Integral equation approach to time-dependent kinematic dynamos in finite domains

Journal Article · Mon Nov 01 04:00:00 EST 2004 · Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

DOI:https://doi.org/10.1103/PhysRevE.70.056305· OSTI ID:20636866

Mingtian, Xu; Stefani, Frank; Gerbeth, Gunter ^[1]

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1. Forschungszentrum Rossendorf, P.O. Box 510119, D-01314 Dresden (Germany)

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 well-known picture of the Biot-Savart 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 time-dependent kinematic dynamos, with stationary dynamo sources, in finite domains. The time dependence is restricted to the magnetic field, whereas the velocity or corresponding mean-field sources of dynamo action are supposed to be stationary. For the spherically symmetric {alpha}{sup 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 toroidal field components. The integral equation formulation for spherical dynamos with general stationary velocity fields is also derived. Two numerical examples - the {alpha}{sup 2} dynamo model with radially varying {alpha} and the Bullard-Gellman model - illustrate the equivalence of the approach with the usual differential equation method. The main advantage of the method is exemplified by the treatment of an {alpha}{sup 2} dynamo in rectangular domains.

• OSTI ID: 20636866
• Journal Information: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics Journal Issue: 5 Vol. 70; ISSN PLEEE8; ISSN 1063-651X
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BIOT-SAVART LAW
DIFFERENTIAL EQUATIONS
INTEGRAL EQUATIONS
INTERPLANETARY MAGNETIC FIELDS
INTERSTELLAR MAGNETIC FIELDS
MAGNETOHYDRODYNAMICS
MEAN-FIELD THEORY
SCALARS
TIME DEPENDENCE