ABSTRACT
This chapter presents the asymptotic methods necessary for the investigation of mean field dynamos in astrophysical objects with small aspect ratio ε. The two particular geometries are considered thin discs appropriate to galaxies and accretion discs, and thin shells which model the dynamo active regions in stars, for example, the convection zone or possibly the tachocline in the Sun. That state is characterised by a localised finite amplitude Parker wave, which emerges smoothly at low latitude and terminates abruptly across a front at high latitude. At lowest order the frequency and front location are fixed by the Dee and Langer criterion that the group velocity vanishes ahead of the front. The conditions for local dynamo action do not guarantee global dynamo action, in the sense that should provide a true eigenfunction of the complete eigenvalue problem. A related approach, which has had some success, is the asymptotic investigation of rapidly growing kinematic dynamo modes at large dynamo number.
