Spatially periodic dynamos

Abstract

It is established analytically that, in a precisely defined sense, almost all steady spatially periodic motions of a homogeneous conducting fluid will give dynamo action at almost all values of the conductivity. The same result is obtained for motions periodic in space-time. The asymptotic form of the growing field, for an arbitrary initial field of finite energy, is also presented. Dynamo action is first shown to require that for some real vector there is a magnetic field solution of the form B= H exp (pt+ij. x), where H is a complex function of position (or of position and time) with the same periodicity as the motion, and p has positive real part, indicating growth. This number p is an eigenvalue of a linear differential operator on the space of admissible functions H. The first term of a power series in j for the eigenvalues/) which vanish to zero order is studied. It is thus proved sufficient for dynamo action that the determinant of the symmetric part of a certain 3 x 3 tensor, a function of the motion and conductivity, is non-zero. Finally, it is shown that this determinant is an analytic function of the conductivity, and is non-zero in a small conductivity limit for nearly all motions. This proves the stated result.