  The structure and dynamics of the solar corona is dominated by the magnetic
field. In most areas in the corona magnetic forces are so dominant that all
non-magnetic forces like plasma pressure gradient and gravity can be neglected
in the lowest order. This model assumption is called the force-free field
assumption, as the Lorentz force vanishes. This can be obtained by either
vanishing electric currents (leading to potential fields) or the currents are
co-aligned with the magnetic field lines. First we discuss a mathematically
simpler approach that the magnetic field and currents are proportional with one
global constant, the so-called linear force-free field approximation. In the
generic case, however, the relation between magnetic fields and electric
currents is nonlinear and analytic solutions have been only found for special
cases, like 1D or 2D configurations. For constructing realistic nonlinear
force-free coronal magnetic field models in 3D, sophisticated numerical
computations are required and boundary conditions must be obtained from
measurements of the magnetic field vector in the solar photosphere. This
approach is currently of large interests, as accurate measurements of the
photospheric field become available from ground-based (for example SOLIS) and
space-born (for example Hinode and SDO) instruments. If we can obtain accurate
force-free coronal magnetic field models we can calculate the free magnetic
energy in the corona, a quantity which is important for the prediction of
flares and coronal mass ejections. Knowledge of the 3D structure of magnetic
field lines also help us to interpret other coronal observations, e.g.,
EUV-images of the radiating coronal plasma.

Solar and Stellar Astrophysics