Homotopic deformations of the Arnold tongue patterns in the MHD α²-dynamo spectrum

The spectrum of an MHD α²-dynamo has been studied under perturbations of the α-profile as well as under smooth changes of the boundary conditions (BCs). Basic ingredient was a set of analytically (exactly) calculable bi-orthogonal eigenfunctions for constant α-profiles α₀=const. These eigenfunctions have been used as input for both perturbative and numerical Galerkin (weighted residual) analyses with BCs imposed on them as one-parameter homotopic family which smoothly interpolates between idealized (Dirichlet) BCs and physically realistic (Robin) BCs.

For constant α-profiles the spectrum is purely real and depending on the BCs it changes from a spectral mesh (living as line structure on the doubly ruled surface of a hyperbolic paraboloid) into a countably infinite set of parabolic branches. Under inhomogeneous perturbations of the α-profile the spectrum deforms in such a way that complex spectral sectors form from a subset of intersection points of the spectral mesh. With increasing perturbation strength these complex sectors widen and have the well known form of partially merging Arnold tongues. Explicit analytical approximations are derived for the Arnold tongues. Technically the approximations are based on the perturbation theory of multiple eigenvalues applied to the double eigenvalues (diabolical points) at the nodes of the spectral mesh. A detailed analysis is presented of the interplay of inhomogeneous α-perturbations and homotopic BC deformations with the formation of a wealth of Arnold tongues and partially invariant spectral branch patterns. A good correspondence between the numerical and the perturbative approaches is observed.