Mass continuity equation

The continuity equation is expressed (in terms of the dry hydrostatic mass, $\mu_d$) as

$${\frac{\partial\mu_d}{\partial t}}+ \nabla \cdot \left({\bf V} \mu_d\right) + {\frac{\partial{\dot\eta}\mu_d}{\partial\eta}} = 0$$ (28)

where the total hydrostatic mass is related to the dry mass simply through

$$\mu_d = \mu \,\varphi_q$$ (29)

so that dry mass is conserved. Many models use either the conservation of dry hydrostatic mass or dry density (in particular, for models using height-based coordinate systems) in order to avoid the additional complexity of including moisture source and sink terms in the continuity equation, along with other moisture adjustments during a time step. But the introduction of moisture into the continuity equation would take the form as shown in Eq. 28 but with $\mu$ instead of $\mu_d$ and would require the inclusion of a ${\dot{S}}_{\mu}$ type term which would include diabatic processes (mass changes owing to moisture changes in time). In addition, owing to the flux form of the equations, such a term would appear in all of the flux form prognostic equations.