An Integral Equation Approach to Kinematic Dynamo Models

The paper deals with dynamomodels in which the induction effects act within a bounded region surrounded by an electrically conducting medium at rest. Instead of the induction equation, an equivalent integral equation is considered, which again poses an eigenvalue problem. The eigenfunctions and eigenvalues represent the magnetic field modes and corresponding dynamo numbers. In the simplest case, that is for homogeneous conductivity and steady fields, this integral equation follows immediately from the Biot-Savart law. For this case, numerical results are presented for some spherical and elliptical axisymmetric 2!-dynamo models. For a large class of models the interesting feature of dipole-quadrupole degeneration is found. Using Green’s function of a Helmholtz-type equation, we derive a more general integral equation, which applies to time-dependent magnetic field modes, too, and gives us some insight into the spectral properties of the integral operators involved. In particular, for homogeneous conductivity the operator is compact and thus bounded, which leads to a necessary condition for dynamo action.