Magnetohydrodynamic equilibria and cusp formation at an X-type neutral line by footpoint shearing

Some results concerning two-dimensional (∂/∂y = 0) magnetohydrodynamic equilibria in the presence of an X-type neutral line are presented. It is shown that in the framework of ideal MHD the latter is structurally unstable: shearing of an initial potential poloidal magnetic field results in the splitting of an X-point into a current sheet with cusp-points (or Y-points for some particular cases) at its ends. At the same time, one finds the formation of current sheets all along the separatrices even for a footpoint shearing displacement that is finite as the separatrix is approached. Scale-invariant or similarity solutions for the poloidal magnetic field inside the cusp are found. They are valid in the vicinity of a cusp-point on lengths much smaller than the global scale. These solutions predict fractional power-law singularities in the current density at the separatrix as well as a universal behavior for the poloidal magnetic field strength B_p near a cusp point, namely B_p ~∼ r^1-3. Discussion is presented regarding the global geometry of the poloidal magnetic field produced by different types of footpoint shearing motions.