TITLE

A nonlinear eigenvalue problem for self-similar spherical force-free magnetic fields

AUTHOR(S)
Lerche, I.; Low, B. C.
PUB. DATE
October 2014
SOURCE
Physics of Plasmas;2014, Vol. 21 Issue 10, p1
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
An axisymmetric force-free magnetic field B(r, θ) in spherical coordinates is defined by a function r sin θBφ=Q(A) relating its azimuthal component to its poloidal flux-function A. The power law r sin θBφ=aA|A|1/n, n a positive constant, admits separable fields with A=An(θ)/rn, posing a nonlinear boundary-value problem for the constant parameter a as an eigenvalue and An (θ) as its eigenfunction [B. C. Low and Y. Q Lou, Astrophys. J. 352, 343 (1990)]. A complete analysis is presented of the eigenvalue spectrum for a given n, providing a unified understanding of the eigenfunctions and the physical relationship between the field's degree of multi-polarity and rate of radial decay via the parameter n. These force-free fields, self-similar on spheres of constant r, have basic astrophysical applications. As explicit solutions they have, over the years, served as standard benchmarks for testing 3D numerical codes developed to compute general force-free fields in the solar corona. The study presented includes a set of illustrative multipolar field solutions to address the magnetohydrodynamics (MHD) issues underlying the observation that the solar corona has a statistical preference for negative and positive magnetic helicities in its northern and southern hemispheres, respectively; a hemispherical effect, unchanging as the Sun's global field reverses polarity in successive eleven-year cycles. Generalizing these force-free fields to the separable form B=H(θ,φ)/rn+2 promises field solutions of even richer topological varieties but allowing for φ -dependence greatly complicates the governing equations that have remained intractable. The axisymmetric results obtained are discussed in relation to this generalization and the Parker Magnetostatic Theorem. The axisymmetric solutions are mathematically related to a family of 3D time-dependent ideal MHD solutions for a polytropic fluid of index γ = 4/3 as discussed in the Appendix.
ACCESSION #
99235725

 

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