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1990 A model for a simple dynamo system

Christophe LETELLIER
11/05/2009
François Lusseyran & Jean-Pierre Brancher
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François Lusseyran

Homopolar dynamo models are used for the understanding of spontaneous magnetic field generation in magnetohydrodynamic flows. In 1979, Moffatt [1] proposed a heuristic model of the disk dynamo of Edward Bullard (1907-1980) taking into account the field exclusion process necessary to satisfy the Alfven theorem of flux conservation. This self-consistent model is obtained by a segmentation of the disk, leaving the possibility of azimuthal currents to exclude the magnetic field.

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Sketch of the dynamo

In 1964, Christian Rioux [2] studied a homopolar dynamo to create current impulses. This machine, to be described later, works exclusively in a nonstationary regime. The copper cylinder is first rotated above the critical starting velocity, then connected to the main coil through liquid contacts, and then the kinetic energy of the wheel is transformed to the electrical form. If the conductivity is not too high, the field exclusion process is negligible, and we obtain only one current peak of constant sign.

- The system

In the case of high conductivities, the field exclusion must be taken into account, and we proposed the use of a segmented dynamo model which produces to more or less complicated oscillations. The set of differential equations [3]


\left\{
    \begin{array}{l}
      \dot{x} = \alpha \left[ \dis (z-1)x +(z+\beta)y \right] \\[0.3cm]
      \dot{y} = \alpha \left[ \dis (1-z)x -(z+\gamma)y \right] \\[0.3cm]
      \dot{z} = -xy - x^2 +C -\nu z \\[0.3cm]
    \end{array}
  \right.
is such that the time is dimensionless according to  \frac{Rl}{L}. Parameters are


  \left\{
    \begin{array}{l}
      \displaystyle \alpha = \frac{LL'}{LL' - M^2} = 1.01 \\[0.3cm]
      \displaystyle \beta = \frac{R'M^2}{RL'^2 } = 0.1136 \\[0.3cm]
      \displaystyle \gamma = \frac{R'L}{RL' } = 11.25 \\[0.3cm]
      \displaystyle C = \frac{ML}{R^2} \frac{T}{4 \pi^2 I_2} =93.5 \\[0.3cm]
      \displaystyle \nu = 3.0 
    \end{array}
  \right.

With the parameters given above, this dynamo system produces a chaotic attractor that is topologically equivalent to the Lorenz attractor (Fig. 1).

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Fig. 1 : Chaotic "Lorenz" attractor solution to the simple dynamo system.

When parameters are changed with \alpha=1.78 and \nu=0.0, the chaotic attractor solution to the simple dynamo system becomes topologically equivalent to the "Burke and Shaw" attractor (Fig. 2).

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Fig. 2 : Chaotic "Burke and Shaw" attractor solution to the simple dynamo model.

[1] H. K. Moffat, A self-consistent treatment of simple dynamo systems, Geophysical and Astrophysical Fluid Dynamics, 14, 147-166, 1979.

[2] C. Rioux, Etude et réalization d’une dynamo unipolaire à régime impulsionnel, Thèse, Paris XI, 1964.

[3] F. Lusseyran \& J. P. Brancher, Some results on simple dynamo systems, IEEE Transactions on Magnetics, 26 (5), 2875-2879, 1990.

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