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 $r_i/r_o \geq 0.55$ and $\beta \leq O(1)$. 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 $\lambda_0 = O(10^2 km^2 s^{-1}$). 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.