The rest of this section provides an brief introduction to MHD and how choices of the material properties of the plasma lead to the diverse family of MHD models describing the physics of flux emergence.
Although we choose not to review the assumptions of MHD nor pursue a full derivation of the equations, it is nevertheless instructive to remind ourselves of the physical principles central to MHD. They are:
- the Principle of mass conservation,
- the Principle of momentum conservation,
- the Principle of energy conservation, and
- Faraday’s law of electromagnetic induction.
We present the MHD equations in the context of continuum mechanics, which is a general framework for the study of continuous media (i.e., at the macroscopic level) under various conditions. For most theories of continuum mechanics (e.g., Newtonian fluids, the study of elastic materials), the governing equations reflect the first three principles listed above. In the following, denotes the Lagrangian derivative of a quantity. While the Eulerian version of the equations is often more convenient for implementation in numerical codes, the Lagrangian description has also been used extensively in the context of magnetic flux emergence. For instance, the Lagrangian MHD equations are the basis for the thin flux tube approximation, which model the evolution of discrete, individual flux tubes rising through the bulk of the solar convection zone. For details about the foundations of the thin flux tube approximation and its uses, we refer the reader to reviews by Moreno-Insertis (1997*) and Fan (2009b*). For this review, we prefer to present a Lagrangian description since it serves as the most useful framework for developing a physical picture of what happens to magnetic flux as it rises through the solar interior and eventually emerges into the atmosphere.
In a Lagrangian fluid description, the principle of mass conservation issolely because of converging or diverging flow.
The principle of momentum conservation is expressed by Cauchy’s 1st law of motion
The principle of energy balance can be stated in terms of the specific entropy , so that6*) can be combined with the continuity equation (1*) to give the evolution equation for internal energy (see, e.g., Section 2.3 of Priest, 1982*), viz.
Faraday’s induction equation is2.2.3 and 3.8).
The dot product of with the momentum equation leads to an evolution equation for the kinetic energy density . Similarly, the dot product of with the induction equation leads to an evolution equation for the magnetic energy density . Combining these two equations with the equation for specific internal energy leads to an equation for the total energy density, (), as is commonly presented in other texts that introduce the MHD equations.
Equations (1*), (2*), (7*), and (8*) capture the physical principles of mass, momentum and energy balance as well as Faraday’s law of induction. They apply to a wide variety of astrophysical and laboratory conditions but these four equations alone do not comprise a complete set of equations that govern an MHD system. They must be supplemented by so-called constitutive relations, which are statements about the material properties of the plasma.
In a variety of MHD applications (both analytical and numerical), the plasma is assumed to follow adiabatic evolution. This should be recognized as an assumption (justified or otherwise) about the material properties of the plasma. When all of the following, including thermal conduction, radiative cooling/heating, Joule and viscous dissipation are neglected, it can be shown that the initial specific entropy of a fluid element is conserved (i.e., , see Section 24 of Mihalas and Weibel-Mihalas, 1984). In this regime, the plasma temperature and pressure change only due to compression or expansion (i.e., ).
An example of a constitutive relation is the stress tensor . Implicit in the form for in Eq. (4*) are assumptions or choices about how the plasma behaves. The existence of the term is a statement that there exists an isotropic gas pressure. The specific form of the isotropic gas pressure (usually called equation of state, EOS) is yet another constitutive relation. For a cold plasma, . As will be discussed in more detail in the following sections, a large number of numerical studies of flux emergence from the solar interior into the corona assume an EOS for a perfect, monatomic ideal gas such that the ratio of specific heats . This assumes that the plasma is either completely neutral or completely ionized everywhere and at all times. In the case of plasma in the near-surface layers of the convection zone and overlying photosphere and chromosphere, the degree of ionization ranges from to and the gas pressure is a function of both and (e.g., Stein and Nordlund, 1998*; Vögler et al., 2005*). In the photosphere and below, the assumption of local thermodynamic equilibrium (LTE) is appropriate and thermodynamic variables such as mass density, internal energy density and specific entropy are state variables that are functions of local gas temperature and pressure (and vice versa). In the tenuous chromosphere, the time scales for recombination of hydrogen ions with free electrons are comparable, or longer than the typical time between the passage of shock waves which ionize hydrogen atoms (Kneer, 1980*; Leenaarts et al., 2007*). As such, studies that aim to examine the thermodynamic response of the chromosphere in response to photospheric and coronal evolution will most likely need to solve the rate equations for the hydrogen atom (together with advection terms due to flows) to determine the local pressure and temperature.
In addition to the equation of state, the radiative and thermal conductive properties of the plasma are constitutive relations that must be appropriately chosen for the given problem. In the evolution equation for specific internal energy (, Eq. (7*)), the form of the volumetric heating/cooling term depends on these choices. For instance, in the flux emergence model of Shibata et al. (1990a*), a radiative term with a specified cooling time scale was imposed in the photospheric layers to mimic the cooling of photospheric plasma by radiation to the higher atmospheric layers. This simple treatment was sufficient to demonstrate that such cooling would lead to the developing of downflows that convectively intensify emerged flux from a few hundred G strength to beyond 1 kG (see Abbett, 2007*; Fang et al., 2010*, for a more sophisticated parametric treatment of radiatively cooling in the solar atmosphere). For studies that attempt to realistically model the temperature structure of the photosphere, 3D radiative transfer calculations must be carried out to solve for the radiative flux (e.g., Stein and Nordlund, 2006*; Cheung et al., 2007a*; Yelles Chaouche et al., 2009; Martínez-Sykora et al., 2008*, 2009*; Tortosa-Andreu and Moreno-Insertis, 2009*). In such cases, is dominated by the term in layers where the plasma transitions from being optically thick to optically thin.
The choice of the thermal conductivity of the plasma is also a constitutive relation. In the solar corona, electrons are strongly magnetized and the transport coefficients are such that thermal conduction is predominantly aligned along magnetic field lines. The presence of such field-aligned thermal conduction leads to efficient energy transport from a reconnection region in the corona to chromospheric footpoints of the field. This can lead to chromospheric evaporation jets which feed mass into the corona (e.g., Yokoyama and Shibata, 2001*). This topic will be discussed in more detail in Section 3.7.2.
The form of the electric field given by Eq. (9*) assumes that the plasma is a perfect electrical conductor. When the electrical conductivity of the plasma is finite, non-ideal terms will appear in the expression for . If a scalar electrical conductivity is assumed (i.e., a scalar Ohm’s law), the corresponding electric field is1 The additional non-ideal term captures the effect of Ohmic diffusion of the magnetic field, which is the key ingredient for magnetic reconnection. The majority of non-ideal solar MHD models use this form of the electric field in the induction equation. Since the estimated magnetic diffusivity of solar plasma is often much too small for numerical simulations to resolve structures at the diffusive scale, modelers often adopt spatially varying forms of such that diffusive effects are minimal outside of reconnection layers and sufficient within these layers permit reconnection without introducing spurious numerical effects. One example of such a treatment is the so-call anomalous resistivity (Yokoyama and Shibata, 1994*). This model for the resistivity assumes , where is the magnitude of the current density. This dependence was found to permit the onset of Petchek-type fast magnetic reconnection in simulations of emerging flux interacting with pre-existing coronal field. Other choices of the imposed magnetic resistivity such as the so-called hyper-diffusivity scheme (Caunt and Korpi, 2001) are invoked to do a similar job as anomalous resistivity, namely to permit magnetic diffusion where it is most needed. This type of spatially-varying magnetic diffusivity appears to be an important ingredient in models that create fast reconnection jets and plasmoid ejections that accompany flux emergence into the solar corona (see Section 3.7.1 for a discussion how resistivity models affect models of flux emergence).
In weakly ionized plasmas, interactions between neutral and charged species leads to an electric field with additional terms describing ambipolar diffusion (also called the Pedersen effect) and the Hall effect. The inclusion of these effects can lead to evolution of the emerging magnetic field that is distinctly different from more traditional MHD models that simply use Eq. (12*) for the electric field. We will discuss this in more detail in Section 3.8.
Not all MHD models are created equal. While the basic principles of physics captured by MHD are general, the freedom for the modeler to choose the constitutive relations that describe the material properties of the plasma is the main reason for the diversity the MHD models that exist for modeling flux emergence.
Figure 5* lists some of the choices a modeler makes before performing a theoretical/numerical study of the flux emergence process. Some models are set up for studying certain physical effects in idealized conditions, while other models attempt to treat all (known) relevant effects at once. Some are concerned with interactions at certain length-scales (e.g., scale of granulation) while others try to capture entire active region scales. Some models use magnetic field configurations that mimic specific instances of observed episodes of emerging flux episodes, while others are not concerned with reproducing the observed patterns with high fidelity. As displayed in this figure, MHD models of flux emergence can be roughly divided into three categories: idealized, ‘realistic’, and data-driven models. These categories need not be mutually exclusive. While the division into the three categories is by no means unique, it serves as a guide for us to navigate the literature on MHD modeling of flux emergence.
Idealized models are so-called because they simplify the problem by choosing to neglect only certain effects. There are a number of advantages to idealized models. By choosing to neglect certain effects, idealized models often have simpler setups that allow the modeler to studying other physical effects in isolation. For example, many MHD models use plane parallel polytropes to mimic the average stratification of the solar atmosphere without including the necessary physics that lead to self-consistently generated convective motions and magneto-acoustic waves that channel energy into the chromosphere and corona. Such a choice allows modelers to concentrate on studying effects such as magnetic Rayleigh-Taylor instabilities (see Section 3.3) and the expansion of magnetic flux as it emerges from the convection zone into the tenuous layers of the atmosphere. Another benefit of idealized models is that they are often computationally less demanding and allow for a more extensive exploration of parameter space. In the context of flux emergence, exploration of parameter space includes variations in the flux content or twist of an emerging flux tube, the initial configuration of any pre-existing magnetic field in the atmosphere, the number of emerging flux tubes etc. In addition, simplifications in the model often allow for larger computational domains, finer resolution, and/or longer simulation times (in terms of characteristic timescales of the system).
In contrast to idealized models, so-called realistic models attempt to capture as much as possible all physical processes that are known to be important for dynamics in the solar atmosphere and convection zone, as well as crucial for synthetic diagnostics. The most developed of this class of models in the flux emergence context is from the work of Martínez-Sykora et al. (2008*, 2009*). Their model of flux emergence employs a computational domain that captures the top few Mm of the convection zone as well as the photosphere, chromosphere, and corona. Convective flows in the convection zone are driven by radiatively cooling at the surface that results from solving the 3D radiative transfer problem and a realistic equation of state is used to account for changes in ionization degree in the plasma. These effects are also captured in the flux emergence models of Cheung et al. (2007a*), Tortosa-Andreu and Moreno-Insertis (2009*), and Stein et al. (2011*). However, the Martínez-Sykora et al.* model also includes a chromosphere treated with radiative transfer that includes scattering (i.e., the source function is a linear combination of the Planck function and the local radiation field) and coronal physics such as field-aligned thermal conductive and optically thin radiative losses. This allows them to compute synthetic diagnostics from their flux emergence simulations for the photosphere, chromosphere (e.g., see Figure 6*), and corona, and therefore to compare with observations of all these layers. One drawback of realistic models is that they are often too prohibitively expensive for extensive parameter studies. The feedback between the many physical processes also make it a challenge to distill general physical insight. For these reasons, idealized studies remain essential for providing complementary guidance.