How to excite oscillating modes in the kinematic mean-field α2-dynamo: A Krein space-related perturbation approach

We consider equations of the kinematic mean-field alpha^2-dynamo with the spherically-symmetric alpha-profile that depends only on the radial coordinate. We study spectrum of the non-self-adjoint boundary eigenvalue problem for the corresponding operator matrix with the boundary conditions associated either with a perfect electrically conducting surrounding or with an insulating one. In the first case we demonstrate that the operator is self-adjoint in a Hilbert space with the indefinite metric (i.e. in the Krein space). Moreover, if, additionally, the alpha-profile is constant (the problem A), the eigenvalues are real-valued linear functions of alpha; hence, only non-oscillatory instability is possible. However, with non-constant alpha-profiles and insulating boundary conditions, oscillatory dynamo regimes can become dominant (the problem B). With the use of the perturbation theory of multiple eigenvalues we explicitly demonstrate how from the real spectrum of the problem A one can get the complex eigenvalues of the problem B due to variation of alpha-profile and interpolation between the two types of boundary conditions.