On purely toroidal dynamo magnetic fields caused by conductivity variations.

Solutions of the induction equation with a purely toroidal magnetic field in a spherical conductor are of special interest since they disclose the possibility of a dynamo operating in a celestial body which remains invisible to an external observer. A necessary condition for such solutions to exist are flow fields and conductivity distributions constrained in such a way that no poloidal field can arise. Otherwise these quantities are assumed to be arbitrary; we allow, in particular, non-radial conductivity variations in order to facilitate dynamo action. This article presents evidence that such solutions do not exist: based on a solution of the constraint a nonlinear evolution equation for the toroidal scalar T is derived. It is proved that sufficiently regular solutions of this equation cannot increase (in time) in the norm [image omitted], where R is the radius of the spherical conductor. Steady solutions are likewise excluded. [ABSTRACT FROM AUTHOR]