Dynamos driven by modified Beltrami flows, and a search for related planar flow dynamos

Abstract

Bachtiar, Ivers and James [Planar velocity dynamos in a sphere. Proc. R. Soc. Lond. 2006, A462, 2439–2456]showed that a planar fluid velocity v can support dynamo action in a conducting sphere, however their modelp1Y22DM12 exhibited slow convergence. Seeking more planar flow dynamos, they also considered convertingsome historical flows to planar flows by a process they termed ‘planarising‘. In particular they studied oneof many flows considered by Pekeris, Accad and Shkoller [Kinematic dynamos and the earth's magnetic field. Phil. Trans. R. Soc. Lond. 1973, A275, 425–461]. This PAS flow was chosen because it produced adynamo at low truncation levels and low critical magnetic Reynolds number R c, properties enhanced bythe flow being a helical Beltrami flow. The original aim of the present project was to find a planar flowdynamo with faster convergence than p1Y22DM12. Whilst the PAS flow can only be partly planarised, wehave constructed two modifications, labelled biPAS and quasiPAS, which can be fully planarised. We havestudied 128 models using the PAS, biPAS, quasiPAS flows, and their part and full planarisations. We haveverified 12 known dynamos, found 84 new dynamos, but found no dynamos with fully planar flows. Thepartly-planarised PAS, and the biPAS- and quasiPAS-based flows are not Beltrami. But 20 (normalisedusing rms( v ), or 22 using max | v |) of the associated dynamos have lower R c than the original Beltrami PASdynamos from which they were derived, showing that the Beltrami property is not essential for low R c. Acomparison is also made herein with optimal enstrophy-normalised dynamos.