Eigenvalue Problem for a Reduced Dynamo Model in Thick Astrophysical Discs

Abstract

Magnetic fields of different astrophysical objects are generated by the dynamo mechanism. Dynamo is based on the alpha-effect and differential rotation, which are described using a system of parabolic equations. Their solution is an important problem in magnetohydrodynamics and mathematical physics. They can be solved assuming exponential growth of the solution, which leads to an eigenvalue problem for a differential operator connected with spatial coordinates. Here, we describe a system of equations connected with the generation of magnetic field in discs, which are associated with galaxies and binary systems. For an ideal case of an infinitely thin disc, the eigenvalue problem can be precisely solved. If we take into account the finite thickness of the disc, the problem becomes more difficult. The solution can be found using asymptotical methods based on perturbations of the eigenvalues. Here, we present two different models which describe field evolution for different cases. For the first, we find eigenvalues taking into account linear and quadratic terms for the perturbations in the eigenvalue problem. For the second, we find eigenvalues using only linear terms; this is quite sufficient. Results were verified through numerical modeling, and basic computational tests show proper correspondence between different methods.