{'fraction_non_alphanumeric': 0.046889089269612265, 'fraction_numerical': 0.010820559062218215, 'mean_word_length': 4.81151832460733, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A nonlinear eigenvalue problem for self-similar spherical force-free magnetic fields

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 se...