List of Footnotes

1 Just to be clear on nomenclature, the term magnetic resistivity is defined as σ− 1 c. Since it is the same as the magnetic diffusivity η but for a constant factor, the two terms are often used interchangeably in the context of MHD.
2 One such exception would be the Gold–Hoyle flux tubes, which is a class of twist flux tube structures for which the Lorentz force vanishes (Gold and Hoyle, 1960).
3 In Section 3.3, we will discuss the interplay between magnetic buoyancy and stratification and their associated instabilities in some detail.
4 Although 3D convection simulations have become increasingly feasible (and popular) for studying solar and stellar convection, 1D mixing-length theory (MLT; Vitense, 1953; Böhm-Vitense, 1958) remains valuable for many theoretical studies of solar and stellar interiors. For instance, Spruit (1974) developed a MLT model of the solar convection zone that predicted the depth of the convection zone to be d = 198 Mm. This is within the range of uncertainties (d = 200± 2 Mm) from helioseismic measurements by Christensen-Dalsgaard et al. (1991). For similarities and differences between MLT and 3D convection models, we refer the reader to the comparative study by Abbett et al. (1997) and to Nordlund et al. (2009*, Section 3.1.2).
5 We do not mean to imply that CMEs originate from individual granules.
6 Section 3.6 provides a discussion dedicated to the role of magnetic twist in the magnetic flux emergence process. Here, we merely focus on the aspect related to magnetic buoyancy instabilities.
7 We are completely sidestepping the important issue of choosing appropriate temperature gradients and thermal and viscous diffusivities to achieve sufficiently high Rayleigh number Ra to allow convection to occur. For the purpose here, we can safely assume that simulation setups are chosen with sufficiently high Ra to allow for convection. For detailed calculations of the critical Rayleigh numbers for stratified atmospheres with different polytropic indices, we refer the reader to Gough et al. (1976).
8 The operation of a surface dynamo was studied in detail by Vögler and Schüssler (2007) using radiative MHD simulations with an initial seed field of zero net flux. A thorough discussion of solar surface dynamo models is perhaps beyond the scope of this review.
9 They happened to use a profile of the transverse field different than that given by Eq. (41*).
10 To simplify this discussion we are neglecting the effects of rotation.
11 As discussed in Section 3.2, the buoyant rise of magnetic field toward the surface can drive horizontal flows that are an integral part of the flux emergence process. Even when the horizontal flows have no shear (i.e. non-zero (∇ ×v )n), they can still make a non-zero contribution to − (Ap ⋅v)Bn. So although the names ‘emergence’ and ‘shear’ terms are commonly used in the literature, it may be more appropriate to refer to them as the vn and vt terms, respectively. The term − (Ap ⋅v )Bn is called the vt term because the choice of the gauge for Ap implies the dot product only has contributions from the transverse components of v.
12 Indeed, in the photosphere and convection zone the energy transfer by radiation dominates the thermal conduction of the plasma.
13 It should be noted that if we had a uniform oblique field in the domain, the initial flux sheet or tube embedded in the convection is no longer in strict mechanical equilibrium. This effect is negligible, however, if the motion driven by the Lorentz force generated by the superposition of the uniform field and the flux sheet/tube is slower than the emerging motion of the flux sheet/tube driven by the buoyancy.
14 As pointed out by Low (2013), this prescription for the velocity is incompatible with line-tied bottom boundary conditions (v = 0 at ∂V), which is the boundary condition for Parker*’s (1988) field-line braiding theory of coronal heating. Cheung and DeRosa (2012*) pointed out that in the MF framework, the induction equation becomes a non-linear diffusion equation with the same mathematical form as ambipolar diffusion (e.g., see Brandenburg and Zweibel, 1994; Cheung and Cameron, 2012). Treating the governing equation as a non-linear diffusion equation (with an effective Ohm’s law that only includes the ambipolar and Ohmic terms), the boundary conditions used in their model are the components of E transverse to the boundary. This is also how some other flux emergence simulations are driven at the bottom boundary (e.g., Fan and Gibson, 2003; Martínez-Sykora et al., 2008). The difference between the MF model and MHD models is additional compatible boundary conditions for the continuity, momentum, and energy equations must be imposed for the latter.