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A shallow-liquid theory in magnetohydrodynamics

Published online by Cambridge University Press:  28 March 2006

L. E. Fraenkel
L. E. Fraenkel
Affiliation:
Aeronautics Department, Imperial College, London
Most of this work was done while the author was on leave of absence at the Guggenheim Aeronautical Laboratory, California Institute of Technology.
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Abstract

The non-linear and linear ‘shallow-water’ theories, which describe long gravity waves on the free surface of an inviscid liquid, are extended to the case of an electrically conducting liquid on a horizontal bottom, in the presence of a vertical magnetic field. The dish holding the liquid, and the medium outside it, are assumed to be non-conducting. The approximate equations are based on a small ratio of depth to wavelength, on the properties of mercury, and on a moderate magnetic field strength. These equations have a ‘magneto-hydraulic’ character, for in the shallow liquid layer the horizontal fluid velocity and current density are independent of the vertical co-ordinate.

Some explicit solutions of the linear equations are obtained for plane flows and for axi-symmetric flows in which the velocity vector lies in a vertical, meridional plane. The amplitudes of waves in a dish, and the amplitudes behind wave fronts progressing into undisturbed liquid, are found to be exponentially damped, the mechanical energy associated with a disturbance being dissipated by Joule heating.

The approximate non-linear equations for plane flow are studied by means of characteristic variables, and it appears that, because of the magnetic damping effect, there is less qualitative difference between solutions of the non-linear and linear approximate equations at large times than is the case when the magnetic field is absent. In particular, the characteristic curves depart only a finite distance from their ‘undisturbed positions’.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 7 , Issue 1 , January 1960 , pp. 81 - 107
Copyright
© 1960 Cambridge University Press

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