On three dimensional steady super-Alfv\'{e}nic magnetohydrodynamics shocks with aligned fields

Abstract

The coupled motion between the hydrodynamic flow and magnetic field introduces significant complexity into the structure of the magnetohydrodynamic (MHD) equations. A key factor contributing to this complexity is the presence of Alfv\'en waves, which critically influences the character of the flow and makes the problem considerably more challenging. Within the framework where the magnetic field is everywhere parallel to the flow velocity, we give an effective decomposition of the steady MHD equations in terms of the deformation tensor and the modified vorticity, where the modification in the vorticity is to record the effect of the Lorentz force on the velocity field. The existence and structural stability of the super-Alfv\'{e}nic cylindrical transonic shock solutions for the steady MHD equations are established under three-dimensional perturbations of the incoming flow and the exit total pressure (kinetic plus magnetic).