Reconstruction of the solar coronal magnetic field in spherical geometry

Context. High-resolution vector magnetographs either onboard spacecrafts or satellites (HMI/SDO, etc.) or ground based (SOLIS, etc.) now gives access to vector synoptic maps, composite magnetograms made of multiple interactive active regions, and full disk magnetograms. It thus become possible to reconstruct the coronal magnetic field on the full Sun scale. Aims. We present a method for reconstructing the global solar coronal magnetic field as a nonlinear force-free field. It is based on a well-posed Grad-Rubin iterative scheme adapted to spherical coordinates Methods. This method is a natural extension to spherical geometry of the one we previously developed in Cartesian geometry. It is implemented in the code XTRAPOLS, which is a massively parallel code. It allows dealing with the strong constraints put on the computational methods by having to handle the very large amounts of data contained in high-resolution large-scale magnetograms. The method exploits the mixed elliptic-hyperbolic nature of the Grad-Rubin boundary value problem. It uses a finite-difference method for the elliptic part and a method of characteristics for the hyperbolic part. The computed field guarantees to be divergence free up to round-off errors, by introducing a representation in terms of a vector potential satisfying specific gauge conditions. The construction of the latter - called here the restricted DeVore gauge - is described in detail in an appendix. Results. We show that XTRAPOLS performs well by applying it to the reconstruction of a particular semi-analytic force-free field that has already been considered by various authors.