Study Of The Theoretical Solar Magnetic Field Extrapolation Method

Understanding solar magnetic field is the predominant factor in studying solar physics. However, precise measurements of solar magnetic fields so far are still confined to the thin layer of solar photosphere. In order to understand the nature of the coronal magnetic fields, it becomes necessary to extrapolate the coronal magnetic fields based on theoretical models using observed photospheric magnetograms as boundary conditions. Taking force-free assumptions, three kinds of models are currently being used: the potential field model, the linear force-free field model and the non-linear force-free (NLFF) field model. Enclosed in this thesis, I have completed following studies related to the extrapolation of solar magnetic field.1. The progress and development of the solar magnetic field extrapolation methods are reviewed briefly. Basic theoretical hypothesis and equations of extrapolation methods are introduced. Methods of extrapolations based on three models, that are, potential field model, linear force-free field model and nonlinear force-free field model, are introduced, with specific examples given and their astrophysical applications mentioned. Problems and related progress in the extrapolation methods are also briefly addressed.2. A specific nonlinear force-free field model is studied and examined. In 2000, Yan & Sakurai proposed a boundary integral method to extrapolate nonlinear force-free fields. Basing on this work, Yan & Li (2006) proposed a new direct boundary integral formulation. Using an optimal method, they can obtain the numerical solution at any position in semi-space. We first test the existence of the solution of the direct boundary integral equation. Using the semi-analytical solutions of NLFF magnetic field in Low & Lou (1990) as the known solution, we use the BROYDEN function and the NEWTON function in IDL to solve the inverse function of the direct boundary equation and get the values of the pseudo force-free factors. We find that we can always find pseudo force-free factors, putting which into the direct boundary equation recovers the analytical solution. So the existence of the equation is validated.3. We further test the dependence of the solution on the four main parameters used in the calculation. The results show that the solution mainly depends on the initial values of the pseudo force-free factor, and there is no strong dependence on other parameters.4. We give an overall accuracy of this method by comparing the numerical results with the exact analytical solution. We conclude that this method provides better solutions at the lower layers than those at higher ones.