The spectral form of the magnetic induction equation

The equations derived by Bullard & Gellman (1954) for the kinematic spherical dynamo problem are rederived by using Wigner 3 - j and 9 - j coefficients. The new method is more general than that of Bullard & Gellman in that compressible flows are permitted and the spectral form of the induced electric field is obtained as an intermediate step. The method clearly shows that the interaction between magnetic field and fluid velocity is governed by just one coupling integral, special cases of which are the various coupling integrals considered by other authors. All the selection rules and symmetries attributed to these other integrals are examples of the selection rules and symmetries of the new integral. The new integral is proportional to a certain 9 – j coefficient and a complete table of these coefficients is included. For magnetic fields frozen-in to a fluid core, the results reduce to a simple set of constraints on the fluid velocity spectral coefficients at the core-surface.