave the latter’s lapse rate unchanged.

Furthermore, since the added constituent adopts a lapse rate different from that of the original constituent—and since that difference persists despite the constituents’ being intimately mixed—it’s not self-evident that thermal coupling would cause the silver wire’s temperature difference to equal the gas column’s. Nor, since the different gas constituents’ lapse rates cause no heat flow down their respective temperature gradients, can we conclude that whatever temperature gradient prevails at equilibrium in the silver wire would necessarily cause heat to flow within it.

In short, although what we think we know about Fourier’s law would seem to dictate that any temperature gradient at all would cause some heat to flow through a heat-conductive medium, we find if we reason from first principles that gravity modifies that conclusion. Before we apply a physical law, that is, it’s important to know the assumptions on which it is based.

I cannot discuss thermal transfer with a man who seriously claims that if two things at different temperatures are connected by a silver wire, that “heat would not flow” through the wire.

If two objects at different temperatures are connected thermally, by radiation or by a silver wire or an iron bar, heat will spontaneously flow from the warm object to the cold object until their temperatures equalize.

My immediately previous remarks were directed specifically to those contentions. I described a simulation you can run that is the basis for my belief that Fourier’s law is actually only a zero-gravity result, i.e., that with gravity Fourier’s Law is off a skosh, so that subject to gravity the wire could have a small temperature gradient without heat flow.

Specifically, I described simulating an equilibrium composite gas comprising two constituent gases of respective, different molecular weights. If you set those constituents’ molecules in motion in a gravitational field and let them randomly collide as they do in an equilibrium gas, what you’d find is that the constituents not only have respective non-zero (but small) lapse rates at equilibrium but also have different mean molecular kinetic energies at every non-zero altitude.

Since such a simulation would show that in the presence of gravity no net heat flow occurs despite a non-zero gradient within each (thermally conductive) equilibrium-gas constituent and despite a temperature difference between the two constituents at non-zero altitudes, we are entitled at least to speculate that a thermally conductive wire, like the thermally conductive gas, can have a temperature gradient at equilibrium, i.e., without heat flow, when it’s subject to gravity.

I’m open to being shown I’m wrong; my conclusion is clearly unconventional. But your idea of refutation seems to be to call me names and demand that I quote you even though I clearly have been doing so. That isn’t as persuasive as you may imagine.

Joe Born,
I would like to understand your ideas more fully. This thread is jumbled so I hope I am stitching into the weave at the correct point 😉
Starting here “For the benefit of readers with a somewhat broader perspective, though, I’ll explain it thus:
If you simulate a monatomic gas comprising two constituents, one consisting of N/2 molecules of mass m and another consisting of N/2 molecules of mass 2m, all randomly traveling in one dimension for the sake of simplicity subject to a gravitational field and among them having total (kinetic + potential) energy NE_{avg}, what you find after a long period of “thermalization” is that at altitude NE_{avg}/2mg the first constituent’s average molecular kinetic energy is \frac{NE_{avg}}{2(3N-2)}, while the second constituent’s is zero”

I take it that by the second constituent is zero you are referring to the more massive monatomic gas 2m and that its particle velocity will be zero? In essence are you describing the effect on particle kinetics of vertical escape velocity in a gravitational field? Let’s assume for the purpose of the model that the vertical gravity field is constant (not true for real planets) then starting at the surface for a given kinetic energy the massive particle 2m will reach a lower apex point of zero upward velocity than the lighter particle of mass m will because for a given temperature in a mixed gas the low mass particles must necessarily have a higher velocity to share the same momentum as the more massive gas. For a planet with a given gravity field and a given degree of solar heating a low mass, fast moving gas, such as Helium, will be lost to space while a slow moving, high mass gas, such as Argon will be retained.

Joe,
Thanks, Now for the next bit about lapse rates:-
“Furthermore, since the added constituent adopts a lapse rate different from that of the original constituent—and since that difference persists despite the constituents’ being intimately mixed”
In the Earth’s mixed Nitrogen/Oxygen atmosphere dry air has a mixed molecular weight (MW) of 28.9. Water vapour however is a light molecule with a MW of 18. Moist air therefore has a different lapse rate to dry air. In addition to this mass difference, ascending moist air cools more slowly because of the release of latent heat associated with the two phase changes of water that can occur. The first change is from vapour to liquid (water droplet clouds) and the second phase change is from liquid to solid forming ice crystal clouds. All of this would be irrelevant if it were not for the physical separation of the water falling under gravity from the storm clouds in the form of solid crystal snow (or hail) at altitude and/or liquid rain lower down. Once the rain has fallen from the cloud the now dried air remains aloft and can only descend at the dry air adiabatic lapse rate. It is this difference in lapse rates that accounts for the Chinook (snow eater) winds that are the end stage of an advection process that transports the moist air from the Pacific across the Rockies, bringing snow to the mountains, and then as the air descends at the dry adiabatic lapse rate transforms it to reach the Canadian Prairies at surface altitudes of 3000 feet where it arrives with a higher air temperature than that of its initial sea level Pacific air mass origin.

Lapse rates matter on Earth because of water, the condensing fluid and its latent heat and rainfall. Here the lapse rates are not the same for ascending moist air parcels and descending dry air parcels.

It appears that the site’s software sometimes interprets R’s assignment operator as part of an HTML tag or something and eats sections of the script. So, although the original script runs fine, when I post it to this site it gets mangled. I’m trying to get around that by replacing the usual assignment operator with equals signs, but I’ve yet to have it go through unmolested. I’ll post a workable version if I find a workaround.

In the meantime, I apologize to anyone who tried to run it.

I again apologize for flubbing the script upload, but I’ve despaired of getting the script to post unmolested.

What you would have seen if you’d been able to run the script is that it simulates a one-dimensional gas in which (for computational purposes, a necessarily small number of) molecules in a gravitational field sometimes collide with and sometimes pass through each other:

http://i68.tinypic.com/9091f6.jpg

If you simulate something like a million seconds each for two-, three-, four-, and five-molecule systems, and if for each system regress one molecule’s kinetic energy (temperature) against its altitude, you get the following illustration that the lapse rate falls as the number of molecules increases.

http://i68.tinypic.com/5odw69.png

By extrapolation, that lapse rate would be exceedingly small for the number of molecules in the atmosphere, but it would be some finite non-zero value.

As to tags, I did try “pre” tags and, later, “code” tags, and neither seemed to work; I’d run the code fine in my office, copy that exact code between the tags, post it, copy the resultant published code, and get a ton of errors when I ran the code I’d copied. Maybe the size of the script had something to do with it. (I believe I mentioned, though, that I sent it by email to Mr. Watts, if that’s helpful.) Anyway, I have long since exhausted that exercise’s amusement value.

Still, I was able to post the output graphs, which in essence are the elevator pitch, although they omit some nuance.

Now to Dr. Brown’s post.

Okay. Once more, with feeling. I did, as seems to have escaped your attention, quote Dr. Brown above, to the effect that a temperature difference would have to cause heat flow in any situation at all where there’s thermal conductivity. That’s what I dispute. We’ll go through why step by step.

First, Dr. Brown based that contention on Fourier’s Law: Heat flow is proportional to temperature gradient, normal area, and conductivity—or, if you prefer, temperature difference, normal area, and conductivity and inversely proportional to orthogonal distance. So, if there’s any temperature difference at all, no matter how small, Fourier’s Law seems to require that at least some heat will flow occur if there’s any thermal conductivity (as there is in a silver wire—and, due to colliding molecules, there necessarily is in a gas).

Second, Fourier’s Law is not restricted to silver wires; it applies to other things, too, like gases.

Third, an ideal gas is a collection of molecules that collide and, if they’re disposed in a gravitational field, otherwise follow ballistic trajectories.

Fourth, if we model an ideal gas consisting of colliding ballistic molecules and find, as the graphs above illustrate for a monatomic gas restricted to one dimension, that at equilibrium—i.e., at no average heat flow—it has a non-zero kinetic-energy gradient (non-zero lapse rate), then we can conclude, contrary to what we may have thought, that Fourier’s Law actually is only a limit as the number of molecules approaches infinity. For a finite number, that is, the law is off a bit, although for the number of molecules in a macroscopic object it’s so close that as a practical matter we’d never detect the difference—and, even in principle, detecting it would take quite a while.

Fifth, if that’s true of Fourier’s Law as applied to gases, we are entitled to suspect that it may be true for silver wires, too. We therefore cannot assume that what little temperature difference is imposed upon a silver wire by an equilibrium-lapse-rate-caused temperature difference between coupling altitudes will necessarily cause heat to flow through the wire.

But, sixth, the fifth step is actually superfluous, since in the fourth step we already found a non-zero lapse rate in an equilibrium gas, which is what Dr. Brown was trying to refute.

Basically, I have no problem with the conclusion of Dr. Brown’s proof if it’s qualified to mean that the lapse rate is so close to zero that as a practical matter we’d never detect the difference. But he rejected qualification:

[H]eat will flow through the silver for any difference in temperature until there is no difference in temperature. . . .

Look, I know this seems really far out: it seems to be denying Fourier’s Law. But in light of the simulation (and, in truth, some statistical mechanics that I won’t go into here) maybe it’s just interpreting Fourier’s Law more correctly.