In the dynamo problem, the calculation of induced modes is of vital importance, because the interaction of fluid motions with the magnetic field induces specific types of fields which are, in many cases, different either from the type of velocity field or from the original magnetic field. This special induction relationship, known as “selection rules”, has so far been derived by calculating Adams-Gaunt integrals and Elsasser integrals.
In this paper, we calculate the induced modes in a more direct way, expressing the magnetic fields and the velocity in a spherical harmonic series. By linearizing the product terms of spherical harmonic functions, which appear in interaction terms between the velocity and the magnetic field, into a simple spherical harmonic series, we have derived the induced magnetic modes in a simple general form.
When the magnetic field and the velocity are expressed by toroidal and poloidal modes, four kinds of interaction are conceivable between the velocity and the magnetic field. By each interaction, two modes, the poloidal and toroidal, are induced, except in the interaction of the toroidal velocity with the toroidal magnetic field, which induces only the toroidal mode. In spite of the diversity of interaction processes, the induced modes have been found to be expressed simply by two types. For a velocity of degree l and order k interacting with a magnetic field of degree n and order m, one type is the mode with degree and order of n+l-2t, |m±k| for an integer t, and the other with n+l-2t-1, |m±k|.