Blackwell Publishing Journal Backfiles 1879-2005
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 combination of the kinetic energy (magnetic Reynolds number) and a smoothness norm on the magnetic field to produce efficient (low magnetic Reynolds number) well-converged (smooth magnetic field) solutions. We illustrate the optimization method by designing two dynamos based on a Kumar-Roberts velocity parametrization; a specific choice of the velocity parameters, KR, sustains a 3-D kinematic model of the geodynamo. Compared with KR, one of our new models, LG1, is designed to have a higher magnetic Reynolds number but smoother magnetic field, and the other, LG2, a lower magnetic Reynolds number and somewhat rougher magnetic field. We suggest that dynamo efficiency, defined by the magnetic Reynolds number, is achieved 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 geophysically realistic flows. It can be extended to the dynamic dynamo problem and, by changing the constraints, be used to design dynamos with good numerical convergence, which match the observed geomagnetic surface field morphology and which place useful quantitative constraints on the physical nature of the geodynamo.
Type of Medium: