Macrodynamics of α2dynamos

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Abstract
Two distributions of the α-effect in a sphere are considered. The inviscid limit is approached both by direct numerical solution and by solution of a simpler nonlinear eigenvalue problem deriving from asymptotic boundary layer analysis for the case of stress-free boundaries. The inviscid limit in both cases is dominated by the need to satisfy the Taylor constraint which states that the integral of the Lorentz force over cylindrical (geostrophic) contours in a homogeneous fluid must tend to zero. For a small supercritical range in α, this condition can only be met by magnetic fields which vanish as the viscosity goes to zero. In this range, the agreement of the two approaches is excellent. In a portion of this range, the method of finite amplitude perturbation expansion is useful, and serves as a guide for understanding the numerical results. For larger α, evidence from the nonlinear eigenvalue problem suggests both that the Taylor state exists, and that the transition from small to large amplitude can require a finite amplitude (oscillatory) instability in accord with the findings of Soward and Jones (1983). However, solutions of the full equations have not been found which are independent of viscosity at larger values of α.

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