Modeling of a linear force-free magnetic field in a bounded domain

A method for the reconstruction of the linear force-free magnetic field in a bounded domain (as a rectangular box, Ω) is presented here. The Dirichlet boundary-value problem for the Helmholtz equation is solved for the B_z component specified at the Ω boundary. Chebyshev's iteration method with the optimal rearrangement of the iteration parameters sequence was used. The solution is obtained as for the positive-definite, and for the non-sign-definite difference analogue of the differential operator ∇²u + α²u. Specifying two scalar functions, B_x and B_y on the intersection of the lateral part of the Ω boundary with one selected plane z = constant, and using B_z inside the Ω, we have found B_x and B_y throughout Ω. The algorithm was tested with the numerical procedure which gives the analytic solution B of the linear force-free field (LFFF) equations for the dipole in a half-space. The root-mean-square deviation of the analytic solution B from the calculated B′ does not exceed 1.0%. Boundary conditions for the B′ calculation were taken as given by the analytic LFFF solution B. Comparison of B′ with B″, which was calculated by the potential non-pholospheric boundary conditions, show that they differ significantly. Thus, the specification of boundary conditions at non-photospheric boundaries of the volume under consideration is of particular importance when modeling the LFFF in a bounded volume. The algorithm proposed here allows one to use the information about magnetic fields in the corona for the modeling of LFFF in a limited domain above an active region on the Sun.