In this review, we tried to give an overview of force-free magnetic fields, particularly model assumptions, which are important for understanding the physics of the solar corona. While the underlying mathematical equations describe stationary states and look relatively simple, solving them is by no means a trivial problem because of the nonlinear nature of the problem. Exact solutions are only available for further simplifications, like linearizing the equations or to restrict to 1D/2D for the nonlinear case. For force-free configurations in 3D, we know that (for given flux distributions in the photosphere) the magnetic field energy is bounded from below by a potential field. An upper-limit for the energy is more difficult to obtain. While the Aly–Sturrock conjecture (Section 5.3) claims that the upper limit is for the configurations with all magnetic field lines open, Choe and Cheng (2002) constructed solutions with energies above the Aly–Sturrock limit. These configurations contain discontinuities and the debate of the validity of the Aly–Sturrock limit is ongoing (Hu, 2004; Wolfson et al., 2012).

For practical computations of the 3D-field in the solar corona, one has to use numerical computations and several codes have been developed, compared, and applied. As input these codes require measurements of the magnetic field vector in the solar photosphere. However, the transverse field component contains an ambiguity in the azimuth, which has to be resolved before the data can be used for coronal magnetic field modeling. The accuracy of photospheric measurements is lower for the transverse field component compared with the line-of-sight field, and in weak field regions measurements and azimuth ambiguity removal are less trustworthy. Consequently the majority of coronal force-free field models are carried out in active regions, although methods for full-disk computations have been developed too. A further complication of using photospheric measurements as the boundary condition for force-free extrapolations is that the photospheric magnetic field is not necessarily consistent with the force-free assumption. Possible solutions are to use only the vertical magnetic field and the vertical electric current as boundary conditions, as done for the Grad–Rubin approach, to preprocess the photospheric measurements with the aim to make them compatible with force-free and other physical requirements, or to allow changes of the transverse magnetic field during the iteration of a force-free field. The latter approach has been implemented in the optimization approach and allows us to take measurement errors into account.

A major source for future research on force-free fields is SDO/HMI, which measures the photospheric magnetic field vector on the full disk, which in principle allows us to compute global coronal models as well as selecting appropriate isolated active regions with a sufficiently large field-of-view. Research on Stokes inversion, azimuth ambiguity removal, and force-free modeling for SDO/HMI data is ongoing. Another important aspect on coronal modeling is the comparison of force-free models as extrapolated from photospheric measurements with coronal images as observed, for example, with the Atmospheric Imaging Assembly (AIA; Lemen et al., 2012) on SDO. On the one hand, such a comparison is important to validate the models (see DeRosa et al., 2009, for details), and, on the other hand, the 3D models help to interpret the observations. With the 3D structure of magnetic loops from the models in hand, one has important tools for modeling of plasma loops, and may gain understanding of coronal heating and plasma flows along the loops. Further steps on the research of eruptive phenomena like flares and CMEs are planned with time-dependent MHD simulations. Force-free models are planned to be used as initial equilibria, which are disturbed by photospheric plasma flows (which can be deduced, e.g., from measurements with SDO/HMI). The temporal evolution and the potential occurrence of eruptions can be investigated with ideal or resistive MHD simulations in comparison with observations. Questions are if or to which extent the configurations remain approximately force-free during eruptions, the role of thin current sheets and discontinuities, and the energy and helicity content. We aim to report about the progress in these aspects in an update of this review in due time.

Living Rev. Solar Phys. 9, (2012), 5
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