Integral Equations for Kinematic Dynamo Models

A new technique for the treatment of the kinematic dynamo problem is presented. The method is applicable when the dynamo is surrounded by a medium of finite conductivity and is based on a reformulation of the induction equation and boundary conditions at infinity into an integral equation. We show that the integral operator involved here is compact in the case of homogeneous conductivity, which is important for both mathematical and numerical treatment. A lower bound for the norm of then yields a necessary condition for the generation of magnetic fields by kinematic dynamos. Numerical results are presented for some simple ²-dynamo models. The far-field asymptotics for stationary and time-dependent field modes are discussed.