Knowledge regarding the coronal magnetic field is important for the understanding of many phenomena, like flares and coronal mass ejections. Because of the low plasma beta in the solar corona, the coronal magnetic field is often assumed to be force-free and we use photospheric vector magnetograph data to extrapolate the magnetic field into the corona with the help of a nonlinear force-free optimization code. Unfortunately, the measurements of the photospheric magnetic field contain inconsistencies and noise. In particular, the transversal components (say B x and B y ) of current vector magnetographs have their uncertainties. Furthermore, the magnetic field in the photosphere is not necessarily force free and often not consistent with the assumption of a force-free field above the magnetogram. We develop a preprocessing procedure to drive the observed nonforce-free data towards suitable boundary conditions for a force-free extrapolation. As a result, we get a data set which is as close as possible to the measured data and consistent with the force-free assumption.

The resonances that appear in the linear compressible MHD formulation of waves are studied for equilibrium states with flow. The conservation laws and the jump conditions across the resonance point are determined for ID cylindrical plasmas. For equilibrium states with straight magnetic field lines and flow along the field lines the conserved quantity is the Eulerian perturbation of total pressure. Curvature of the magnetic field lines and/or velocity field lines leads to more complicated conservation laws. Rewritten in terms of the displacement components in the magnetic surfaces parallel and perpendicular to the magnetic field lines, the conservation laws simply state that the waves are dominated by the parallel motions for the modified slow resonance and by the perpendicular motions for the modified Alfv6n resonance.The conservation laws and the jump conditions are then used for studying surface waves in cylindrical plasmas. These waves are characterized by resonances and have complex eigenfrequencies when the classic true discontinuity is replaced by a nonuniform layer. A thin non-uniform layer is considered here in an attempt to obtain analytical results. An important result related to earlier work by Hollweg et al. (1990) for incompressible planar plasmas is found for equilibrium states with straight magnetic field lines and straight velocity field lines. For these equilibrium states the incompressible and compressible surface waves have the same frequencies at least in the long wavelength limit and there is an exact correspondence with the planar case. As a consequence, the conclusions formulated by Hollweg et al. still hold for the straight cylindrical case. The effects of curvature are subsequently considered.

A basic procedure is presented for dealing with the resonance problems that appear in MHD of which resonant absorption of waves at the Alfv~n resonance point is the best known example in solar physics. The procedure avoids solving the full fourth-order differential equation of dissipative MHD by using connection formulae across the dissipation layer.

The structure and dynamics of the solar corona is dominated by the magnetic field. In most areas in the corona magnetic forces are so dominant that all non-magnetic forces like plasma pressure gradient and gravity can be neglected in the lowest order. This model assumption is called the force-free field assumption, as the Lorentz force vanishes. This can be obtained by either vanishing electric currents (leading to potential fields) or the currents are co-aligned with the magnetic field lines. First we discuss a mathematically simpler approach that the magnetic field and currents are proportional with one global constant, the so-called linear force-free field approximation. In the generic case, however, the relation between magnetic fields and electric currents is nonlinear and analytic solutions have been only found for special cases, like 1D or 2D configurations. For constructing realistic nonlinear force-free coronal magnetic field models in 3D, sophisticated numerical computations are required and boundary conditions must be obtained from measurements of the magnetic field vector in the solar photosphere. This approach is currently of large interests, as accurate measurements of the photospheric field become available from ground-based (for example SOLIS) and space-born (for example Hinode and SDO) instruments. If we can obtain accurate force-free coronal magnetic field models we can calculate the free magnetic energy in the corona, a quantity which is important for the prediction of flares and coronal mass ejections. Knowledge of the 3D structure of magnetic field lines also help us to interpret other coronal observations, e.g., EUV-images of the radiating coronal plasma.

We develop a new methodology that can determine magnetic helicity flux as well as Poynting flux across the photosphere based on magnetograph observation. By applying this method, we study the injection mechanism of magnetic helicity and magnetic free energy into the solar corona. In order to derive the helicity and energy fluxes, first the velocity tangential to the solar surface is constructed by applying a correlation tracking technique on the magnetic observation, and second, the velocity component normal to the photosphere is derived from the condition that the magnetic evolution must be consistent with the induction equation. Through this procedure, we can determine the helicity and energy fluxes separately for the shear motion effect and for the flux emergence effect. Based on this new method, NOAA Active Region 8100 was analyzed from 1997 November 1 to 5 using data observed by the Solar and Heliospheric Observatory Michelson Doppler Interferometer and the vector magnetograph at the National Astronomical Observatory of Japan (NAOJ) in Tokyo. The results indicate that the photospheric shear motion and the flux emergence process have equally contributed to the helicity injection and have supplied magnetic helicity of opposite signs into this active region.