TY - JOUR
T1 - A Krein space related perturbation theory for MHD α2-dynamos and resonant unfolding of diabolical points
AU - Günther, Uwe
AU - Kirillov, Oleg
PY - 2006/7/26
Y1 - 2006/7/26
N2 - The spectrum of the spherically symmetric α2-dynamo is studied in the case of idealized boundary conditions. Starting from the exact analytical solutions of models with constant α-profiles, a perturbation theory and a Galerkin technique are developed in a Krein space approach. With the help of these tools, a very pronounced α-resonance pattern is found in the deformations of the spectral mesh as well as in the unfolding of the diabolical points located at the nodes of this mesh. Non-oscillatory as well as oscillatory dynamo regimes are obtained. An estimation technique is developed for obtaining the critical α-profiles at which the eigenvalues enter the right spectral half-plane with non-vanishing imaginary components (at which overcritical oscillatory dynamo regimes form). Finally, Fréchet derivative (gradient) based methods are developed, suitable for further numerical investigations of Krein space related setups such as MHD α2-dynamos or models of {\cal P}{\cal T} -symmetric quantum mechanics.
AB - The spectrum of the spherically symmetric α2-dynamo is studied in the case of idealized boundary conditions. Starting from the exact analytical solutions of models with constant α-profiles, a perturbation theory and a Galerkin technique are developed in a Krein space approach. With the help of these tools, a very pronounced α-resonance pattern is found in the deformations of the spectral mesh as well as in the unfolding of the diabolical points located at the nodes of this mesh. Non-oscillatory as well as oscillatory dynamo regimes are obtained. An estimation technique is developed for obtaining the critical α-profiles at which the eigenvalues enter the right spectral half-plane with non-vanishing imaginary components (at which overcritical oscillatory dynamo regimes form). Finally, Fréchet derivative (gradient) based methods are developed, suitable for further numerical investigations of Krein space related setups such as MHD α2-dynamos or models of {\cal P}{\cal T} -symmetric quantum mechanics.
U2 - 10.1088/0305-4470/39/32/S08
DO - 10.1088/0305-4470/39/32/S08
M3 - Article
VL - 39
SP - 10057
EP - 10076
JO - Journal of Physics A: Mathematical Nuclear and General
JF - Journal of Physics A: Mathematical Nuclear and General
SN - 1751-8113
IS - 32
ER -