10 - The Fast Dynamo

Summary

The ‘fast dynamo’ is introduced through the ‘stretch-twist-fold’ mechanism whereby the magnetic field in a circular flux tube may be doubled in intensity. This apparently provides a dynamo whose growth rate is independent of diffusivity η in the limit η —> 0. However, the writhe generated by the stretch-twist-fold cycle is compensated by the appearance of twist within the tube, leading to the development of a pathological structure under infinite iteration of the cycle. A related mechanism for a helicity cascade to fine scales is proposed. The non-existence of smooth fast dynamos is proved, with the inference that fast-dynamo eigenfunctions have fine structure on scales O(η1/2) as η —> 0, non-differentiable in the limit. The homopolar disc dynamo is shown to be slow, rather than fast, when azimuthal current in the disc is taken into account. The Ponomarenko dynamo with discontinuous velocity is shown to be fast with the above pathological structure. Similarly, it is shown that certain space-periodic dynamos can be fast, the magnetic field being concentrated in narrow cigar-shaped regions. The manner in which the space-periodic Archontis dynamo evolves from slow to fast is described.