Contributions to the theory of a two-scale homogeneous dynamo experiment

Abstract

The principle of the two-scale dynamo experiment at the Forschungszentrum Karlsruhe is closely related to that of the Roberts dynamo working with a simple fluid flow which is, with respect to proper Cartesian coordinates x, y, and z, periodic in x and y and independent of z. A modified Roberts dynamo problem is considered with a flow more similar to that in the experimental device. Solutions are calculated numerically, and on this basis an estimate of the excitation condition of the experimental dynamo is given. The modified Roberts dynamo problem is also considered in the framework of the mean-field dynamo theory, in which the crucial induction effect of the fluid motion is an anisotropic alpha effect. Numerical results are given for the dependence of the mean-field coefficients on the fluid flow rates. The excitation condition of the dynamo is also discussed within this framework. The behavior of the dynamo in the nonlinear regime, i.e., with backreaction of the magnetic field on the fluid flow, depends on the effect of the Lorentz force on the flow rates. The quantities determining this effect are calculated numerically. The results for the mean-field coefficients and the quantities describing the backreaction provide corrections to earlier results, which were obtained under simplifying assumptions.