Journal of Mathematical Physics
The equations of ideal magnetohydrodynamics (MHD) with an external gravitational potential—a ‘‘magnetoatmosphere’’—are examined in detail as a singular nonlinear eigenvalue problem. Properties of the spectrum are discussed with specific emphasis on two systems relevant to solar magnetohydrodynamics. In the absence of a gravitational potential, the system reduces to that of importance in MHD and plasma physics, albeit in a different geometry. This further reduces to a form isomorphic to that derived in the study of plasma oscillations in a cold plasma, Alfvén wave propagation in an inhomogeneous medium, and MHD waves in a sheet pinch. In cylindrical geometry, the relevant model equations are those for a diffuse linear pinch. The full system, including gravity, has been applied to the study of flare‐induced coronal waves, running penumbral waves in sunspots, and linear wave coupling in a highly inhomogeneous medium. The structure of the so‐called MHD critical layer and its contribution to the continuous spectrum is examined in detail for a model magnetoatmosphere, based on properties of the hypergeometric differential operator. The relationship of this singular region to critical layers in classical linear hydrodynamic stability theory is also discussed in the light of a specific model (in the Appendix).
Original Publication Citation
Adam, J. A. (1989). A nonlinear eigenvalue problem in astrophysical magnetohydrodynamics: Some properties of the spectrum. Journal of Mathematical Physics, 30(3), 744-756. doi:10.1063/1.528392
Adam, John A., "A Nonlinear Eigenvalue Problem in Astrophysical Magnetohydrodynamics: Some Properties of the Spectrum" (1989). Mathematics & Statistics Faculty Publications. 156.