A self-consistent treatment of simple dynamo systems

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Abstract
In Part I, the simple homopolar disc dynamo of Bullard (1955) is discussed, and it is shown that the conventional description is over-simplified and misleading in an important respect, in that it suggests the possibility of exponential growth of the magnetic field even in the limit of perfect disc conductivity, whereas, from fundamental considerations, it is known that the flux of magnetic field across the disc must in this limit remain constant. This contradiction is resolved through consideration of the effect of the azimuthal current distribution which is, in general, inevitably induced in the disc when the conditions for dynamo action are satisfied. By considering a refined model, it is shown that the field growth rate then tends to zero as the disc conductivity tends to infinity. The stability characteristics of this model are determined. In Part II, an analogous contradiction arising in fluid dynamo theory is identified, viz. that whereas the α2-dynamo in a spherical geometry suggests the possibility of exponential field growth even in the limit of perfect conductivity, fundamental considerations (Bondi and Gold, 1950) show that the dipole strength of the field is, in this limit, permanently bounded. Two possible ways of resolving this contradiction are discussed. The first, following a suggestion of Kraichnan (1979), involves consideration of an inhomogeneity layer on the surface of the sphere within which α (and the eddy diffusivity β) falls to zero; diffusion of flux across this layer is analogous to diffusion of flux across the rim of the disc in the simpler disc dynamo context. The second involves introduction of a time lag in the conventional linear relationship between mean field and mean electromotive force; this represents in a crude way the process by which flux has to diffuse into the interior of each helical eddy within which the fundamental field regeneration effect occurs. By either means, compatibility with the Bondi and Gold result may be achieved.

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