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[Phys-l] Temperture profile in a graviational field



Here is an interesting question that I have been seeing in the context
of climate science and the "Greenhouse effect". (I may have more points
to make later, since this topic is interesting and important -- for
science and for society.)


Suppose you have a perfectly insulated column of air. Let's minimize
concerns about IR by making the inner walls of the container highly
reflect and making the gas N2 (which emits/absorbs minimal amounts of
IR). Suppose the column is a few km tall, with the base at the surface
of the earth.

1) What will be the temperature profile? Certainly there is a pressure
gradient and a density gradient in the column. I would say there is a
temperature gradient as well. On a microscopic scale, between every
collision, if the molecule gains altitude it will gain PE and lose KE
(ie it will cool). Any molecule moving downward will warm. On a
macroscopic level, this can discussed in terms of the "dry adiabatic
lapse rate".
http://en.wikipedia.org/wiki/Lapse_rate#Dry_adiabatic_lapse_rate and
the "potential temperature"
http://en.wikipedia.org/wiki/Potential_temperature

In either case, it is clear to me that the equilibrium condition (both
in this ideal column and in the real atmosphere) would be a temperature
gradient (cooling ~ 10K/km). Do others agree?


2) If this is true, how can this best be squared with the 2nd law of
thermodynamics? If the top and bottom of the column were held at a the
same temperature, there would be a continuous flow of energy from top to
bottom, even though they are the same temperature. Even if the top were
slightly cooler than the bottom, there would be a downward flow. This
would violate a standard statement of the 2nd law, since we have
spontaneous heat from cool to warm.

I've been trying to think of a good way to explain that this is not
indeed a violation. I suspect the best explanation will have to involve
the more fundamental statement of the 2nd law -- that entropy tends to a
maximum. The adiabatic lapse rate leads to an isentropic gas and a
constant potential temperature.