Spherical, Oscillatory $α^2$-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 $α^2$-dynamo model consisting of an inner electrically conducting core (magnetic diffusivity $λ_i$ and radius $r_i$) with $α= 0$ surrounded by an electrically conducting spherical shell (magnetic diffusivity $λ_o$ and radius $r_o$) with a constant $α$ is shown to exhibit oscillatory behavior for values of $β= λ_i/λ_o$ and $r_i/r_o$ relevant to the solar dynamo. Time-dependent dynamo solutions require $r_i/r_o \geq 0.55$ and $β\leq O(1)$. For the Sun, $r_i/r_o$ is about 0.8 and $β\approx 10^{-3}$. The time scale of the oscillations matches the 22 year period of the sunspot cycle for $λ_0 = O(10^2 km^2 s^{-1}$). It is unnecessary to hypothesize an $αω$-dynamo to obtain oscillatory dynamo solutions; an $α^2$-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.