Mass conservation and the anelastic approximation

Numerical anelastic models solve a diagnostic elliptic equation for the pressure field using derivative boundary conditions. The pressure is therefore determined to within a function proportional to the base-state density field with arbitrary amplitude. This ambiguity is removed by requiring that the total mass be conserved in the model. This approach enables one to determine the correct temperature field that is required for the microphysical calculations. This correct, mass-conserving anelastic model predicts a temperature field that is an accurate approximation to that of a compressible atmosphere that has undergone a hydrostatic adjustment in response to a horizontally homogeneous heating or moistening. The procedure is demonstrated analytically and numerically for a one-dimensional, idealized heat source and moisture sink associated with moist convection. Two-dimensional anelastic simulations compare the effect of the new formulation on the evolution of the flow fields in a simulation of the ascent of a warm bubble in a conditionally unstable model atmosphere. In the Boussinesq case, the temperature field is determined uniquely from the heat equation despite the fact that the pressure field can only be determined to within an arbitrary constant. Boussinesq air parcels conserve their volume, not their mass.