Numerical search of eigenvalues for analysis of perturbations of field in one problem of magnetohydrodynamics

Numerical search of eigenvalues for analysis of perturbations of field in one problem of magnetohydrodynamics

Frolova M.V. (MSU, Moscow, Russia)
Mikhailov E.A. (MSU, Moscow, Russia; FIAN, Moscow, Russia)

Abstract

Magnetohydrodynamic processes play a crucial role in numerous physical phenomena and engineering processes involving the flow of conducting fluids. Of great importance in astrophysics is the excitation of a magnetic field due to the properties of turbulent motion – the so-called dynamo mechanism. It is typically described by averaging the equations of magnetohydrodynamics and introducing the alpha effect. The possibility of generating a magnetic field in this case is determined by the growth of its small perturbations over time. This requires linearizing the problem, which leads to an evolution equation with a differential operator on the right-hand side. The growth or decay of the magnetic field is determined by the sign of the real parts of its eigenvalues (or the sign if they are real). One of the most interesting problems is the possibility of excitation of a magnetic field in a disk. This model well describes the behavior of magnetic fields in galaxies and in accretion disks that surround compact astrophysical objects (black holes, white dwarfs, and neutron stars). In the simplest case of a disk of negligible thickness, the eigenvalue problem may even have an analytical solution. However, for disks of finite thickness, this leads to the possibility of solving the problem only using asymptotic approximations. If the disk expands toward its edges (which is quite typical for both galaxies and accretion disks), then this problem can apparently only be solved numerically. This paper presents the results of a numerical search for eigenvalues for the excitation of a magnetic field in an expanding disk. For this purpose, the differential operator is replaced by a finite-difference operator, and its eigenvalues are found using the inverse power-law method. In this case, the problem can be solved using the nonmonotonic sweep method. The leading eigenvalues and eigenfunctions for various disk expansion models are presented.

Keywords

magnetic field; non-monotonic sweep method; eigenvalue problem; dynamo.

Edition

Proceedings of the Institute for System Programming, vol. 38, issue 3, part 4, 2026, pp. 217-228

ISSN 2220-6426 (Online), ISSN 2079-8156 (Print).

DOI: 10.15514/ISPRAS-2026-38(3)-57

For citation

Frolova M.V., Mikhailov E.A. Numerical search of eigenvalues for analysis of perturbations of field in one problem of magnetohydrodynamics. Proceedings of the Institute for System Programming, vol. 38, issue 3, part 4, 2026, pp. 217-228 DOI: 10.15514/ISPRAS-2026-38(3)-57.

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