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On the structure of inertial waves produced by an obstacle in a deep, rotating container

Published online by Cambridge University Press:  19 April 2006

K. Stewartson  and
H. K. Cheng
K. Stewartson
Affiliation:
Department of Mathematics, University College London
H. K. Cheng
Affiliation:
Department of Aerospace Engineering, University of Southern California, Los Angeles
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Abstract

The inviscid flow above an obstacle in slow transverse motion inside a rotating vessel is analysed to study the influence of the container depth on the basic steady flow structure. An asymptotic theory is presented for an arbitrarily small Rossby number Ro = U0/2Ωl under a fixed H = hRo/l (where Ω is the angular velocity of the container, U0 the obstacle velocity relative to the vessel, h the depth of the container, and l a body length measured transversely to the rotation axis). The equations when linearized for a thin obstacle or shallow topography take on the form of the inertial-wave equation; their solutions for non-vanishing H are obtained for obstacles of three-dimensional as well as ridge-like two-dimensional shapes. In all cases analysed, the solution possesses a bimodal structure, of which one part is column-like with a vorticity proportional to the body elevation (or ground topography). The other part is confined mainly to a region enclosing the body, extending a distance O(H½) upstream of the obstacle and behind a wedge-shaped caustic front at large distances; its contribution consists of lee waves similar to that discussed by Cheng (1977) for an infinite depth. The field associated with the lee waves is then biased on the downstream side, but there is little indication of any tendency to tilting in the sense of Hide, Ibbetson & Lighthill (1968).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 91 , Issue 3 , 11 April 1979 , pp. 415 - 432
Copyright
© 1979 Cambridge University Press

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References

Batchelor, G. K. 1967 Fluid Dynamics, p. 567. Cambridge University Press.
Boyer, D. L. 1970a Trans. A.S.M.E. J. Bas. Eng. D 92, 430.
Boyer, D. L. 1970b J. Fluid Mech. 50, 675.
Cheng, H. K. 1977 Z. angew. Math. Phys. 28, 753.
Grace, S. F. 1926 Proc. Roy. Soc. A 113, 46.
Greenspan, H. P. 1969 The Theory of Rotating Fluids, p. 51. Cambridge University Press.
Hardy, G. H. & Littlewood, J. E. 1913 Proc. London Math. Soc. (2), 11, 411.
Hide, R. & Ibbetson, A. 1966 Icarus 5, 279.
Hide, R., Ibbetson, A. & Lighthill, M. J. 1968 J. Fluid Mech. 32, 251.
Huppert, H. E. 1975 J. Fluid Mech. 67, 397.
Ingersoll, A. P. 1969 J. Atms. Sci. 26, 744.
Jacobs, S. J. 1964 J. Fluid Mech. 20, 581.
James, I. N. 1977 (Private communication.)
Lamb, H. 1932 Hydrodynamics, 6th ed., p. 434. Cambridge University Press.
Long, R. R. 1953 J. Met. 10, 197.
Lighthill, M. J. 1967 J. Fluid Mech. 27, 725.
Mason, P. J. 1975 J. Fluid Mech. 71, 577.
Maxworthy, T. 1968 J. Fluid Mech. 31, 643.
Maxworthy, T. 1970 J. Fluid Mech. 40, 453.
Maxworthy, T. 1977 Z. angew. Math. und Phy. 28, 853.
Prandtl, L. 1952 Fluid Dynamics, p. 353. New York: Hafner Publishing Co.
Proudman, J. 1916 Proc. Roy. Soc. A 92, 408.
Stewartson, K. 1953 Quart. J. Mech. Appl. Math. 6, 141.
Stewartson, K. 1967 J. Fluid Mech. 30, 357.
Taylor, G. I. 1922 Proc. Roy. Soc. A 102, 114.
Taylor, G. I. 1923 Proc. Roy. Soc. A 104, 213.
Walker, J. D. A. & Stewartson, K. 1972 Z. angew. Math. Phys. 23, 745.
Walker, J. D. A. & Stewartson, K. 1974 J. Fluid Mech. 66, 767.