New: Lapse Rate by Gravitation: Loschmidt or Boltzmann/Maxwell?
Written by Professor Claes Johnson
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.
- 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
- dp = -g rho dx or Vdp = – gdx
- (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).
- 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.