The kinematic dynamo problem, part I: analytical treatment with the Bullard–Gellman formalism

Abstract

This paper is dedicated to the description of kinematic dynamo action in a sphere and its analytical treatment with the Bullard–Gellman formalism. One goal of dynamo theory is to answer the question: Can magnetic fields of stellar objects be generated or sustained due to (fluid) motion in the interior? Bullard and Gellman were among the first to study this question, leading the way for many subsequent studies, cf. Bullard (Philos Trans R Soc A 247(928):213–278, 1954). In their publication the differential equations resulting from a toroidal–poloidal decomposition of the velocity and magnetic field are stated without an in-depth discussion of the employed methods and computation steps. This study derives the necessary formalism in a compact and concise manner by using an operator-based approach. The focus lies on the mathematical steps and necessary properties of the considered formalism. Prior to that a derivation of the induction equation is presented based on rational continuum electrodynamics. As an example of the formalism the decay of two magnetic fields is analyzed.