A.2 The anelastic equations

The stratification throughout most of the solar envelope is very nearly adiabatic, a result which is expected theoretically (because efficient convection tends to mix entropy) and which is supported by helioseismic inversions, interpreted in the context of solar structure models (e.g., Gough and Toomre, 1991 ; Christensen-Dalsgaard, 2002 ). We may quantify the degree of adiabaticity as follows:

where $CP$ is the specific heat per unit mass at constant pressure, and $HT$ is the temperature scale height (see Appendix A.1). The final equality in Equation (35

) is valid for a perfect gas in hydrostatic equilibrium where the adiabatic temperature gradient is given by $(@T /@r)ad = - g/CP$ .

The magnitude of $e$ can be estimated using a standard solar structure model such as model S of Christensen-Dalsgaard (1996 ), which is available online at Christensen-Dalsgaard (2003 ). This yields $- 4 e < 10$ in all but the outer few percent of the convection zone, $r > 0.97Ro .$ ( $-2 e < 10$ for $r < 0.995Ro .$ ), implying that the thermodynamic variations induced by convection in the envelope must be small relative to the background stratification. Since the advection terms in the momentum equations are likely to be comparable to or less than the perturbation pressure gradients, $rv2 < P$ , this in turn implies low Mach numbers throughout most of the convective envelope and radiative interior: $v/cs« 1$ , where $cs$ is the sound speed. The Alfvénic Mach number, $vA/cs$ , is also thought to be much less than unity, based on theoretical arguments and upper limits obtained from helioseismology (Section 3.7). These conditions represent a necessary and sufficient justification for the anelastic approximation in which thermodynamic variations are treated as perturbations relative to a spherically symmetric background or reference state (Gough, 1969 ; Gilman and Glatzmaier, 1981 ; Lantz and Fan, 1999 ).

We consider here a perfect gas in a rotating spherical shell under the anelastic approximation. We denote reference state quantities with overbars and we assume hydrostatic equilibrium such that

We also assume a perfect gas equation of state:

where $CP$ and $CV$ are the specific heats per unit mass at constant pressure and volume and are here taken to be constant in space and time. For a perfect gas we may also define the specific entropy such that

where $P0$ and $r0$ are arbitrary fiducial values for the pressure and density.

More sophisticated equations of state may also be used within the anelastic framework but this is a convenient and reasonably accurate approximation in all but the outermost layers of the solar interior (e.g., Christensen-Dalsgaard and Däppen, 1992 ; Basu et al., 1999 ).

In many applications the reference state is allowed to evolve in time in response to dynamically-induced stratification changes. This is particularly important for numerical models of penetrative convection where the structure of the overshoot region changes significantly due to mass and energy transport by convective motions (e.g., Miesch et al., 2000 ).

The lowest-order perturbation equations include the equations of mass continuity,

$\~/ .(rv) = 0, (39)$

momentum conservation,

$( ) r- @v-+ (v . \~/ )v = - \~/ P - rg^r - 2r(_O_0 × v) + 1--( \~/ × B) × B + \~/ .D, (40) @t 4p$

internal energy,

$--(@S ) -- dS-- [ ( dT- )] 4pj rT ---+ v . \~/ S = -r-Tvr--- + \~/ . krrCP \ ~/ T + ---~ ^r + P + --2-J 2, (41) @t dr dr c$

and magnetic induction

$@B--= \~/ × (v × B) - \~/ × (j \~/ × B) . (42) @t$

The linearized ideal-gas equation of state is

The viscous stress tensor and heating rate are given by

$1- Dij = -2rn(eij - 3 ( \~/ .v)dij), (44)$

and

$-- 1 2 P = 2n r(eijeij- 3-( \~/ .v) ), (45)$

where $eij$ is the strain rate tensor. The current density $J$ is given by

$c J = -- \~/ × B. (46) 4p$

In the anelastic system described here we have assumed that the rotation rate of the coordinate system $_O_0$ and the gravitational acceleration $g$ are constant in time. Furthermore, we neglect the centrifugal force so the gravitational acceleration is purely radial: $g = - g(r)^r$ . This is a good approximation in the solar envelope where the centrifugal force is about five orders of magnitude smaller than the gravitational force. However, it precludes certain dynamics such as Eddington-Sweet circulations which depend on the centrifugal force (Section 4.4). We have also neglected nuclear energy generation which is significant only in the deep core of the Sun.

Boundary conditions are often taken to be stress-free which has the advantage that angular momentum is conserved. In some situations, a latitudinal differential rotation may also be applied on the upper or lower boundary of the shell. Typically the heat flux or the entropy gradient is specified on the lower boundary and the entropy is fixed on the upper boundary. When magnetism is included, the lower boundary is often taken to be perfectly conducting and the upper boundary is matched to a potential external field which decays as $r --> oo$ .