Moist-entropic vertical adiabatic lapse rates: the standard cases and   some lead towards inhomogeneous conditions
  This note is a companion of Marquet and Geleyn (2013, {arXiv:1401.2379 [ao-ph]}), where adiabatic lapse rates $\Gamma_{ns}$ and $\Gamma_{sw}$ are derived for non-saturated ($\Gamma_{ns}$) or saturated ($\Gamma_{sw}$) parcel of moist-air. They are computed in terms of the vertical derivative of the moist-air entropy potential temperature $\theta_s$ defined in Marquet (2011, {arXiv:1401.1097 [ao-ph]}). The saturated value $\Gamma_{sw}$ is rewritten in this note so that a more compact formulation is obtained. The new formulation for $\Gamma_{sw}$ is expressed in term of a weighting factor $C$. This factor may represent the proportion of an air parcel being in saturated conditions. 
Jean-Fran\c{c}ois Geleyn and Pascal Marquet
physics.ao-ph physics.flu-dyn physics.geo-ph