Consider a horizontal closed insulated 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 2nd Law of Thermodynamics states that under adiabatic and isentropic transformation (no external heat source and no turbulent dissipation):
- c_vdT + pdV =0
- pdV + Vdp = RdT
- dp = -g rho dx or Vdp = - gdx
- (c_v + R) dT = -gdx or c_p dT/dx = - g,
We thus find still air solutions with 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).
- p(x) ~ (1 - gx)^(a+1)
- rho(x) ~ (1 - gx)^a
- T(x) ~ (1 - gx)
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.
For the Euler/Navier-Stokes equations for a compressible gas subject to gravitation, see Computational Thermodynamics and the chapter Climate Thermodynamics in Slaying the Sky Dragon.