On the invisible dynamo

Abstract

Solutions of the kinematic dynamo problem with a purely toroidal magnetic field in a spherical volume of conducting fluid are called invisible, since the magnetic field is completely confined to the fluid volume and vanishes at the conductor-insulator interface, thus being invisible to an external observer who is not able to penetrate the interface. This paper presents evidence that such solutions do not exist. More precisely, it is shown that the toroidal scalar T, if it satisfies a certain regularity condition, decays monotonically with respect to the norm in the case of a spherical fluid volume with radius R and with respect to the norm in the case of a plane layer. The result is valid for nonsteady radially varying conductivity, possibly moving boundaries and a nonsteady, compressible flow field constrained in such a way that no poloidal magnetic field is generated. If the mixed norm is replaced by the uniform (but weaker) L ₁-norm, the decay result implies boundedness of T—apart from a constant factor—by its initial value. Note that the decay result neither provides decay rates nor excludes steady solutions. The latter, however, can be excluded by a different argument.

Key Words:

• Kinematic dynamo
• antidynamo theorem
• invisible dynamo