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Dynamo waves in semi-infinite and finite domains. (English) Zbl 0877.76084

The behaviour of dynamo waves (introduced by Parker to model the solar sunspot cycle) is discussed in detail. The dynamo waves originate in two processes: the alpha-effect generates poloidal field from a toroidal one, and the omega-effect generates a toroidal field from a poloidal one. In their simplest realization these waves propagate in latitude and are plane waves. Now the authors show that spatial inhomogeneities have an anomalously important effect, independently of their actual cause (variations in the driving alpha-effect, or imposition of boundaries on latitude which simulate the pole and the equator). This makes it necessary to revise the usual notions of weakly nonlinear bifurcation theory.
Two distinct cases are studied. In the first case, the propagation domain is infinite, the alpha-effect is antisymmetric about the equator, and both dipole and quadrupole states are present. Here the linear problem reveals that the frequency spectrum is continuous, and the dynamo number plays the role of a neutral curve in the spatial domain. The critical dynamo number remains equal to that for the unbounded homogeneous layer. In the weakly nonlinear regime, the dynamo waves arise from instabilities and become slowly modulated in space. They can be described by an evolution equation of Ginzburg-Landau type with space and time variables interchanged.
The second case studies a large but finite domain and admits slow spatial variation of the alpha-effect. Here a sustained dynamo possesses a higher dynamo number than in the first case, and the onset of dynamo action is localized equatorially. The dynamo number here is independent in the leading order of the domain size and of the boundary conditions at the poles.

MSC:

76W05 Magnetohydrodynamics and electrohydrodynamics
85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics
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