Mean-field coefficients for helical flow fields

We have performed kinematic simulations of dynamo action driven by a small scale helical flow of a conducting fluid. We considered different Roberts-flow like configurations with and without a vertical mean flow in a periodic Cartesian domain. Mean field coefficients that allow a parameterization of induction effects of the small scale flow in terms of an alpha- and beta-effect are computed with the so-called test-field method. The validity of the test-field method is checked by comparing the numerical solutions obtained with a fully resolved velocity field and the solutions from a low resolution mean field model based on the corresponding alpha- and beta coefficients.

Below and slightly above the dynamo threshold our results show a good agreement in terms of growth-rates and vertical field wavenumber. In the strongly overcritical regime, however, field modes with a larger vertical wave number emerge and require a careful consideration of the scale dependence of the alpha-effect. As expected, the agreement between fully resolved models and mean field models is better for increasing scale separation, i.e., for smaller horizontal flow scales.

The behavior of the beta-effect, that essentially describes an enhancement of the magnetic diffusivity by the action of the small scale flow, is determined by the ratio of horizontal to vertical velocity components. For a small vertical flow amplitude, beta scales proportional to the square of the local magnetic Reynolds number whereas for a large relation between vertical and horizontal flow component, beta becomes proportional to the cube root of Rm. This behavior is in rough agreement with measurements of the turbulent beta-effect from Frick et al. (PRL 2010, 105 (18), 184502).

In models with mean flow we have also taken into account internal rods and/or walls that lie in the center of individual eddies and/or provide a separation of the eddies from each other. These flow guiding fixtures can be made of soft iron with a relative permeability much larger than one. The associated inhomogeneity significantly reduces the critical magnetic Reynolds number. However, the problem of modelling the induction effects caused by the non-uniform permeability distribution within the framework of mean field coefficients is yet to be solved.

Without additional effects of permeability, our results allow an easy extrapolation to arbitraryly large systems that consist of an extremely large number of individual helical eddies. Such flows cannot be resolved in direct simulations, but the corresponding mean field models allow to assess whether a certain configuration provides for dynamo action. Such estimations are indeed useful because the typical flow in the cores of fast reactors consists of a forced helical Motion around each individual fuel rod. The occurrence of dynamo action may cause a significant pressure drop in the system with unpredictable consequences for the cooling of the core.