Another look at the moist baroclinic Ertel-Rossby invariant with mass forcing

Due to the importance of the mass forcing induced by precipitation and condensation in moist processes, the Lagrangian continuity equation without a source/sink term utilized to prove the Ertel-Rossby invariant (ERI) and its conservation property is re-derived considering the mass forcing. By introducing moist enthalpy and moisture entropy, the baroclinic ERI could be adapted to moist flow. After another look at the moist ERI, it is deployed as the dot product between the generalized velocity and the generalized vorticity in moist flow, which constitutes a kind of generalized helicity. Thus, the baroclinic ERI is further extended to the moist case. Moreover, the derived moist ERI forumla remains formally consistent with the dry version, no matter whether mass forcing is present. By using the Weber transformation and the Lagrangian continuity equation with a source/sink effect, the conservation property of the baroclinic ERI in moist flow is revisited. The presence or absence of mass forcing in the Lagrangian continuity equation determines whether or not the baroclinic ERI in moist flow is materially conserved. In other words, it would be qualified as a quasi-invariant but only being dependent on the circumstances. By another look at the moist baroclinic ERI, it is surely a neat formalism with a simple physical explanation, and the usefulness of its anomaly in diagnosing atmospheric flow is demonstrated by case study.