A three-dimensional kinematic dynamo

Abstract
A number of steady (marginal) solutions of the induction equation governing the magnetic field created by a particular class of threedimensional flows in a sphere of conducting fluid surrounded by an insulator are derived numerically. These motions possess a high degree of symmetry which can be varied to confirm numerically that the corresponding asymptotic limit of Braginsky is attained. The effect of altering the spatial scale of the motions without varying their vigour can also be examined, and it is found that dynamo action is at first eased by decreasing their characteristic size. There are, however, suggestions that the regenerative efficiency does not persistently increase to very small length scales, but ultimately decreases. It is further shown that time varying motions, in which the asymmetric components of flow travel as a wave round lines of latitude, can sustain fields having co-rotating asymmetric parts. It is demonstrated that, depending on their common angular velocity, these may exist at slightly smaller magnetic Reynolds numbers than the corresponding models having steady flows and fields. The possible bearing of the integrations on the production of the magnetic field of the Earth is considered, and the implied ohmic dissipation of heat in the core of the Earth is estimated for different values of the parameters defining the model.

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