Analytic solution of an oscillatory migratory alpha^2 stellar dynamo

TL;DR

This paper presents an analytic solution demonstrating that oscillatory migratory alpha^2 dynamos can exist in bounded domains with constant alpha, challenging previous assumptions about their impossibility in such conditions.

Contribution

It provides the first analytic solution for oscillatory alpha^2 dynamos in a bounded domain with constant alpha and mixed boundary conditions.

Findings

Oscillatory solutions occur with Dirichlet and von Neumann boundary conditions.

The magnetic field migrates away from the perfect conductor boundary.

Solution serves as a benchmark and pedagogical example for dynamo theory.

Abstract

Analytic solutions of the mean-field induction equation predict a nonoscillatory dynamo for homogeneous helical turbulence or constant alpha effect in unbounded or periodic domains. Oscillatory dynamos are generally thought impossible for constant alpha. We present an analytic solution for a one-dimensional bounded domain resulting in oscillatory solutions for constant alpha, but different (Dirichlet and von Neumann or perfect conductor and vacuum) boundary conditions on the two boundaries. We solve a second order complex equation and superimpose two independent solutions to obey both boundary conditions. The solution has time-independent energy density. On one end where the function value vanishes, the second derivative is finite, which would not be correctly reproduced with sine-like expansion functions where a node coincides with an inflection point. The field always migrates away from the perfect conductor boundary toward the vacuum boundary, independently of the sign of alpha. The obtained solution may serve as a benchmark for numerical dynamo experiments and as a pedagogical illustration that oscillatory migratory dynamos are possible with constant alpha.