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