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Withdrawal of a stratified fluid from a rotating channel

Published online by Cambridge University Press:  26 April 2006

N. Robb McDonald  and
Jörg Imberger
N. Robb McDonald
Affiliation:
Department of Civil and Environmental Engineering and Centre for Water Research, University of Western Australia, Nedlands 6009, WA, Australia Present address: Robert Hooke Institute, The Observatory, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK.
Jörg Imberger
Affiliation:
Department of Civil and Environmental Engineering and Centre for Water Research, University of Western Australia, Nedlands 6009, WA, Australia
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Abstract

The flow of a stratified fluid toward a line sink in a rotating channel of finite width and depth is studied. The withdrawal flow is shown to be established by a set of Kelvin shear waves trapped within a distance of Nh/fn from the right-hand side wall (f > 0) looking in the direction of propagation, where n = 1, 2,… is the vertical mode number. In addition there are a set of waves (Poincaré modes) which propagate away from the sink with a cross-channel modal structure. The withdrawal flow has a boundary-layer structure: far from the right-hand wall the flow resembles that of McDonald & Imberger (1991), whereas close to the right-hand wall the development of the vertical structure of the withdrawal flow resembles that of the non-rotating case due to the presence of Kelvin shear waves. In a narrow channel Kelvin shear waves dominate the establishment of the withdrawal flow. The withdrawal flow is investigated for large times compared to the inertial period, where it is shown that the width of the boundary layer is of the same order as the distance downstream from the sink. The flow within the boundary layer is unsteady as the withdrawal layer thickness δ continues to collapse indefinitely, while outside the boundary layer it is steady with δ ∼ fL/N, L being the horizontal lengthscale downstream from the sink. A scaling analysis is developed for the narrow channel case in which the cross-channel velocity can be ignored. The results are applied to actual field data, where it is shown that the effect of rotation may explain why previous non-rotating theories have been inaccurate in predicting withdrawal layer thickness.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 235 , February 1992 , pp. 643 - 664
Copyright
© 1992 Cambridge University Press

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References

Bretherton, F. P. 1967 The time dependent motion due to a cylinder moving in an unbounded rotating or stratified fluid J. Fluid Mech. 28, 545–570.Google Scholar
Condie, S. A. & Ivey, G. N. 1988 Convectively driven coastal currents in a rotating basin. J. Mar. Res. 46, 473–494.Google Scholar
Gill, A. E. 1976 Adjustment under gravity in rotating channel. J. Fluid Mech. 77, 603–601.Google Scholar
Herman, A. J., Rhines, P. B. & Johnson, E. R. 1989 Nonlinear Rossby adjustment in a channel: beyond Kelvin waves. J. Fluid Mech. 205, 469–502.Google Scholar
Hogg, N. G. 1985 Multilayer hydraulic control with application to the Alboran Sea circulation. J. Phys. Oceanogr. 15, 454–466.Google Scholar
Imberger, J. 1972 Two-dimensional sink flow of a stratified fluid contained in duct. J. Fluid Mech. 53, 329–349.Google Scholar
Imberger, J. 1980 Selective withdrawal: a review. In 2nd Intl Symp. on Stratified Flows, Trondheim, p. 381.
Imberger, J. & Patterson, J. C. 1990 Physical limnology. Adv. Appl. Mech. 27, 303–475.Google Scholar
Imberger, J., Thompson, R. & Fandry, C. 1976 Selective withdrawal from a finite rectangular tank. J. Fluid Mech. 78, 489–512.Google Scholar
Ivey, G. N. & Blake, S. 1985 Axisymmetric withdrawal and inflow in a density stratified container. J. Fluid Mech. 161, 115–137.Google Scholar
Ivey, G. N. & Imberger, J. 1978 Field investigation of selective withdrawal. J. Hydraul. Div. ASCE 9, 1225–1236.Google Scholar
Kranenburg, C. 1980 Selective withdrawal from a rotating two layer fluid. In IAHR Proc. Second Intl Symp. on Stratified Flows, Trondheim, Norway, vol. 1, pp. 411–423.Google Scholar
McLachlan, N. W. 1955 Bessel Functions for Engineers. Oxford. 239 pp.
McDonald, N. R. 1990 Far-field flow forced by the entrainment of a convective plane plume in a rotating stratified fluid. J. Phys. Oceanogr. 20, 1791–1798.Google Scholar
McDonald, N. R. & Imberger, J. 1991 A line sink in a rotating stratified fluid. J. Fluid Mech. 233, 349–368 (referred to herein as MI).Google Scholar
Monismith, S. G., Imberger, J. & Billi, G. 1988 Unsteady selective withdrawal from a line sink. J. Hydraul. Div. ASCE 114, 1134–1152.Google Scholar
Monismith, S. G. & Maxworthy, T. 1989 Selective withdrawal and spin up of a rotating stratified fluid. J. Fluid Mech. 199, 377–401.Google Scholar
Pao, H.-S. & Kao, T. W. 1974 Dynamics of establishment of selective withdrawal of a stratified fluid from a line sink. Part 1. Theory. J. Fluid Mech. 65, 657.Google Scholar
Pedlosky, J. 1979 Geophysical Fluid Dynamics. Springer. 624 pp.
Roberts, G. E. & Kaufman, H. 1966 Table of Laplace Transforms. Saunders. 367 pp.
Stakgold, I. 1967 Boundary Value Problems of Mathematical Physics, Vol. 1. Macmillan. 340 pp.
Whitehead, J. A. 1980 Selective withdrawal of a rotating stratified fluid. Dyn. Atmos. Oceans 5, 123.Google Scholar
Whitehead, J. A. 1985 A laboratory study of gyres and uplift near the Strait of Gibraltar. J. Geophys. Res. C 90, 7045–7060.Google Scholar
Wong, K. K. & Kao, T. W. 1970 Stratified flow over extended obstacles and its application to topographical effect on vertical wind shear. J. Atmos. Sci. 27, 884–889.Google Scholar