DYNAMO WAVES IN SEMIINFINITE AND FINITE DOMAINS

In order to model the solar sunspot cycle, Parker introduced the conce pt of dynamo waves. These waves owe their existence to two processes, the generation of poloidal field from toroidal by the alpha-effect, an d the generation of toroidal field from the poloidal by the Omega-effe ct. In their simplest realization these waves are plane waves that pro pagate in latitude. In this paper we show that spatial inhomogeneities , whether due to variations in the driving alpha-effect, or simply thr ough the imposition of boundaries in latitude to simulate the solar po le and Equator, have an anomalously important effect, and make it nece ssary to revise the usual notions of weakly nonlinear bifurcation theo ry. Two distinct cases are considered. In the first, the solution doma in is infinite, but the alpha-effect is antisymmetric about the Equato r at x = 0. Both dipole and quadrupole states are considered. The Line ar problem reveals an unusual feature: the frequency spectrum is conti nuous and the neutral curve N(omega), where N is the dynamo number, pl ays the role of the more usual neutral curve in the spatial domain. Ho wever, the critical dynamo number remains equal to that for an unbound ed homogeneous layer. In the weakly nonlinear regime the resulting ins tability takes the form of dynamo waves that are slowly modulated in s pace, and is described by an evolution equation of Ginzburg-Landau typ e but with space and time interchanged. The second case, perhaps more appropriate to the Sun, deals with large but finite domains while incl uding possible slow spatial variations of the alpha-effect. Surprising ly, in this case sustained dynamo action can only occur when the dynam o number exceeds a value somewhat higher than that for an infinite lay er. The onset mode is now localized equatorially. At leading order thi s dynamo number is independent of the domain size and of the boundary conditions imposed at large \x\, i.e. at the poles. This feature of th e problem is related to the distinction between convective and absolut e instabilities.