 Saturated vapor column on a dry adiabat

Prescribe a dry adiabat for a background ideal gas A,where $R_A$ is its specific gas constant, $c_{pA}$ its constant specific heat capacity at fixed pressure, and $T_s,p_s$ the surface temperature and pressure. A dilute saturated vapour B has specific gas constant $R_B$, constant latent heat $L$, and saturation vapor pressure $e_s(T)$. Put $ϵ=R_A/R_B$ and $\beta =L/(R_B{T}_s)$.

The ideal-vapour Clausius-Clapeyron relation integrates toIts dilute vapour mass fraction is approximately $ϵe_s/p$. Under constant-$g$ hydrostatic balance, the vapour mass path is thereforeFor the second equality, set $\zeta =\log (p_s/p)$ and then $y=\beta e^{x\zeta }$. Thus the exponential integral follows from the combination of adiabatic cooling and rapidly falling saturation vapor pressure.

For $\beta \gg 1$, integration by parts gives $e^{\beta }{E}_1(\beta )\sim {\beta }^{-1}(1-{\beta }^{-1}+2{\beta }^{-2}-\cdots )$, henceThe column is weighted toward its warm lower part. Diluteness must hold throughout the column. This calculation prescribes the dry adiabat; if condensation heating changes that profile, a consistent moist adiabat is needed instead.