New: Lapse Rate by Gravitation: Loschmidt or Boltzmann/Maxwell?
by Professor Claes Johnson
Will an atmosphere under the action of gravity assume a linear temperature profile with slope equal to the dry adiabatic lapse rate? Loschmidt said yes, while Boltzmann and Maxwell claimed that the atmosphere would be isothermal. Graeff (2007) has made experiments supporting Loschmidt and so it is natural to seek a theoretical explanation.
Consider a horizontal tube filled with still air at uniform temperature. Let the tube be turned into an upright position. Alternatively, we may consider a vertical tube with gravitation gradually being turned on from zero, or a horizontal tube being rotated horizontally starting from rest. During increasing gravitational force the air will be compressed and knowing that compression of air causes heating, we expect to see a temperature increase. How big will it be? Well, the 1st Law of Thermodynamics states that under adiabatic transformation (no external heat source):
- c_vdT + pdV =0
where c_v is heat capacity under constant volume, dT is change of temperature T, p is pressure and dV is change of volume V. Recalling the differentiated form of the gas law pV = RT with R a gas constant
- pdV + Vdp = RdT
and the equilibrium equation in still air with x a vertical coordinate
- dp = -g rho dx or Vdp = – gdx
where g is the gravity constant, rho = 1/V is density, we find
- (c_v + R) dT = -gdx or c_p dT/dx = – g,
where c_p = c_v + R is heat capacity under constant pressure.
We thus find that stationary still air solutions must have a dry adiabatic lapse rate dT/dx= – g/c_p = – g with c_p = 1 for air, as a consequence of compression by gravitation, using
- work by compression stored as heat energy
- pressure balancing gravity (hydrostatic balance).
We thus find a family of stationary still air solutions of the form (assuming x = 0 corresponds to the bottom of the tube):
- p(x) ~ (1 – gx)^(a+1)
- rho(x) ~ (1 – gx)^a
- T(x) ~ (1 – gx)
with a >0 a constant. In the absence of heat conduction such solutions may remain as stationary still air solutions. We thus find support of Loschmidt’s conjecture of still air solutions with the dry adiabatic lapse rate, in the absence of heat conduction. In the presence of (small) heat conduction, it appears that a (small) external source will be needed to maintain the lapse rate. Of course, in planetary atmospheres external heat forcing from insolation is present.
Returning the tube to a horizontal position would in the present set up without turbulent dissipation, restore the isothermal case. Turning the tube upside down from the vertical position would then establish a reverse lapse rate passing through the horizontal isothermal case.
Further, it seems that without heat source, the isothermal case of Boltzmann/Maxwell will take over under the action of heat conduction, with p(x) ~ exp( – cx) and rho(x) ~ exp ( – cx) with c>0 a constant.
For the Euler/Navier-Stokes equations for a compressible gas subject to gravitation, see the Computational Thermodynamics and the chapter Climate Thermodynamics in Slaying the Sky Dragon.
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