Moist-entropic vertical adiabatic lapse rates: the standard cases and some lead towards inhomogeneous conditions

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This note is a companion of Marquet and Geleyn (2013, arXiv:1401.2379 [ao-ph]), where adiabatic lapse rates Γns and Γsw are derived for non-saturated (Γns) or saturated (Γsw) parcel of moist-air. They are computed in terms of the vertical derivative of the moist-air entropy potential temperature θₛ defined in Marquet (2011, arXiv:1401.1097 [ao-ph]). The saturated value Γsw is rewritten in this note so that a more compact formulation is obtained. The new formulation for Γsw is expressed in term of a weighting factor C. This factor may represent the proportion of an air parcel being in saturated conditions.