Anelastic magnetohydrodynamic equations for modeling solar and stellar convection zones

The anelastic approximation has strong advantages for numerical simulations of stellar and solar convection zones. The chief and generally known one is that it suppresses acoustic modes, permitting larger simulated time steps to be taken than would be possible in a fully compressible model. This paper clarifies and extends previous work on the anelastic approximation by presenting a new vorticity-based formulation that can be used for two- and three-dimensional MHD simulations. In the new formulation, all fluctuating thermodynamic variables except the entropy are eliminated from the equations. This shows in the plainest way how the anelastic approximation generalizes the Boussinesq approximation, which appears as a special limit. The roots of both models are traced to the mixing-length theory of convection, which establishes the scaling parameters for "deep" (weakly superadiabatic) convection at low Mach numbers. The Ogura & Phillips and the Glatzmaier derivations of the anelastic model are broadened to include a possible depth dependence in the thermodynamic properties of constituent gases. This permits a variable state of gas ionization, for example, which is important for stars like the Sun, in which the convecting regions coincide with the ionization zones of hydrogen and helium. Tests with the new model are presented, in which it is shown that the new model is capable of reproducing earlier results in the linear and nonlinear stages of convection.