A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. I. Theory

We derive an analytical approximation of nonlinear force-free magnetic field solutions (NLFFF) that can efficiently be used for fast forward-fitting to solar magnetic data, constrained either by observed line-of-sight magnetograms and stereoscopically triangulated coronal loops, or by 3D vector-magnetograph data. The derived NLFFF solutions provide the magnetic field components $B_x(\mathbf{x})$, $B_y(\mathbf{x})$, $B_z(\mathbf{x})$, the force-free parameter $\alpha (\mathbf{x})$, the electric current density $\mathbf{j}(\mathbf{x})$, and are accurate to second-order (of the nonlinear force-free $\alpha$-parameter). The explicit expressions of a force-free field can easily be applied to modeling or forward-fitting of many coronal phenomena.