Two Time-dependent Solutions of Magnetic field Annihilation in Two Dimensions

In this paper, two classes of exact analytic time-dependent soultion of magnetic annihilation for incompressible magnetic fluid, have been obtained by solving the magnetohydrodynamic (MHD) equations directly. The solutions derived here possess scaling property with time $t$ as the scale factor. Based on these two solutions, we find that, for some given inflow fields, the evolution of the annihilating magnetic field can be described by the solutions of certain ordinary differential equations whose variables are dilated simply by time $t$. The relevant evolution characteristics in the process of magnetic annihilation are also revealed.