For the bulk of the solar convection zone, the fluid stratification is very close to being adiabatic with , where is the non-dimensional superadiabaticity with and denoting the actual and the adiabatic logarithmic temperature gradient of the fluid respectively, and the convective flow speed is expected to be much smaller than the sound speed : (see Schwarzschild, 1958; Lantz, 1991). Furthermore, the plasma defined as the ratio of the thermal pressure to the magnetic pressure () is expected to be very high () in the deep convection zone. For example for flux tubes with field strengths of order , which is significantly super-equipartition compared to the kinetic energy density of convection, the plasma is of order . Under these conditions, a very useful computational approach for modeling subsonic magnetohydrodynamic processes in a pressure dominated plasma is the well-known anelastic approximation (see Gough, 1969; Gilman and Glatzmaier, 1981; Glatzmaier, 1984; Lantz and Fan, 1999). The main feature of the anelastic approximation is that it filters out the sound waves so that the time step of numerical integration is not limited by the stringent acoustic time scale which is much smaller than the relevant dynamic time scales of interest as determined by the flow velocity and the Alfvén speed.
Listed below is the set of anelastic MHD equations (see Gilman and Glatzmaier, 1981; Lantz and Fan, 1999, for details of the derivations):nearly adiabatic stratification, and velocity , magnetic field , thermodynamic fluctuations , , , and are the dependent variables to be solved that describe the changes from the reference state. The quantity is the viscous stress tensor given by
Fully compressible MHD simulations have also been applied to study the dynamic evolution of a magnetic field in the deep solar convection zone using non-solar but reasonably large values such as to . In several cases comparisons have been made between fully compressible simulations using large plasma and the corresponding anelastic MHD simulations, and good agreement was found between the results (see Fan et al., 1998a; Rempel, 2002). Near the top of the solar convection zone, neither the TFT model nor the anelastic approximation are applicable because the active region flux tubes are no longer thin (Moreno-Insertis, 1992) and the velocity field is no longer subsonic. Fully compressible MHD simulations are necessary for modeling flux emergence near the surface (Section 8).
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