An example of the resistive MHD equations used in the paper of Lionello et al. (2009) for constructing a steady-state solution are:

where are the magnetic field, electric current density, electric field, plasma density, velocity, pressure, temperature, gravitational acceleration, resistivity, kinematic viscosity, radiative losses, electron and proton number densities, polytropic index, coronal heating term, heat flux, and wave pressure, respectively. Different formulations of the MHD equations may be seen in the papers of DeVore and Antiochos (2008), Downs et al. (2010), and Feng et al. (2012) where key differences are the inclusion of resistive or viscous terms and the use of adiabatic or non-adiabatic energy equations.To date, non-eruptive 3D global MHD simulations have been used to model the solar corona through two distinct forms of simulation. Firstly, there is the construction of steady state coronal solutions from fixed photospheric boundary conditions (Riley et al., 2006; Vásquez et al., 2008; Lionello et al., 2009; Downs et al., 2010). To compute these solutions, the system is initialised by (i) specifying the photospheric distribution of flux (often from observations), (ii) constructing an initial potential magnetic field and, finally, (iii) superimposing a spherically symmetric solar wind solution. Equations (44) – (50) (or their equivalents) are then integrated in time until a new equilibrium is found. A key emphasis of this research is the direct comparison of the resulting coronal field and plasma emission with that seen in observations (Section 4.4). In the paper of Vásquez et al. (2008) a quantitative comparison of two global MHD models (Stanford: Hayashi, 2005, and Michigan: Cohen et al., 2007), with coronal densities determined through rotational tomography was carried out. In general the models reproduced a realistic density variation at low latitudes and below , however, both had problems reproducing the correct density in the polar regions. In contrast to this construction of steady state MHD solutions, advances in computing power have recently enabled global non-eruptive MHD simulations with time-dependent photospheric boundary conditions. These boundary conditions have been specified both in an idealised form (e.g., Lionello et al., 2005; Edmondson et al., 2010) and from synoptic magnetograms (Linker et al., 2011). An initial application of these models has been to simulate the Sun’s open flux and coronal holes: see Section 4.2.

To quantify the difference between the magnetic field produced in global MHD simulations and PFSS extrapolations, Riley et al. (2006) carried out a direct comparison of the two techniques. The results from each technique are compared in Figure 12 for periods of low activity (top) and high activity (bottom). An important limitation of this study was that, for both the PFSS model and MHD model, only the radial magnetic field distribution at the photosphere was specified in order to construct the coronal field. Under this assumption, a good agreement was found between the two fields where the only differences were (i) slightly more open flux and larger coronal holes in the MHD simulation, and (ii) longer field lines and more realistic cusp like structures in the MHD simulations. While the two set of simulations closely agree, as discussed by the authors it is important to note that if vector magnetic field measurements are applied on a global scale as an additional lower boundary condition, then significant differences may result.

While global MHD models have mostly been applied to the Sun, Cohen et al. (2010) applied the models to observations from AB Doradus. Through specifying the lower boundary condition from a ZDI magnetic map and also taking into account the rotation (but not differential rotation) of the star the authors showed that the magnetic structure deduced from the MHD simulations was very different from that deduced from PFSS models. Due to the rapid rotation a strong azimuthal field component is created. They also considered a number of simulations where the base coronal density was varied and showed that the resulting mass and angular momentum loss, which are important for the spin-down of such stars, may be orders of magnitude higher than found for the Sun.

Living Rev. Solar Phys. 9, (2012), 6
http://www.livingreviews.org/lrsp-2012-6 |
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