# Numerical Simulations of the onset of dynamo action with a hybrid finite volume/ boundary element method (FV-BEM)

## Numerical Simulations of the onset of dynamo action with a hybrid finite volume/ boundary element method (FV-BEM)

Giesecke, A.; Stefani, F.

The experimental realization of dynamo excitation as well as theoretical and numerical examinations of the induction equation have shown the relevance of boundary conditions and material properties for a self-sustaining dynamo. Generally, in non-spherical geometry typical insulating boundary conditions are described by elaborated schemes (e.g. solving of the Laplace equation in an extended domain) or by simplifying approximations (pseudo vacuum). A different approach is provided by a modified integral equation procedure, commonly known as the boundary element method (BEM). Integrating the Laplace equation on the boundaries allows to overcome the difficulties of the non-local character of insulating boundary conditions and the direct computation of the magnetic field next to an insulator becomes possible. However, within the interior of a field producing domain geometric constraints or varying material properties (e.g. electrical conductivity of the container walls or localized high-permeability material) might also play a role. For this problem, a more flexible approach utilizing a local discretization like the constraint transport (CT) method as a well known realization of a finite volume scheme is recommendable. The CT-scheme ensures a fast, robust and accurate solution of the kinematic dynamo problem and intrinsically maintains the solenoidal character of the magnetic field. Combining both methods in a hybrid FV-BEM scheme offers the flexibility of a local discretization with a stringent treatment of insulating magnetic boundary conditions in arbitrary geometries. Preliminary simulations of an alpha^2-Dynamo in a cartesian box and/or the decay of an initial magnetic field demonstrate the applicability of the approach.
Further examinations are intended to understand the behavior of the VKS-dynamo experiment where the field producing flow is driven by ferrous propellers and the dynamo mechanism probably is dominated by this high permeability material.
Other applications of the hybrid scheme may also be important for the evaluation of forthcoming dynamo experiments
for which a precise knowledge of the critical magnetic Reynolds number is essential.

• Lecture (Conference)
10th MHD Days, 26.-29.11.2007, Garching, Germany