科研之友
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摘要
An axisymmetric force-free magnetic field B(r, theta) in spherical coordinates is defined by a function r sin theta Bu-phi Qd(A) relating its azimuthal component to its poloidal flux-function A. The power law r sin theta Bu-phi aA vertical bar A vertical bar(1/n), n a positive constant, admits separable fields with A = An(theta)/r(n), posing a nonlinear boundary-value problem for the constant parameter a as an eigenvalue and A(n)(theta) as its eigen-function [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 eigen-functions 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(theta,phi)/r(n+2) promises field solutions of even richer topological varieties but allowing for u-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 gamma-4/3 as discussed in the Appendix.
- 出版日期2014-10
