2.2 MHD simulations

The thin flux tube (TFT) model described above is physically intuitive and computationally tractable. It provides a description of the dynamic motion of the tube axis in a three-dimensional space, taking into account large scale effects such as the curvature of the convective envelope and the Coriolis force due to solar rotation. The Lagrangian treatment of each tube segment in the TFT model allows for preserving perfectly the frozen-in condition of the tube plasma. Thus there is no magnetic diffusion in the TFT model. However, the TFT model ignores variations within each tube cross-section. It is only applicable when the flux tube radius is thin (Section 2.1) and the tube remains a cohesive object (Section 5.4). Clearly, to complete the picture, direct MHD calculations that resolve the tube cross-section and its interaction with the surrounding fluid are needed. On the other hand, direct MHD simulations that discretize the spatial domain are subject to numerical diffusion. The need to adequately resolve the flux tube – so that numerical diffusion does not have a significant impact on the dynamical processes of interest (e.g. the variation of magnetic buoyancy) – severely limits the spatial extent of the domain that can be modeled. So far the MHD simulations cannot address the kinds of large scale dynamical effects that have been studied by the TFT model (Section 5.1). Thus the TFT model and the resolved MHD simulations complement each other.

For the bulk of the solar convection zone, the fluid stratification is very close to being adiabatic with δ ≪ 1, where δ ≡ ∇ − ∇ad is the non-dimensional superadiabaticity with ∇ = dln T∕d lnp and ∇ = (dln T∕d ln p) ad ad denoting the actual and the adiabatic logarithmic temperature gradient of the fluid respectively, and the convective flow speed vc is expected to be much smaller than the sound speed cs: 1∕2 vc∕cs ∼ δ ≪ 1 (see Schwarzschild, 1958Lantz, 1991). Furthermore, the plasma β defined as the ratio of the thermal pressure to the magnetic pressure (β ≡ p∕(B2 ∕8π )) is expected to be very high (β ≫ 1) in the deep convection zone. For example for flux tubes with field strengths of order 105 G, which is significantly super-equipartition compared to the kinetic energy density of convection, the plasma β is of order 105. 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, 1969Gilman and Glatzmaier, 1981Jump To The Next Citation PointGlatzmaier, 1984Lantz and Fan, 1999Jump To The Next Citation Point). 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, 1981Lantz and Fan, 1999, for details of the derivations):

[ ∇ ⋅ (ρ0v)] = 0, (10 ) ∂v 1 ρ0 ---+ (v ⋅ ∇)v = − ∇p1 + ρ1g + --(∇ × B ) × B + ∇ ⋅ Π, (11 ) [ ∂t ] 4π ∂s1- -1- 2 ρ0T0 ∂t + (v ⋅ ∇ )(s0 + s1) = ∇ ⋅ (K ρ0T0 ∇s1 ) + 4π η|∇ × B| + (Π ⋅ ∇ ) ⋅ v, (12 ) ∇ ⋅ B = 0, (13 ) ∂B-- ∂t = ∇ × (v × B) − ∇ × (η∇ × B), (14 ) ρ1 p1 T1 ---= ---− ---, (15 ) ρ0 p0 T0 s1 T1- γ −-1p1- cp = T0 − γ p0, (16 )
where s0(z), p0(z), ρ0(z), and T0 (z ) correspond to a time-independent, background reference state of hydrostatic equilibrium and nearly adiabatic stratification, and velocity v, magnetic field B, thermodynamic fluctuations s1, p1, ρ1, and T1 are the dependent variables to be solved that describe the changes from the reference state. The quantity Π is the viscous stress tensor given by
( ) Π ≡ μ -∂vi + ∂vj-− 2(∇ ⋅ v )δ , ij ∂xj ∂xi 3 ij
and μ, K and η denote the dynamic viscosity, and thermal and magnetic diffusivity, respectively. The anelastic MHD equations (10View Equation) – (16View Equation) are derived based on a scaled-variable expansion of the fully compressible MHD equations in powers of δ and β− 1, which are both assumed to be quantities ≪ 1. To first order in δ, the continuity equation (10View Equation) reduces to the statement that the divergence of the mass flux equals to zero. As a result sound waves are filtered out, and pressure is assumed to adjust instantaneously in the fluid as if the sound speed was infinite. Although the time derivative of density no longer appears in the continuity equation, density ρ1 does vary in space and time and the fluid is compressible but on the dynamic time scales (as determined by the flow speed and the Alfvén speed) not on the acoustic time scale, thus allowing convection and magnetic buoyancy to be modeled in the highly stratified solar convection zone. Fan (2001aJump To The Next Citation Point) has shown that the anelastic formulation gives an accurate description of the magnetic buoyancy instabilities under the conditions of high plasma β and nearly adiabatic stratification.

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 β values such as β ∼ 10 to 1000. In several cases comparisons have been made between fully compressible simulations using large plasma β and the corresponding anelastic MHD simulations, and good agreement was found between the results (see Fan et al., 1998aJump To The Next Citation PointRempel, 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, 1992Jump To The Next Citation Point) 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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