An attempt is made to find the boundary tangential components of potential magnetic fields without constructing solutions in the entire domain. In our procedure, the magnetic energy is expressed as a functional of tangential and normal magnetic fields at the boundary and is minimized by the variational principle. This paper reports a preliminary study on two dimensional potential fields above a plane.

We have numerically simulated the evolution of supernova remnant VRO 42.05.01(G166.0+43) in order to understand its morphology m radio and X-ray observations as well as the dynamical interaction of the supernova remnant with the surrounding interstellar medium. The Radio and ROSAT observations suggest that the unusual morphology of VRO 42.05.01 is caused by a supernova blast wave breaking out of the cloud boundary where the supernova event occurred, and expanding into another cloud across a much less dense and presumably hot cavity. Our numerical results are consistent with the interactions of a passing supernova blast wave with the clouds across the cavity. We discuss the implications of our results on the dynamics of VRO 42.05.01 such as the reverse shocks, the rejuvenation of the hot cavity, and the reflected shocks formed by the reverse shock collisions. We have theoretically mapped the radio continuum(1.4 GHz) and X-ray(0.1~2 keV) morphology which are consistent with the observational ones.

Sobolev approximation can be adopted to a macroscopic supersonic motion comparatively larger than a random (thermal) one. It has recently been applied not only to the winds of hot early type stars, but also to envelopes of late type giants and/or supergiants. However, since the ratio of wind velocity to stochastic one is comparatively small in the winds of these stars, the condition for applying the Sobolev approximation is not fulfilled any more. Therefore the validity of the Sobolev approximation must be checked. We have calculated exact P Cygni profiles with various velocity ratios, V∞/Vsto, using the accelerated lambda iteration method, comparing with those obtained by the Sobolev approximation. While the velocity ratio decrease, serious deviations have been occured over the whole line profile. When the gradual increase in the velocity structure happens near the surface of star, the amount of deviations become more serious even at the high velocity ratios. The investigations have been applied to observed UV line profile of CIV in the Copernicus spectrums of ʃ Puppis and NV of τ Sco. In case of τ Sco which has an expanding envelope with the gradual velocity increase in the inner region, The Sobolev approximation has given the serious deviations in the line profiles.