How to Impose the Boundary Conditions Operatively in Force-Free Field Solvers

To construct a coronal force-free magnetic field, we must impose the boundary normal current density (or three components of magnetic field) as well as the boundary normal field at the photosphere as boundary conditions. The only method that is known to implement these boundary conditions exactly is the method devised by Grad and Rubin (1958). However, the Grad-Rubin method and all its variations (including the fluxon method) suffer from convergence problems. The magnetofrictional method and its variations are more robust than the Grad-Rubin method in that they at least produce a certain solution irrespective of whether the global solution is compatible with the imposed boundary conditions. More than often, the influence of the boundary conditions does not reach beyond one or two grid planes next to the boundary. We have found that the 2D solenoidal gauge condition for vector potentials allows us to implement the required boundary conditions easily and effectively. The 2D solenoidal condition is translated into one scalar function. Thus, we need two scalar functions to describe the magnetic field. This description is quite similar to the Chandrasekhar-Kendall representation, but there is a significant difference between them. In the latter, the toroidal field has both Laplacian and divergence terms while in ours, it has only a 2D Laplacian term. The toroidal current density is also expressed by a 2D Laplacian. Thus, the implementation of boundary normal field and current are straightforward and their effect can permeate through the whole computational domain. In this paper, we will give detailed math involved in this formulation and discuss possible lateral and top boundary conditions and their meanings.