Integral equation approach to timedependent kinematic dynamos in finite domains
Integral equation approach to timedependent kinematic dynamos in finite domains
Xu, M.; Stefani, F.; Gerbeth, G.
The homogeneous dynamo effect is at the root of cosmic magnetic field generation. With only a very few exceptions, the numerical treatment of homogeneous dynamos is carried out in the framework of the differential equation approach. The present paper tries to facilitate the use of integral equations in dynamo research. Apart from the pedagogical value to illustrate dynamo action within the wellknown picture of BiotSavart's law, the integral equation approach has a number of practical advantages. The first advantage is its proven numerical robustness and stability. The second and perhaps most important advantage is its applicability to dynamos in arbitrary geometries. The third advantage is its intimate connection to inverse problems relevant not only for dynamos but also for technical applications of magnetohydrodynamics. The paper provides the first general formulation and application of the integral equation approach to timedependent kinematic dynamos in finite domains. For the spherically symmetric alpha^2 dynamo model it is shown how the general formulation is reduced to a coupled system of two radial integral equations for the defining scalars of the poloidal and the toroidal field components. The integral equation formulation for spherical dynamos with general velocity fields is also derived. Two numerical examples, the alpha^2 dynamo model with radially varying alpha, and the BullardGellman model illustrate the equivalence of the approach with the usual differential equation method.

Physical Review E 70(2004), 56305 ff
DOI: 10.1103/PhysRevE.70.056305
Cited 25 times in Scopus
Permalink: https://www.hzdr.de/publications/Publ5610
Publ.Id: 5610