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On free-surface oscillations in a rotating paraboloid

Published online by Cambridge University Press:  28 March 2006

John W. Miles  and
F. K. Ball
John W. Miles
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra
F. K. Ball
Affiliation:
C.S.I.R.O. Division of Meteorological Physics, Aspendale, Victoria, Australia
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Abstract

Lamb's analysis of small-amplitude, shallow-water oscillations in a rotating paraboloid, interpreted by him in the inconsistent context of an approximately plane free surface, is re-interpreted to obtain results that are valid for $0 \le \omega^2l|2g \; \textless \;1$ (ω = rotational speed, l = latus rectum of paraboloid); no equilibrium is possible for ω2l/2g > 1. It is shown that the frequencies of the dominant modes for the azimuthal wave numbers 0 (axisymmetric motion) and 1 are independent of ω for an observer in a non-rotating reference frame and that the frequencies of all other axisymmetric modes are decreased by rotation (Lamb concluded that they would be increased). An axisymmetric mode of zero frequency, which was over-looked by Lamb, also is found.

Exact solutions to the non-linear equations of motion, which reduce to the aforementioned dominant modes for small amplitudes, are determined. The axisymmetric solution is inferred from similarity considerations and is found to contain all harmonics of the fundamental frequency. The finite motion of azimuthal wave-number 1 is a quasi-rigid displacement of the liquid and is found to be simple harmonic except for a second-harmonic component of the free-surface displacement (but the horizontal velocity at a given point remains simple harmonic).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 17 , Issue 2 , October 1963 , pp. 257 - 266
Copyright
© 1963 Cambridge University Press

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References

Ball, F. K. 1962 An exact theory of simple finite shallow water oscillations on a rotating earth. Proc. Conference on Fluid Mechanics and Hydraulics, University of Western Australia, Nedlands, 1962, p 293. Pergamon Press.
Ball, F. K. 1963 Some general theorems concerning the finite motion of a shallow rotating liquid lying on a paraboloid. J. Fluid Mech. 17, 240.Google Scholar
Fultz, Dave 1962 Trans. N. Y. Acad. Sci 24, 421–46.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Miles, J. W. 1963 Free-surface oscillations in a slowly rotating liquid. J. Fluid Mech. (In the press.)Google Scholar
Murty, T. S. 1962 Gravity modes in a rotating paraboloidal dish. M. S. Dissertation, University of Chicago.
Platzman, G. W. 1962 Unpublished; cited by Fultz l.c. ante (1962).