Implementation of Boundary Conditions for Nonlinear Force-Free Field Computations Using Vector Potentials

A nonlinear force-free field is solely determined by the normal components of magnetic field and current density on the entire boundary of the domain. Methods using three components of magnetic field suffer from either overspecification of boundary conditions and/or a nonzero divergence B problem. A vector potential formulation eliminates the latter issue, yet poses difficulties in imposing normal components of both magnetic field and current density on the boundary. This challenge arises due to the inability to fix all three components of the vector potential at the boundary while the vector potential within the interior domain undergoes iterative changes. This paper explores four distinct boundary treatment approaches within the vector potential formulation. In two methods, the normal component of the vector potential is adjusted to satisfy the prescribed boundary-normal component of current density at each iteration, while the tangential components remain fixed throughout. In the other two methods, the tangential components of the vector potential are modified at each iteration, while the normal component remains fixed. Each group comprises both first-order and second-order boundary treatments. We conduct a comparative analysis of these methods against our established poloidal-toroidal formulation code and the optimization code in Solar Software. While none of the four methods outperform our poloidal-toroidal formulation code, they demonstrate comparable or superior performance compared to the latter. This research is regarded to provide insights into optimizing boundary conditions for data-driven simulations using vector potentials.