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.:
Verkley WTM., Gerkema T, 2003. On Maximum Entropy Profiles. Journal of atmospheric sciences.
Akmaev RA., 2008. On the energetics of maximum‐entropy temperature profiles. Quarterly Journal of the Royal Meteorological Society
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.