On The Low-Lou Model For The Nonlinear Force-free Magnetic Field

Nonlinear force-free magnetic field is an important mathematical model in astrophysics, a group of nonlinear partial deferential equations, which is frequently used in the theoretical research of solar and stellar magnetic field. In1990, Low and Lou constructed two semi-analytical solutions for the nonlinear force-free magnetic field equations, often used to test the validness of relevant calculations. Based on Low and Low's achievements, this thesis presents a study concerning two aspects of the nonlinear force-free magnetic field.Targeting the ordinary differential equation of second order with an unknown parameter, we have proposed a “parametric shooting method”, being capable of calculating all the eigenvalues in the shooting interval at one time. In addition, the accuracy of the results is examined by exploiting the force-free and divergence-free merits. The parametric shooting method can be regarded as the completion and extension for the solutions due to Low and Lou, enabling to offer more optional numerical force-free magnetic fields. In particular, we have recognized a double helix structure from the newly calculated solutions.We have proposed a new merit for measuring the helical structure of the solar magnetic field, i.e. a geometrical helicity by virtue of torsion and curvature on the field lines. In addition, we reveal the intrinsic relations between the geometrical helicity and the classical helicity. In particular, by exploiting the cylindrical nonlinear force-free magnetic field model, we have proposed a “local grid method” in order to accurately calculate the numerical torsion and curvature on the field lines. The geometrical helicity may be exploited to study the inherent structure in the Low-Lou model.