4.1 Governing equations

In order to understand the phenomena described in Section 3, we must consider the equations of magnetohydrodynamics (MHD) which express the conservation of mass, energy, and momentum in a magnetized plasma. Although the dynamics is made more complex by the presence of density stratification, rotation, magnetic fields, and spherical geometry, there is at least one property of the motions which may be safely exploited in order to simplify the equations of motion somewhat: they possess a low Mach number (this can usually be verified a posteriori in any numerical or theoretical model). In other words, the kinetic energy of the convection is small relative to the internal energy of the plasma. Furthermore, the ratio of magnetic to internal energy is also small (implying an Alfvénic Mach number $« 1$ ). Under such conditions, it is valid to adopt the anelastic approximation. The anelastic approximation is justified throughout the solar interior with the exception of the near-surface layers ( $r > 0.98Ro .$ ) where velocities associated with granulation can exceed the sound speed and where radiative transfer and ionization effects must be taken into account.

In the anelastic approximation the velocity, magnetic fields, and thermodynamic variations induced by convection (or by other means) are treated as perturbations relative to a spherically-symmetric background or reference state. The resulting system of equations is given in Appendix A.2. In numerical applications, the anelastic equations can be much more computationally efficient to implement than the fully compressible MHD equations because they filter out high-frequency acoustic waves which would otherwise severely limit the time step required to maintain numerical stability. Furthermore, from a theoretical standpoint, the anelastic equations are generally more analytically tractable, partly because the velocity field can be expressed in terms of scalar streamfunctions and velocity potentials, thus eliminating one velocity variable (e.g., Glatzmaier, 1984 ).

In the remainder of this paper, we will use the anelastic equations described in Appendix A.2 to illustrate a few fundamental aspects of solar interior dynamics, the first being energy balance. The reference state density, pressure, specific entropy, and temperature are represented by overbars: $-- r$ , $-- P$ , $-- S$ , and $-- T$ . These same symbols without overbars denote fluctuations about the reference state. For more on notation, see Appendixes A.1 and A.2.

Figure 5:

Schematic diagram illustrating the energy flow in an anelastic model. The thermal energy incorporates both the internal energy of the plasma and the gravitational potential energy as described in the text. The buoyancy force and compression can transfer energy among the thermal and kinetic energy reservoirs while the Lorentz force can transfer energy among the kinetic and magnetic energy reservoirs. Viscous and Ohmic heating can also convert kinetic and magnetic energy to thermal energy.