OPTIMIZED KINEMATIC DYNAMOS

A numerical optimization approach is introduced to the subject of dyna mo theory. Conventional kinematic dynamo studies treat the induction e quation as an eigenvalue problem by choosing a candidate velocity fiel d and solving for a marginally stable solution of magnetic field and c ritical magnetic Reynolds number. The conventional approach has told u s something about dynamo action and magnetic field morphology for spec ific velocities, but the arbitrary choice of fluid flow is a hit-or-mi ss affair; not all velocities sustain dynamo action, and of those that do, few yield mathematically tractable solutions. As a result, progre ss 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 held. The induction equation is discretized following the Bullard-Gellman forma lism and the resulting algebraic equations solved by an iterative, glo bally convergent, Newton-Raphson method. The particular choice of opti mization constraints allows one to design a dynamo which satisfies cer tain conditions; in this paper we minimize a linear combination of the kinetic energy (magnetic Reynolds number) and a smoothness norm on th e magnetic field to produce efficient (low magnetic Reynolds number) w ell-converged (smooth magnetic held) solutions. We illustrate the opti mization method by designing two dynamos based on a Kumar-Roberts velo city parametrization; a specific choice of the velocity parameters, KR , sustains a 3-D kinematic model of the geodynamo. Compared with KR, o ne of our new models, LG1, is designed to have a higher magnetic Reyno lds number but smoother magnetic field, and the other, LG2, a lower ma gnetic Reynolds number and somewhat rougher magnetic field. We suggest that dynamo efficiency, defined by the magnetic Reynolds number, is a chieved through reduced differential rotation and a favourable spatial distribution of the helicity. These examples demonstrate the value of the optimization method as a tool for exploring dynamo action with ge ophysically realistic flows. It can be extended to the dynamic dynamo problem and, by changing the constraints, be used to design dynamos wi th good numerical convergence, which match the observed geomagnetic su rface field morphology and which place useful quantitative constraints on the physical nature of the geodynamo.