Optimization approach for the computation of magnetohydrostatic coronal equilibria in spherical geometry

Abstract

Context. This paper presents a method which can be used to calculate models of the global solar corona from observational data. Aims. We present an optimization method for computing nonlinear magnetohydrostatic equilibria in spherical geometry with the aim to obtain self-consistent solutions for the coronal magnetic field, the coronal plasma density and plasma pressure using observational data as input. Methods. Our code for the self-consistent computation of the coronal magnetic fields and the coronal plasma solves the non-forcefree magnetohydrostatic equilibria using an optimization method. Previous versions of the code have been used to compute nonlinear force-free coronal magnetic fields from photospheric measurements in Cartesian and spherical geometry, and magnetostaticequilibria in Cartesian geometry. We test our code with the help of a known analytic 3D equilibrium solution of the magnetohydrostatic equations. The detailed comparison between the numerical calculations and the exact equilibrium solutions is made by using magnetic field line plots, plots of density and pressure and some of the usual quantitative numerical comparison measures. Results. We find that the method reconstructs the equilibrium accurately, with residual forces of the order of the discretisation error of the analytic solution. The correlation with the reference solution is better than 99.9% and the magnetic energy is computed accurately with an error of <0.1%. Conclusions. We applied the method so far to an analytic test case. We are planning to use this method with real observational data as input as soon as possible.

A first step in reconstructing the solar corona self-consistently with a magnetohydrostatic model during solar activity minimum

“…by using coronagraph images from different viewpoints, as provided by the STEREOmission, for a tomographic inversion. Wiegelmann & Neukirch (2006) and Wiegelmann et al (2007) developed codes to solve the MHS-equations (1)-(3) numerically in cartesian and spherical geometry, respectively. The spherical MHS-code generalizes the global nonlinear force-free code developed by Wiegelmann (2007).…”

The magnetic field in the solar atmosphere

“…This allows a self-consistent modeling of magnetic field and plasma e.g., in the high-β regimes containing the photosphere and lower chromosphere, and beyond the source surface in global simulations. Generally, these equilibria require the computation of non-linear equations, which are numerically even more challenging (and slower converging) than the set of NLFF equations, in particular, in a mixed-β plasma (see Wiegelmann and Neukirch 2006;Wiegelmann et al 2007, for an implementation in cartesian and spherical geometry, respectively).…”

Three-dimensional solutions of the magnetohydrostatic equations: rigidly rotating magnetized coronae in cylindrical geometry

“…Only very few analytical solutions are known and even using numerical methods for calculating 3D MHS solutions is usually far from straightforward (e.g. Wiegelmann & Neukirch 2006;Wiegelmann et al 2007). …”