5 Nonlinear Force-free Fields in 3D

In the following section, we briefly discuss some general properties of force-free fields, which are relevant for solar physics, like the magnetic helicity, estimations of the minimum and maximum energy a force-free field can have for certain boundary conditions and investigations of the stability. Such properties are assumed to play an important role for solar eruptions. The Sun and the solar corona are of course three-dimensional and for any application to observed data, configurations based on symmetry assumptions (as used in Section 3) are usually not applicable. The numerical treatment of nonlinear problems, in particular in 3D, is significantly more difficult than linear ones. Linearized equations are often an over-simplification which does not allow the appropriate treatment of physical phenomena. This is also true for force-free coronal magnetic fields and has been demonstrated by comparing linear force-free configurations (including potential fields, where the linear force-free parameter α is zero).

Computations of the photospheric α distribution from measured vector magnetograms by Equation (14View Equation) show that α is a function of space (see, e.g., Pevtsov et al., 1994; R egnier et al., 2002Jump To The Next Citation Point; DeRosa et al., 2009Jump To The Next Citation Point). Complementary to this direct observational evidence that nonlinear effects are important, there are also theoretical arguments. Linear models are too simple to estimate the free magnetic energy. Potential fields correspond to the minimum energy configuration for a given magnetic flux distribution on the boundary. Linear force-free fields contain an unbounded magnetic energy in an open half-space above the photosphere (Seehafer, 1978), because the governing equation in this case is the Helmholtz (wave) equation (Equation (17View Equation)) whose solution decays slowly toward infinity. Consequently both approaches are not suitable for the estimation of the magnetic energy, in particular not an estimation of the free energy a configuration has in excess of a potential field.

 5.1 Magnetic helicity
 5.2 Energy principles
 5.3 Maximum energy
 5.4 Stability of force-free fields
 5.5 Numerical stability investigations

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