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I'm trying to understand why there are differing predictions of the atmospheric temperature profile. It is well established that the dry adiabatic lapse rate (DALR) is:

$$ \frac{\mathrm{d}T}{\mathrm{d}z} = -\frac{g}{c_p} \approx -9.8\ \mathrm{K/km} $$

This is derived by assuming adiabatic process and hydrostatic pressure gradient:

$$ \mathrm{d}s = c_p\mathrm{d}\ln{T} - R\mathrm{d}\ln{p}\quad(= \frac{\delta q}{T} = 0)\\ \quad \frac{\mathrm{d}p}{\mathrm{d}z} = -\rho' g $$

where $\rho'$ is density of ambient air, and pressure of an air parcel is the same as ambient pressure ($p = p'$). It is essenitally the cooling an air parcel will experience due to change in ambient pressure when risen adiabatically in the atmosphere.

However, when using the principle of maximum entropy (i.e. looking for the equilibrium profile) we get isothermal profile, as predicted classically by Gibbs and Boltzmann.

Apparently, the actual atmospheric profile (where there is no moisture condensing) is consistent with the dry adiabatic lapse rate much more than with the isothermal profile. The actual profile is of course subject to continuous thermal cooling of the atmospheric layers, thermal warming by radiation from the Earth's surface, heating from the surface by conduction (by turbulence and molecular diffusion), and during the day to solar heating. These factors can obviously affect any equilibrium.

My question is:

  • What causes the discrepancy? Is there a consensus?

  • If there was initially a DALR profile, would it eventually turn into an isothermal profile if there was no influence (no radiation, no surface, no dynamic phenomena).

My impression is that when radiative processes (thermal cooling and thermal heating from surface) are included in the maximum entropy calculation, one might get some sort of lapse rate, i.e. something between a pure thermal radiative profile and the isothermal profile.

I found some papers discussing the issue, esp.:

where they treat potential temperature as something which is conserved when applying the maximum entropy principle in order to reach a profile with a lapse rate. But it is not clear to me why such an assumption should be made, or if there is any general consensus that this is the right approach.

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The two articles you found are spot-on. An isothermal atmosphere is indeed the condition that maximizes entropy for a given amount of energy. Yet a positive lapse rate is almost always observed. Temperature typically decreases with increasing altitude. The key to resolving this apparent paradox is how heat flows into and through the Earth's atmosphere. The Earth's atmosphere is a system that is very far from equilibrium.

If the atmosphere isn't in that maximum entropy state (and it never is, at least not from very bottom to very top), heat transfer will set up to put move it toward that state. Very key: That entropy-driven heat transfer rate is rather low. If other heat transfer processes are in play (and they always are), it is very easy to overwhelm that low entropy-driven heat transfer that would nominally equalize the temperature. The Earth's troposphere is heated from below and cooled from above. This alone overwhelms the entropy-driven heat transfer, and by multiple orders of magnitude.

The other factor is how heat flows through the atmosphere. The presence of greenhouse gases in the atmosphere set up the conditions that make a positive lapse rate possible. By way of analogy, imagine a perfect blackbody heat source that generates heat at a rate $Q$ and has a surface area A. If that heat source radiates to the dark sky, the equilibrium temperature will be given by the Stefan-Boltzmann law, $T_\text{top} = (Q/(\sigma A))^{1/4}$. (Note: I'm assuming that $Q$ is much, much greater than the tiny amount of heat coming in from the cosmic microwave background.) If you put a perfect blackbody blanket atop that heat source, the blanket will radiate at a rate $Q$ outward and inward. The heat source will receive heat from above and below, making the equilibrium temperature of the heat source with one blanket equal to $(2Q/(\sigma A))^{1/4}$, or $2^{1/4} T_\text{top}$. Yet another blanket makes the thermal equilibrium equal to $3^{1/4} T_\text{top}$, and so on.

The atmosphere with its greenhouse gases that are optically thick in the thermal infrared acts as a bunch of blankets. Obviously not perfect blackbody blankets, but blankets nonetheless. This is what favors the development of a positive lapse rate.

This favoring isn't universal. There are times when weather creates conditions where the lapse rate is negative, that is, temperature increases with altitude. This turns out to be a very stable condition (the name for this is a "very stable atmosphere"). This negative lapse rate precludes rising and falling parcels of air. With calm air as well, the only thing that can transfer heat is diffusion, which is a very slow process in the atmosphere.

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    $\begingroup$ Great answer. I didn't see it matters how fast radiative heating/cooling is compared to processes causing equilibrium, and entopy maximization has no way of accounting for it. You mention "entropy-driven" heat trasfer - I assume you mean diffusion, but there is also convection, which is fast compared to radiative heating/cooling. But it can only operate when gradient is unstable, which makes me conclude DALR will dominate over radiation where radiative balance predicts unstable gradient. Once DALR is reached, only diffusion is possible, which is too weak to cause isothermal profile anywhere. $\endgroup$ – peter Jan 9 '15 at 18:46
  • $\begingroup$ @peter keep in mind while DALR is neutral to dry motions it is unstable to moist motions which will continue to drive convection that also results in latent heat release as water changes phase, introducing heating to throughout the mid/upper troposphere. $\endgroup$ – casey Jan 10 '15 at 16:53
  • $\begingroup$ @casey I know things are vastly more complicated in reality - I was instead interested in a minimal framework which could explain the difference between the two predictions (in the context of lower troposphere - sorry it was not mentioned in the question). Radiation seems to be the one missing thing to get a reasonable "first-order" approximation, even if it were a simple grey thermal radiative balance model (if my understanding is right). $\endgroup$ – peter Jan 10 '15 at 22:06
  • $\begingroup$ @peter - The minimal framework would be a thermally isolated column of air, for example, a two hundred kilometer tall insulated test tube. That would do quite nicely. This is where Gibbs' isothermal equilibrium applies. Suppose Gibbs was wrong, that the maximum entropy state involves some temperature gradient. If that's the case, it's not hard to conceptually construct a perpetual motion machine, and not just a perpetual motion machine that violates the 2nd law of thermodynamics, but a perpetual motion machine of the worst kind, one that violates the 1st law of thermodynamics. $\endgroup$ – David Hammen Jan 10 '15 at 22:46
  • $\begingroup$ @DavidHammen Yes, the equilibrium calculation itself is correct (given the assumptions). Rather, the assumptions are problematic. So I suppose the answer to the question: "If there was initially a DALR profile, would it eventually turn into an isothermal profile if there was no influence (no radiation, no surface, no dynamic phenomena)." is "yes, it would". $\endgroup$ – peter Jan 11 '15 at 13:35

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