Publications Repository - Helmholtz-Zentrum Dresden-Rossendorf

Usually, homogeneous dynamos are simulated in the framework of the differential equation approach. For purely kinematic models, the induction equation for the magnetic field has to be solved. For spherically shaped dynamo domains, such as planets and stars, the problem of implementing the non-local boundary conditions for the magnetic field is trivially solved by introducing separate boundary conditions for every degree of the spherical harmonics. However, for the simulation of other than spherically shaped dynamos, in particular for galactic and some of the recent laboratory dynamos, the handling of the boundary conditions is a notorious problem. The integral equation approach provides a solution to this problem. Basically, this approach is an application of Biot-Savart's law for dynamos. The magnetic field is produced by currents driven by an electromotive force that, in its turn, depends on the magnetic field. For the case of finite domains, the simple Biot-Savart equation has to be supplemented by a boundary integral equation for the electric potential. If the dynamo becomes time-dependent, yet another equation for the vector potential at the boundary has to be added. First numerical tests of the method for some well-investigated dynamo models in spherical domains have shown convincing agreement with the results of the differential equation method. The prospective advantage of the method, however, is its suitability to handle dynamos in other than spherical geometries.