Notes about collapse in magnetohydrodynamics

Abstract

We discuss a problem about magnetic collapse as a possible process for singularity formation of the magnetic field in a finite time within ideal magneto-hydrodynamics for incompressible fluids. This process is very important from the point of view of various astrophysical applications, in particular, as a mechanism of magnetic filaments formation in the convective zone of the Sun. The collapse possibility is connected with compressibility of continuously distributed magnetic field lines. A well-known example of the formation of magnetic filaments in the kinematic dynamo approximation with a given velocity field, first considered by Parker in 1963, rather indicates that the increase in the magnetic field is exponential in time. In the case of the kinematic approximation for the induction equation, the magnetic filaments formation is shown to occur in areas with a hyperbolic velocity profile.