Spherical, Oscillatory ${\alpha }^2$-Dynamo Induced by Magnetic Coupling Between a Fluid Shell and an Inner Electrically Conducting Core: Relevance to the Solar Dynamo

Abstract

A two-layer spherical ${\alpha }^2$-dynamo model consisting of an inner electrically conducting core (magnetic diffusivity ${\lambda }_i$ and radius $r_i$) with $\alpha =0$ surrounded by an electrically conducting spherical shell (magnetic diffusivity ${\lambda }_o$ and radius $r_o$) with a constant $\alpha$ is shown to exhibit oscillatory behavior for values of $\beta ={\lambda }_i/{\lambda }_o$ and $r_i/r_o$ relevant to the solar dynamo. Time-dependent dynamo solutions require $r_i/r_o\ge 0.55$ and $\beta \le O(1)$. For the Sun, $r_i/r_o$ is about 0.8 and $\beta \approx 10^{-3}$. The time scale of the oscillations matches the 22 year period of the sunspot cycle for ${\lambda }_0=O(10^{2}k{m}^2{s}^{-1}$). It is unnecessary to hypothesize an $\alpha \omega$-dynamo to obtain oscillatory dynamo solutions; an ${\alpha }^2$-dynamo suffices provided the spherical shell region of dynamo action lies above a large, less magnetically diffusive core, as is the case for the solar dynamo.

7 pages, 1 postscript figure

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