  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.
