The hydromagnetic toroidal resonance

The usual solution for the toroidal mode, predicting spatially dependent eigenfrequencies, is shown to be inconsistent. A reinterpretation of this solution, in which the frequency is a constant, violates the magnetohydrodynamic approximation in that it results in an infinite parallel current density. A possible modification of the toroidal equation by introducing an electric field parallel to the ambient magnetic field yields constant eigenvalues that depend on the dimensions of the resonant cavity rather than on the length of individual field lines. The geophysical implications of these results are discussed.