Time-Dependent Kinematic Dynamos in Finite Cylinders as Treated with the Integral Equation Approach


Time-Dependent Kinematic Dynamos in Finite Cylinders as Treated with the Integral Equation Approach

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

Dynamo action is generally accepted as the source of cosmic magnetic fields, such as the fields of planets, stars and galaxies. The understanding of dynamos has considerably progressed during the last decades. Recently, the hydromagnetic dynamo effect has been demonstrated experimentally in large liquid sodium facilities in Riga and Karlsruhe.
The electrically conducting fluid is usually confined to a finite domain which is surrounded by an insulator. For only a few cases, such as spheres or cylinders with infinite length, the boundary conditions for the magnetic field are easily implementable. In most cases, including finite cylinders which are relevant for the liquid sodium experiments in Riga, Karlsruhe, Cadarache and New Mexico, the correct handling of the non-local boundary conditions is far from trivial. We have re-formulated Pre-Maxwell's equations to an integral equation system under the assumption that the velocity field is stationary.
The main advantage of this approach is that the numerical solution of the integral equation system does not require any discretization of the exterior region.
In the present work, the integral equation approach is performed on a group of Beltrami flows (and some modifications) with the topologies s2t2, s2t1 and s1t1 in a finite cylinder. Beltrami flows are characterized by maximum helicity for a given magnetic Reynolds number. Due to the axisymmetry of these flows, the integral equation system in the cylindrical system is reduced to a two-dimensional form.
The calculated results show a good agreement with those obtained by a differential equation approach. This integral equation approach exhibits attractive features such as robustness and stability.
The impact of a stagnant conducting layer surrounding the cylinder is also investigated. Such a layer can reduce the critical magnetic Reynolds number significantly, and it can even transform no-dynamos into dynamos. Interestingly, the presence of a layer can also change an oscillatory dynamo to a steady dynamo.

  • Lecture (Conference)
    7th MHD-Days, Technische Universität Ilmenau, 20.-21.09.2004, Ilmenau, Germany

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Publ.-Id: 6335