A Toroidal Magnetic Field Theorem

In the framework of magnetohydrodynamics the kinematic dynamo problem in a spherical fluid volume as well as in a plane layer is considered. On the premises of a purely toroidal magnetic field a nonlinear evolution equation for the toroidal scalar is derived. In this equation the flow field is constrained in such a way that no poloidal magnetic field can arise, but is otherwise arbitrary; the magnetic diffusivity is assumed to be spherically (horizontally, resp.) symmetric. Solutions of this problem are of particular interest since the magnetic field is confined to the fluid volume and therefore invisible to an external observer. It is proved in this paper that the maximum norm of smooth solutions of this equation decays exponentially fast to zero. Thus, dynamo solutions, i.e. nondecaying solutions, of this type do not exist.