A reevaluation of ice-liquid water potential temperature

A synthesis of previous studies suggests the need for new, more accurate approximations for ice-liquid water potential temperature (θ _il), a thermodynamic variable utilized in some numerical models. Starting from equations presented in a previous study, two new approximate formulations of θ_il are derived, along with their governing equations. The new formulations are significant improvements over previous ones because no terms are dropped during their derivation and no Taylor series approximations are utilized. The governing equations for the new formulations reveal conditions under which θ_il can be considered a conserved variable. Potential temperature lapse rates determined from a reference thermodynamic equation are compared numerically against lapse rates determined from several approximations of θ_il. Many of the findings agree with previous studies. However, the results show that a commonly used formulation does not account for the specific heats of water, and thus has an inherent cold bias. When the tendency to θ_il is assumed to be zero (by approximation), one of the new formulations is superior to all other formulations that have been presented previously. When the full governing equation is used, the new θ_il formulations produce results identical to those from the reference equation.