Nonlinear Force-Free Reconstruction of the Global Solar Magnetic Field: Methodology

We present a novel numerical method that allows the calculation of nonlinear force-free magnetostatic solutions above a boundary surface on which only the distribution of the normal magnetic field component is given. The method relies on the theory of force-free electrodynamics and applies directly to the reconstruction of the solar coronal magnetic field for a given distribution of the photospheric radial field component. The method works as follows: we start with any initial magnetostatic global field configuration (e. g. zero, dipole), and along the boundary surface we create an evolving distribution of tangential (horizontal) electric fields that, via Faraday's equation, give rise to a respective normal-field distribution approaching asymptotically the target distribution. At the same time, these electric fields are used as boundary condition to numerically evolve the resulting electromagnetic field above the boundary surface, modeled as a thin ideal plasma with non-reflecting, perfectly absorbing outer boundaries. The simulation relaxes to a nonlinear force-free configuration that satisfies the given normal-field distribution on the boundary. This is different from existing methods relying on a fixed boundary condition - the boundary evolves toward the a priori given one, at the same time evolving the three-dimensional field solution above it. Moreover, this is the first time that a nonlinear force-free solution is reached by using only the normal field component on the boundary. This solution is not unique, but it depends on the initial magnetic field configuration and on the evolutionary course along the boundary surface. To our knowledge, this is the first time that the formalism of force-free electrodynamics, used very successfully in other astrophysical contexts, is applied to the global solar magnetic field.