Finite Plasma β Three-dimensional Magnetic Field Extrapolation Based on the Magnetohydrodynamic Relaxation Method

The three-dimensional (3D) magnetic field in the solar atmosphere provides crucial information to help understand explosive phenomena such as solar flares and coronal mass ejections. Since it is still hard to determine the 3D magnetic field from direct observation, a nonlinear force-free field (NLFFF) extrapolation is one of the best modeling methods that provides the 3D magnetic field. However, the method is based on a zero-β assumption, i.e., the model ignores the gas-pressure gradient and gravitational force. The magnetic field based on an NLFFF is not well reconstructed in the high-β region, such as in the chromospheric or lower-height layers and in the weak-field region. To overcome this problem, we need to consider the magnetohydrostatic equilibrium. In this study, we developed a finite plasma β magnetic field extrapolation method based on a magnetohydrodynamic relaxation method. In our method, we consider a force balance of the Lorentz force and the gas pressure. We tested three different schemes and extrapolated the 3D magnetic field using an observational photospheric vector magnetic field of one solar active region (AR), which is a quadrupole complex sunspot group and well studied with an NLFFF. The verification of three schemes is performed by comparing the residual force, and we concluded that our methods reduce ∼4% of the residual force of the previous NLFFF. We also examined the plasma β profile along the height and found that, in the core of the AR, plasma β reaches a local minimum of ≈0.01 in the lower corona with β ≈ 1 at the photosphere.