Ribbons characterize magnetohydrodynamic magnetic fields better than lines: a lesson from dynamo theory
Abstract
Blackman and 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 α^{2} dynamo that tracks magnetic helicity and provides distinct scales of largescale magnetic helicity, smallscale magnetic helicity, and kinetic helicity involved in the process. This also motivates our construction of a new `2.5 scale' minimalist generalization of the helicityevolving equations for the α^{2} dynamo that separately allows for these three distinct lengthscales while keeping only two dynamical equations. We solve these equations and, as in previous studies, find that the largescale 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 smallscale current helicity resides to that of the forcing scale, the earlier the nonlinear dynamo quenching occurs, and the weaker the largescale 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 magnetohydrodynamic are better visualized as twodimensional ribbons (or pairs of lines) rather than single lines.
 Publication:

Monthly Notices of the Royal Astronomical Society
 Pub Date:
 August 2014
 DOI:
 10.1093/mnras/stu939
 arXiv:
 arXiv:1403.3445
 Bibcode:
 2014MNRAS.442.1040B
 Keywords:

 accretion;
 accretion discs;
 dynamo;
 MHD;
 stars: magnetic field;
 galaxies: magnetic fields;
 Astrophysics  Solar and Stellar Astrophysics;
 Astrophysics  Cosmology and Nongalactic Astrophysics;
 Astrophysics  Astrophysics of Galaxies;
 Astrophysics  High Energy Astrophysical Phenomena;
 Physics  Plasma Physics
 EPrint:
 11 pages, 10 figure panels, version accepted by MNRAS