Inertial manifolds and completeness of eigenmodes for unsteady magnetic dynamos

Abstract

We prove the emergence of asymptotic spatial patterns in magnetic dynamos generated by unsteady fluid flows. The patterns emerge because solutions of the dynamo equation converge exponentially to a time-dependent inertial manifold. This inertial manifold exists for general time-aperiodic velocity fields under a spectral gap condition on the associated Stokes operator. For time-periodic velocity fields, we show that the inertial manifold is spanned by Floquet eigenmodes that are analogous to the strange eigenmodes observed in the mixing of diffusive tracers. This result gives an affirmative answer to the long-standing question of completeness of Floquet solutions in time-periodic dynamo problems.