Analytical solutions for the current driven by a rotating magnetic field in a spherical plasma

Abstract

The steady currents driven in a spherical plasma by a rotating magnetic field via the Hall effect are studied analytically. The total field is shown to be symmetric across the origin. Integral relationships are obtained between Ohmic dissipation, angular momentum and the oscillating axial current density. The topology of the sum of a Hill's vortex field and a rotating field is documented. Analytical solutions for the driven current are obtained by expansion for the limits corresponding to small rotation frequency, to small number density, to large rotating-field magnitude, to small resistivity, and to small rotating-field magnitude combined with very small resistivity. The latter solution, relevant to the reactor limit, indicates that, with control of the vertical field magnitude, an MHD equilibrium can be generated with total current any fraction of the currentcorresponding to synchronous rotation of the electrons. Oscillating currents sufficient to drive the synchronous current are determined.