Resonance effects in α²-dynamos

First results on a perturbation theory for α²-dynamos with non-constant spherically symmetric α-profiles are presented. The basics of the mathematical technique is demonstrated on a simplified model with idealized boundary conditions (BCs). Such BCs allow for a full semi-analytical treatment of the problem due to the selfadjointness of the dynamo operator in a so called Krein space --- a Hilbert space with an additional indefinite metric structure.
Starting point of the consideration is the model with constant α-profile α₀ which is exactly solvable. The eigenvalues λ of such a model form a mesh-like branch structure in the (α₀,Re λ)-plane. The nodes of this mesh (the intersection points of the branches) are double semi-simple eigenvalues (diabolical points) of algebraic and geometric multiplicity 2.
A Krein space related perturbation theory as well as a Galerkin technique for the numerical treatment of inhomogeneous perturbations of the α-profile have been developed. With the help of these tools, a very pronounced α-resonance pattern has been found in the deformations of the spectral mesh as well as in the unfolding of the diabolical points. The selection rules of this resonance pattern are defined by the Fourier coefficients of the perturbations leading to a strong correspondence between the characteristic length scale of the α-perturbations and the decay rates of the coherently induced field excitations. Such correlations will exist also in models with realistic BCs and may provide a qualitative explanation for some specifics of field reversal processes --- supporting corresponding numerical simulations on more realistic dynamo setups.
Furthermore, an estimation technique has been developed for obtaining optimal α-profiles for which α²-dynamos can become overcritical in oscillatory regimes. This basic technique is demonstrated explicitly for a simplified monopole (l=0) model, where it leads to a bound on the Fourier components of the α-profile. The capability of the used Galerkin approach is demonstrated in extending the strength of the α-perturbations from weakly perturbed regimes to strongly perturbed ones.
Extensions of the presented techniques to spherically symmetric α²-dynamos with realistic boundary conditions are in preparation.