Finite Plasma Beta Three-dimensional Magnetic Field Extrapolation Based on MHD Relaxation Method

TL;DR

This work addresses the limitation of nonlinear force-free field extrapolations in high-β regions by developing a finite-plasma-β magnetohydrostatic extrapolation using an MHD-relaxation framework that balances Lorentz forces and gas pressure. It introduces three numerical schemes to solve the finite-β MHD system and compares them against NLFFF using NOAA AR 12887 data, finding that residual forces are reduced by about 4% relative to NLFFF, with Scheme A performing best. The study reveals a pronounced vertical variation in plasma β, from β ≈ 1 at the photosphere to β ≈ 0.01 in the lower corona for strong-field regions, and β > 10 at higher altitudes in weaker-field regions, underscoring the importance of pressure effects in coronal magnetic-field modeling. Future work will include gravity to capture stratification and will pursue validation against EUV observations, as well as boundary-condition optimizations for finite-β extrapolations, to enhance reliability across the solar atmosphere.

Abstract

Three-dimensional (3D) magnetic field in the solar atmosphere provides crucial information to understand the explosive phenomenon such as solar flares and coronal mass ejections. Since it is still hard that we determine the 3D magnetic field from direct observation, a nonlinear force-free field (NLFFF) extrapolation is one of the best modeling methods that provides 3D magnetic field. However, the method is based on zero-beta 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 high-beta region, such as in chromospheric or lower height layer and in weak field region. To overcome this problem, we need to consider the magnetohydrostatic equilibrium. In this study, we developed a finite plasma beta magnetic field extrapolation method based on 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 3D magnetic field using an observational photospheric vector magnetic field of one solar active region, 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 residual force of the previous NLFFF. We also examined the plasma beta profile along the height and found that, in the core of the active region, plasma beta reaches a local minimum of ~0.01 in the lower corona with beta ~1 at the photosphere.

Figures (7)

• Figure 1: Schematics of three schemes. Horizontal and vertical axes correspond to the numbers of time step of magnetic field evolution and gas pressure evolution, respectively. Red, green, and blue solid arrows show the calculation path of Scheme A, B, and C, respectively. Yellow dashed arrow shows the path of an NLFFF extrapolation.
• Figure 2: Overview of the target AR which we used as test data for magnetic field extrapolation. (a) Full-disk EUV 171 Å image taken by $SDO$/AIA on 2021 October 28 15:58 UT. Red box indicates the region of interest (ROI). (b) EUV 171 Å image of ROI. (c) Magnetic field of ROI. Colored field lines show the NLFFF extrapolation results. The color corresponds to the electric current density. Background grayscale show the normal component of the photospheric magnetic field.
• Figure 3: Temporal evolution of the extrapolated magnetic field structures. (a,b,c) time step of 10000, 20000, and 30000 of the NLFFF as a reference. Color of field lines shows the electric current density normalized by the magnetic field strength. The grayscale at the bottom surface ($z=0$) represents the noromal component of the photospheric magnetic field, the color scale is the same as that of panel (c) in Figure \ref{['fig2']}. (d,e,f) same as (a,b,c) but for Scheme A, respectively. (g,h,i) same as (a,b,c) but for Scheme B, respectively. (j,k,l) same as (a,b,c) but for Scheme C, respectively. (An animation of this figure is available.)
• Figure 4: (a) Temporal evolution of the maximum value of residual force in calculation domain. Black, red, green, and blue plots correspond to the NLFFF and the magnetic field obtained from Schemes A, B, and C, respectively. (b) Temporal evolution of an averaged value of each residual force in calculation domain. Color of the plots is the same as that of panel (a).
• Figure 5: (a) Top view of the magnetic field structure in final-state of Scheme A. Blue field lines are the same as that displayed in Figure \ref{['fig3']}. Orange and cyan field lines correspond to those around the regions A ($100<x<140, y=100$) and B ($10<x<50, y=100$). (b) Same as (a) but for Scheme B. (c) Same as (a) but for Scheme C. (d) 2-dimensional plasma $\beta$ distribution in $x-z$ cross section ($y=100$) for Scheme A. Red, green, blue, and orange lines display the contours of plasma $\beta$ of 10, 1, 0.1, 0.01, respectively. (e) Same as (d) but for Scheme B. (f) Same as (d) but for Scheme C.