Cylindrical Taylor states conserving total absolute magnetic helicity

Abstract

The Taylor state of a three-dimensional (3D) magnetic field in an upright cylindrical domain V is derived from first principles as an extremum of the total magnetic energy subject to a conserved, total absolute helicity Habs. This new helicity [Low, Phys. Plasmas 18, 052901 (2011)] is distinct from the well known classical total helicity and relative total helicity in common use to describe wholly-contained and anchored fields, respectively. A given field B, tangential along the cylindrical side of V, may be represented as a unique linear superposition of two flux systems, an axially extended system along V and a strictly transverse system carrying information on field-circulation. This specialized Chandrasekhar-Kendall representation defines Habs and permits a neat formulation of the boundary-value problem (BVP) for the Taylor state as a constant-α force-free field, treating 3D wholly-contained and anchored fields on the same conceptual basis. In this formulation, the governing equation is a scalar integro-partial differential equation (PDE). A family of series solutions for an anchored field is presented as an illustration of this class of BVPs. Past treatments of the constant-α field in 3D cylindrical geometry are based on a scalar Helmholtz PDE as the governing equation, with issues of inconsistency in the published field solutions discussed over time in the journal literature. The constant-α force-free equation reduces to a scalar Helmholtz PDE only as special cases of the 3D integro-PDE derived here. In contrast, the constant-α force-free equation and the scalar Helmholtz PDE are absolutely equivalent in the spherical domain as discussed in Appendix. This theoretical study is motivated by the investigation of the Sun's corona but the results are also relevant to laboratory plasmas.