The large-scale magnetic Ðelds of stellar and galactic bodies are generally understood to be organized and ampliÐed by motions in the conducting Ñuid media of these bodies. This article examines a mechanism by which continual excitation of the conducting Ñuid by small-scale Ðelds results in production of large-scale Ðelds. The excitation of the induction equation by small-scale Ðelds is parameterized as stochastic forcing, and the crucial role of the nonnormality of the induction operator in determining the spatial and temporal structure of variation in the large-scale Ðelds is emphasized. A cylindrically symmetric helical Ñow is used to provide illustrative examples.

The mechanism by which large-scale fields in stars and galaxies arise remains uncertain, but it is believed that initially small internally generated or primordial seed fields are amplified and organized by motions in the conducting fluid interiors of these bodies. Methods for analyzing this process in the weak field limit are based on the induction equation and fall into two classes: those involving advection of the magnetic field as a passive tracer, and those involving calculation of exponential instabilities. The former is a nonmodal stability analysis, while the latter is essentially modal. In this work these two methods of analysis are synthesized, making use of recent advances in the theory of nonnormal system dynamics. An application of this generalized stability to the helical dynamo model of Lortz is described in which the maximum field growth over prescribed time intervals and the perturbation structures producing this growth are identified.