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Inertial waves and an initial-value problem for a thin spherical rotating fluid shell

Published online by Cambridge University Press:  12 April 2006

J. M. Huthnance
J. M. Huthnance
Affiliation:
Department of Oceanography, University of LiverpoolPresent address: Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead, Merseyside L43 7RA, England.
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Abstract

Natural modes of oscillation of a vanishingly thin spherical rotating fluid shell, with frequencies σ less than twice the angular velocity Ω, were found by Haurwitz (1940). Their validity is, however, put in question by the presence of a singularity at critical co-latitudes θc: 2Ω cosθc = σ in the O(ε) term of an expansion in the relative shell thickness ε (Stewartson & Rickard 1969). The problem is investigated here by considering the evolution of flow from a specified initial distribution. The principal features are as follows:

(i) The O(1) natural modes of Haurwitz, decaying on a time scale Ω−1ε−2.

(ii) Corrections to (i), regular and of magnitude O(ε) except near critical latitudes.

(iii) Essentially transient inertial waves of magnitude O(ε).

(iv) Inertial waves of magnitude O(ε½) with the natural-mode frequencies σ and generated by (i) at critical latitudes.

On a time scale Ω−1ε−1, (iii) and (iv) develop vertical and horizontal length scales ε and propagate throughout the ocean. The continuing energy transfer from (i) to (iv), at a rate O(ε2Ω), appears to be the principal respect in which (i) and (ii) fail to constitute a conventional normal mode.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 86 , Issue 2 , 29 May 1978, pp. 273 - 288
Copyright
© 1978 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Aldridge, K. D. 1973 Axisymmetric inertial oscillations of a fluid in a rotating spherical shell. Mathematika 19, 163–168.Google Scholar
Haurwitz, B. 1940 The motion of atmospheric disturbances on the spherical earth. J. Mar. Res. 3, 254–267.Google Scholar
Longuet-Higgins, M. S. 1964 Planetary waves on a rotating sphere. Proc. Roy. Soc. A 279, 446–473.Google Scholar
Miles, J. W. 1974 On Laplace's tidal equations. J. Fluid Mech. 66, 241–260.Google Scholar
Pekeris, C. L. 1975 A derivation of Laplace's tidal equation from the theory of inertial oscillations. Proc. Roy. Soc. A 344, 81–86.Google Scholar
Stewartson, K. 1971 On trapped oscillations of a rotating fluid in a thin spherical shell. Tellus 23, 506–510.Google Scholar
Stewartson, K. & Rickard, J. A. 1969 Pathological oscillations of a rotating fluid. J. Fluid Mech. 35, 759–773.Google Scholar
Stewartson, K. & Walton, I. C. 1976 On waves in a thin shell of stratified rotating fluid. Proc. Roy. Soc. A 349, 141–156.Google Scholar
Walton, I. C. 1975 On waves in a thin rotating spherical shell of slightly viscous fluid. Mathematika 22, 46–59.Google Scholar