CAN WE EXTRAPOLATE A MAGNETIC-FIELD WHEN ITS TOPOLOGY IS COMPLEX

In order to understand various solar phenomena controlled by the magne tic held, such as X-ray bright points, flares and prominence eruptions , the structure of the coronal magnetic field must be known. This requ ires a precise extrapolation of the photospheric magnetic held. Presen tly, only potential or linear force-free field approximations can be u sed easily. A more realistic modelling of the field is still an active research area because of well-known difficulties related to the nonli near mixed elliptic-hyperbolic nature of the equations. An additional difficulty arises due to the complexity of the magnetic field structur e which is caused by a discrete partition of the photospheric magnetic field. This complexity is not limited to magnetic regions having magn etic nulls (and so separatrices) but also occurs in those containing t hin elongated volumes (called Quasi-Separatrix Layers) where the photo spheric field-line linkage changes rapidly. There is a wide range for the thickness of such layers, which is determined by the character (bi polar or quadrupolar) of the magnetic region, by the sizes of the phot ospheric field concentrations and by the intensity of the electric cur rents. The aim of this paper is to analyse the recent nonlinear force- free held extrapolation techniques for complex coronal magnetic fields .