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

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 $r_i\mathrm{/}{r}_o\ge 0.55$ and $\beta \le O(1)$. For the Sun, $r_i\mathrm{/}{r}_o$ 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 $\lambda_0 = O(10^2 km^2 s^{-1}$)\mathrm{.}Itisunnecessarytohypothesizean$\alpha\omega$-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.