The integral equation approach to kinematic dynamos in finite domains


The integral equation approach to kinematic dynamos in finite domains

Stefani, F.; Xu, M.; Gundrum, T.; Gerbeth, G.

The usual method to treat kinematic dynamos numerically is to solve the induction equation. For celestial bodies of spherical shape, like the Earth and the Sun, the boundary conditions for the magnetic field can be formulated separately for every degree and order of the spherical harmonics. However, there are a number of aspherical dynamos as, e.g., galactic dynamos and the laboratory dynamos in Riga, Karlsruhe and Cadarache, for which the correct handling of the non-local boundary conditions is a notorious problem. Several methods have been used to deal with this problem, including the use of simplified local boundary conditions, the embedding of the very dynamo region into a spherical region with lower conductivity, or the full solution of the Laplace equation in the exterior.
An alternative way to implement the correct boundary conditions is based on Biot-Savart's law. We present the general formulation of the integral equation approach to both steady and time-dependent dynamos. For spherical dynamos we show the numerical equivalence of this approach with the differential equation approach. The suitability of the method to cope with dynamo problems in arbitrary finite domains is exemplified by the treatment of mean-field dynamos in rectangular boxes.
For flows with small magnetic Reynolds numbers, that are exposed to an external magnetic field, the integral equation approach can be cast into a linear inverse problem for the determination of the flow velocity from externally measured induced magnetic fields. This inverse problem is the basis of the "Contactless Inductive Flow Tomography" (CIFT) which has been demonstrated in a recent experiment. In a cylindrical vessel filled with InGaSn a flow is produced by a motor driven propeller. The moving liquid is exposed alternately to an axial and an transversal magnetic field. The two corresponding sets of induced magnetic fields measured at 49 external Hall sensors are processed in an inverse problems solver to give a rough, but reliable, picture of the flow.

  • Lecture (others)
    Joint meeting of COST-P6-Working Group 1 and CNRS-GDR "Dynamo" , 22.-23.01.2004, Paris, France

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Publ.-Id: 6056