Computation of Atmospheric Lapse Rate

To compute the lapse rate (temperature drop with height z) of an atmosphere, we start from the  2nd Law of Thermodynamics in the form presented in Computational Thermodynamics:

                                                c_v dT + P dV     non-negative

where c_v is specific heat under constant volume, which combined with the differentiated form PdV + V dP = R dT of the gas law PV =RT gives 

                                                                   (c_v+R) dT - V dP    non-negative.

Vertical hydrostatic balance is expressed by dP =  - g Rho dz with Rho = 1/V density, from which follows that 

                                                                     dT/dz + g/C_p   non-negative,

where C_p = (c_v+R)/V Rho = (c_v+R)/mass.  The lapse rate is thus bounded below by - g/C_p, which for a dry Earth atmosphere equals  - 10 C/km. 

The observed lapse rate is dT/dz = - 6 C/km, which is partly (because there is also

evaporation/condensation) an effect of turbulent dissipation with c_v dT + P dV  strictly positive: Rising hot air expands under temperature drop which  is counterbalanced by turbulent dissipation reducing the drop. 

The Joule experiment discussed in the previous post, concerns precisely this effect. We have shown that we can simulate this effect by computing turbulent soutions of the Navier-Stokes equations, and thus it should be possible to similarly compute the effective lapse rate in the 

Earth Atmosphere.

In the ideal case of isentropic expansion the 2nd Law is satisfied with equality, from which follows 

                                             c_v dT + R T dV/V = 0.

This gives 

                             T V^a = constant

where   a = R/c_v, which determines P and V (or Rho) as functions of z through T(z).