Poedts, S.; Hermans, D.; Goossens, M. The continuous spectrum of an axisymmetric self-gravitating and static equilibrium with a mixed poloidal and toroidal magnetic field. (English) Zbl 0588.76192 Astron. Astrophys. 151, 16-26 (1985). The linearized MHD-equations and the gravitational Poisson-equation is considered for the equilibrium problem in mixed poloidal and toroidal magnetic field and in ideal fluid. The continuous spectrum of a purely poloidal magnetic field is given by an eigenvalue problem of two uncoupled second order differential equations along the field lines. The continuous spectrum is degenerate with respect to the azimuthal wavenumber and the solutions are polarized in the magnetic surface either perpendicular or parallel to the field lines. The two uncoupled continuous parts are called, with all rights, Alfvén and cusp continuum, respectively; both are affected by toroidicity but the cusp one only is affected by compressibility and gravity. Variational expressions for the continuum frequencies show that the Alfvén continuum is always on the stable side of the spectrum, while the cusp one may extend beyond the stability limit. In the case of mixed poloidal- toroidal field, the spectrum is given by a fourth order system of coupled differential equations, where the coupling is due to the toroidal field, while its degree depends on equilibrium quantities in the magnetic surfaces. The continuous spectrum is non-degenerate, is affected by toroidicity, gravity and compressibility. A variational expression is derived for the continuum frequencies, and it is shown that the equilibrium gravitational field can lead to an unstable continuous spectrum. Reviewer: I.Abonyi MSC: 76W05 Magnetohydrodynamics and electrohydrodynamics 85A05 Galactic and stellar dynamics Keywords:degenerate continuous spectrum; stellar magnetic fields; linearized MHD- equations; gravitational Poisson-equation; equilibrium problem; continuous spectrum of a purely poloidal magnetic field; eigenvalue problem; cusp continuum; Alfvén continuum; stability limit; mixed poloidal-toroidal field; magnetic surfaces; continuum frequencies; equilibrium gravitational field PDFBibTeX XMLCite \textit{S. Poedts} et al., Astron. Astrophys. 151, 16--26 (1985; Zbl 0588.76192)