A class of self-sustaining dissipative spherical dynamos

Abstract The present paper treats rigorously the dynamo equations describing the effects of the internal motion of a bounded volume of incompressible fluid with nonzero ohmic resistivity on the magnetic field produced by electric currents in that fluid. The procedure involves representing an arbitrary solenoidal vector field in terms of two scalars, analogous to the representation of an arbitrary irrotational field as the gradient of a single scalar. The dynamo equations are reduced to scalar heat equations for the two field scalars, the coupling between them taking the form of a heat source term. Precise results about the magnetic field can be obtained from these heat equations with the help of several variational inequalities analogous to Rayleigh's variational estimate for the fundamental frequency of a vibrating system. The main result is the explicit construction of a large class of continuously differentiable fluid velocities capable of indefinitely maintaining or amplifying the dipole moment of the external magnetic field. These motions all involve periods of stasis in the fluid, and cannot, therefore, be expected to occur in the earth's core. It is believed that it will be possible eventually to obtain more exact bounds than those presented here for the magnetic field components with high wave number, thus eliminating the need for such periods of stasis. The fluid motions shown capable of dynamo maintenance are of this sort: a toroidal shear symmetric about the z axis proceeds long enough to produce from P1z1, the lowest poloidal free-decay mode symmetric about that axis, a very large energy in T1z1, the lowest toroidal free-decay mode with such symmetry. During a period of stasis, everything else almost dies out, leaving a field which is largely T1z1. Then almost any velocity which has a radial component and is not axisymmetric about the z axis will regenerate P1z1 and the external dipole moment. A critique of some previous attempts to produce dissipative self-regenerative spherical dynamos is included. The techniques which lead to the existence of self-sustaining dynamos produce other results about the dynamo equations, most of which are to the author's knowledge either new or not previously precisely formulated. These results are listed below. 1. (i) Fluid motions in a sphere can be regarded as bounded linear operators on the Hilbert space of magnetic fields with finite total energy. 2. (ii) The free-decay modes in the rigid sphere are complete in that space. 3. (iii) The magnetic effect of a given fluid motion on a given initial field depends continuously on the resistivity ϱ of the fluid even at ϱ = 0. 4. (iv) The magnetic effect of any motion can be approximated with arbitrary accuracy by replacing it by a series of rapid jerks interpersed with periods of rest. 5. (v) The effect of a rigid rotation of the fluid is to rotate the magnetic field while that field decays as if the fluid were motionless. This effect can be approximated with arbitrary accuracy by rotating all but a sufficiently thin shell at the surface, even if every point on the surface remains fixed and a large shear develops in the shell. 6. (vi) If the fluid velocity has no radial component, the poloidal magnetic field decays as rapidly as if the fluid were at rest. If, further, no poloidal field is initially present, the toroidal field decays as rapidly as if the fluid were at rest. 7. (vii) Dynamo maintenance is impossible if the local strain-rate of the fluid is always and everywhere less than the decay rate of P1z1 when the velocity of the fluid is zero.