Optimized kinematic dynamos

A numerical optimization approach is introduced to the subject of dynamo theory. Conventional kinematic dynamo studies treat the induction equation as an eigenvalue problem by choosing a candidate velocity field and solving for a marginally stable solution of magnetic field and critical magnetic Reynolds number. The conventional approach has told us something about dynamo action and magnetic field morphology for specific velocities, but the arbitrary choice of fluid flow is a hit-or-miss affair; not all velocities sustain dynamo action, and of those that do, few yield mathematically tractable solutions. As a result, progress has been slow. Here we adopt a new approach, a non-linear numerical variational approach, which allows us to solve the induction equation simultaneously for both the magnetic field and the velocity field. The induction equation is discretized following the Bullard-Gellman formalism and the resulting algebraic equations solved by an iterative, globally convergent, Newton-Raphson method. The particular choice of optimization constraints allows one to design a dynamo which satisfies certain conditions; in this paper we minimize a linear combi-nation of the kinetic energy (magnetic Reynolds number) and a smoothness norm on