Thin disc kinematic α ω-dynamo models I. long length scale modes

The assumptions and approximations commonly used in thin disk kinematic dynamo theory are discussed and appraised. Here attention is restricted to those modes, which have a radial length scale long compared to the disk thickness but short compared to the disc radius, upon which the characteristics of the disc vary. One of the commonly employed approximations pivots on the assumption that the radial structure is determined by lateral diffusion in the disc, whereas on those long radial scales the role of the external potential field is more potent. Stix's (1975, 1978) α ω-galactic dynamo in an oblate spheroid with aspect ratio, ∊, is adopted as an illustrative example. He reports numerical results for models with ∊ as small as 1/30. Our perturbation methods are applied to three distinct modes, namely the steady dipole, the steady and unsteady quadrupole. Each has very different characteristics. It is shown that the determination of the radial structure of the solutions on the long radial length scale is extremely delicate in the limit ∊ ↓ 0. The higher order asymptotic theory, which determines these details, appears only to be valid for extremely small values of ∊; certainly far smaller than the value of 1/30 reached by Stix. Nevertheless, despite the limitations of the theory, the lowest order approximation, which determines the critical Dynamo number and the nature of the dynamo itself, gives generally reliable results, while the higher order theory predicts trends.