Abstract
The kinematic dynamo problem poses a non-self-adjoint eigenvalue problem. Corresponding to this eigenvalue system is its adjoint, which has the same eigenvalues but different eigenfunctions that are orthogonal to those of the dynamo system. The differential equations governing the two systems are the same, but the velocity appears with opposite signs; the α-effect (if present) is the same in each; the boundary conditions are different. The adjoint problem is useful in (a) examining why so often dynamos of dipole and quadrupole parity are excited with almost equal ease, (b) providing a check on the accuracy of numerical calculations, and (c) constructing weakly nonlinear solutions of the full MHD dynamo problem, at slightly supercritical magnetic Reynolds numbers, a topic postponed to sequels of this paper.
The adjoint system is deduced here both for spherical dynamos and for dynamos operating in containers of general form. The application of a technique developed by one of us previously (KONO, 1990), that rests on the use of computer algebra, is generalized to include the α-effect. Eigenvalues of both the dynamo system and its adjoint are derived numerically for a number of axisymmetric spherical dynamos of α2- and αω-type. Agreement between the eigenvalues of the two systems, and between those reported in earlier integrations, is demonstrated, as is the closeness of the eigenvalues of dipole and quadrupole type for α2-dynamos; their separation for αω-dynamos is greater.
