A New and Fast Way to Reconstruct a Nonlinear Force‐free Field in the Solar Corona

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Abstract

We reexamine the method of upward integration of a nonlinear force-free field (NFFF), which is, as is well known, an ill-posed problem. It can be modified to a well-posed one by the following means: instead of using finite difference to express partial derivatives, we use smooth continuous functions to approach magnetic field values, write down three field components consisting of amplitude functions multiplying morphology functions, and reduce four basic NFFF equations to ordinary differential ones. They are then solved in an asymptotic manner (zeroth-order, first-order, etc.). Considering the physical meaning of α, we found a self-consistent compatibility condition for the boundary values. Furthermore, a computation algorithm is proposed, similar to the usual time-dependent two-dimensional MHD simulation scheme. This algorithm is steady and robust against the noise in the magnetic field (in particular, the transverse field) measurement and is able to deal with concentrated photospheric currents. The algorithm runs very fast on an ordinary PC and lasts only 6 minutes for the 80 × 60 (x × y) mesh up to a height of 80 (= 216,000 km ∼ 0.3 R☉). So it provides a powerful tool for solar scientists to analyze the magnetic field properties of solar active regions and to make predictions of solar activity.

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