A lagrangian study of dynamics and singularity formation at magnetic null points in ideal three-dimensional magnetohydrodynamics

The ideal three-dimensional incompressible magnetohydrodynamics equations are analyzed at magnetic null points using a generalization of a method from fluid dynamics. A closed system of ordinary differential equations governing the evolution of traces of matrices associated with the fluid velocity and magnetic field gradients are derived using a model for the pressure Hessian. It is shown rigorously that the eigenvalues of the magnetic field gradient matrix are constant in time and that, in the model, a finite time singularity occurs with characteristics similar to the magnetic field-free case.