2021/06/14 by J. -B. Durrive, Jean-Baptiste Durrive, Rony Keppens +2
Physics and Astronomy · #Astrophysics and Star Formation Studies #Classical mechanics #Cosmology and Gravitation Theories #Eigenfunction #Eigenvalues and eigenvectors #Hamiltonian (control theory) #Instability #Magnetohydrodynamic drive #Magnetohydrodynamics #Mechanics #Physics #Plasma #Polytrope #Polytropic process #Quantum mechanics #Solar and Space Plasma Dynamics #astro-ph.GA #physics.flu-dyn #physics.plasm-ph
paper · pdf · doi:10.1093/mnras/stab1726
20 pages, 1 figure, accepted for publication in MNRAS
arxiv created 2021/06/14 · openalex publication_date 2021/06/17 · arxiv updated 2021/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Abstract In this paper, we revisit the governing equations for linear magnetohydrodynamic (MHD) waves and instabilities existing within a magnetized, plane-parallel, self-gravitating slab. Our approach allows for fully non-uniformly magnetized slabs, which deviate from isothermal conditions, such that the well-known Alfvén and slow continuous spectra enter the description. We generalize modern MHD textbook treatments, by showing how self-gravity enters the MHD wave equation, beyond the frequently adopted Cowling approximation. This clarifies how Jeans’ instability generalizes from hydro to magnetohydrodynamic conditions without assuming the usual Jeans’ swindle approach. Our main contribution lies in reformulating the completely general governing wave equations in a number of mathematically equivalent forms, ranging from a coupled Sturm-Liouville formulation, to a Hamiltonian formulation linked to coupled harmonic oscillators, up to a convenient matrix differential form. The latter allows us to derive analytically the eigenfunctions of a magnetized, self-gravitating thin slab. In addition, as an example we give the exact closed form dispersion relations for the hydrodynamical p- and Jeans-unstable modes, with the latter demonstrating how the Cowling approximation modifies due to a proper treatment of self-gravity. The various reformulations of the MHD wave equation open up new avenues for future MHD spectral studies of instabilities as relevant for cosmic filament formation, which can e.g. use modern formal solution strategies tailored to solve coupled Sturm-Liouville or harmonic oscillator problems.