Ribbons characterize magnetohydrodynamic magnetic fields better than lines: a lesson from dynamo theory

Abstract

Blackman & Brandenburg argued that magnetic helicity conservation in dynamo theory can in principle be captured by diagrams of mean field dynamos when the magnetic fields are represented by ribbons or tubes, but not by lines. Here we present such a schematic ribbon diagram for the $\alpha^2$ dynamo that tracks magnetic helicity and provides distinct scales of large scale magnetic helicity, small scale magnetic helicity, and kinetic helicity involved in the process. This also motivates our construction of a new "2.5 scale" minimalist generalization of the helicity-evolving equations for the \alpha^2 dynamo that separately allows for these three distinct length scales while keeping only two dynamical equations. We solve these equations and, as in previous studies, find that the large scale field first grows at a rate independent of the magnetic Reynolds number R_M before quenching to an R_M dependent regime. But we also show that the larger the ratio of the wavenumber where the small scale current helicity resides to that of the forcing scale, the earlier the non-linear dynamo quenching occurs, and the weaker the large scale field is at the turnoff from linear growth. The harmony between the theory and the schematic diagram exemplifies a general lesson that magnetic fields in MHD are better visualized as two-dimensional ribbons (or pairs of lines) rather than single lines.