Dynamo Waves in Semi-Infinite and Finite Domains

Details Dynamo Waves in Semi-Infinite and Finite Domains Dynamo Waves in Semi-Infinite and Finite Domains

Jan 1997

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1997rspsa.453..119w&link_type=abstract

Proceedings: Mathematical, Physical and Engineering Sciences, Volume 453, Issue 1956, pp. 119-143

Physics

19

Scientific paper

In order to model the solar sunspot cycle, Parker introduced the concept of dynamo waves. These waves owe their existence to two processes, the generation of poloidal field from toroidal by the α -effect, and the generation of toroidal field from the poloidal by the Ω -effect. In their simplest realization these waves are plane waves that propagate in latitude. In this paper we show that spatial inhomogeneities, whether due to variations in the driving α -effect, or simply through the imposition of boundaries in latitude to simulate the solar pole and Equator, have an anomalously important effect, and make it necessary to revise the usual notions of weakly nonlinear bifurcation theory. Two distinct cases are considered. In the first, the solution domain is infinite, but the α -effect is antisymmetric about the Equator at x = 0. Both dipole and quadrupole states are considered. The linear problem reveals an unusual feature: the frequency spectrum is continuous and the neutral curve N(ω ), where N is the dynamo number, plays the role of the more usual neutral curve in the spatial domain. However, the critical dynamo number remains equal to that for an unbounded homogeneous layer. In the weakly nonlinear regime the resulting instability takes the form of dynamo waves that are slowly modulated in space, and is described by an evolution equation of Ginzburg-Landau type but with space and time interchanged. The second case, perhaps more appropriate to the Sun, deals with large but finite domains while including possible slow spatial variations of the α -effect. Surprisingly, in this case sustained dynamo action can only occur when the dynamo number exceeds a value somewhat higher than that for an infinite layer. The onset mode is now localized equatorially. At leading order this 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 the problem is related to the distinction between convective and absolute instabilities.

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