ABC flows then and now

Abstract

We review cellular space-periodic dynamos without scale separation, starting with early work in the 1980s on ABC flows with prescribed steady velocity fields u  = (A sin z + C cos y, B sin x + A cos z, C sin y + B cos x). These naturally led to work done in the 1990s together with Mike Proctor on 2-D time-dependent versions which gave strong numerical evidence for the existence of fast dynamos growing on the flow turnover timescale. Similar calculations were subsequently performed for a spherical shell geometry jointly with Rainer Hollerbach. Also in the 1990s other studies began to take into account the back reaction of the Lorentz force when the flow rather than being prescribed was instead allowed to evolve in response to a forcing of the above ABC form. The dynamos that resulted were mostly filamentary and showed a disconcerting tendency to equilibrate with total magnetic energy much less than total kinetic energy in the low diffusivity limit relevant for astrophysics. The remarkable discovery by Archontis in 1999 of a non-filamentary dynamo with almost equal magnetic and kinetic energies showed that the unfavourable scalings for the filamentary case can be overcome; this dynamo used an ABC forcing with the cosines left out. Since then several authors have been struggling with partial success to understand just how this state of affairs comes about. Most recently efforts have been made to produce other examples of this type of dynamo, to investigate why the Archontis case is robust over a wide range of magnetic Prandtl numbers ν/η, and above all to understand its remarkable stability at very low diffusivities when non-magnetic flows are almost always unstable.

Keywords:

• Dynamo theory
• MHD

Acknowledgements

I am grateful to Robert Cameron for his persistence in convincing me that non-filamentary dynamos were possible and for his collusion in understanding how they come about, and to David Hughes and Charlie Macaskill for help with graphical issues. I would also like to thank Mike Proctor for three most enjoyable collaborations dealing with magnetoconvection, hexagons and fast dynamos, the last of which I have built into this article. I hope more are in store!