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

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 model p1Y22DM12 exhibited slow convergence. Seeking more planar flow dynamos, they also considered converting some historical flows to planar flows by a process they termed ‘planarising‘. In particular they studied one of 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 a dynamo at low truncation levels and low critical magnetic Reynolds number R c, properties enhanced by the flow being a helical Beltrami flow. The original aim of the present project was to find a planar flow dynamo with faster convergence than p1Y22DM12. Whilst the PAS flow can only be partly planarised, we have constructed two modifications, labelled biPAS and quasiPAS, which can be fully planarised. We have studied 128 models using the PAS, biPAS, quasiPAS flows, and their part and full planarisations. We have verified 12 known dynamos, found 84 new dynamos, but found no dynamos with fully planar flows. The partly-planarised PAS, and the biPAS- and quasiPAS-based flows are not Beltrami. But 20 (normalised using rms( v ), or 22 using max | v |) of the associated dynamos have lower R c than the original Beltrami PAS dynamos from which they were derived, showing that the Beltrami property is not essential for low R c. A comparison is also made herein with optimal enstrophy-normalised dynamos.