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The Quarterly Journal of Mechanics and Applied Mathematics 1957 10(4):385-405; doi:10.1093/qjmam/10.4.385
© 1957 by Oxford University Press
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TWO-DIMENSIONAL PROBLEMS OF THE DECAY OF MAGNETIC FIELDS IN MAGNETOHYDRODYNAMICS

T. G. COWLING and A. HARE

( The University Leeds )

The normal modes of decay of a magnetic field in the presence of steady motions in a fluid conductor are studied in the two-dimensional case. Slow motions are considered by perturbation methods, which show that to a first approximation the motion always increases the rate of decay in the lowest mode. For fast motion, there exists a limited class of modes little affected by the motion, in which the lines of force and streamlines nearly coincide. In the general class of modes the motion profoundly affects the decay; these are studied in two special cases, those of stream-ing in a fluid slab between parallel walls and of non-uniform rotation in a circular cylinder. The motion is found to transport and distort the field, and to increase the rate of decay in the lower modes, roughly proportional to the 2/3 power of the velocity. For the slab problem, velocities of intermediate magnitude are studied by numerical methods. The extension of the results to more general classes of two-dimensional motion is briefly considered.


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