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Withdrawal of a stratified fluid from a rotating channel

  • N. Robb McDonald (a1) (a2) and Jörg Imberger (a1)

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.

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References

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Bretherton, F. P. 1967 The time dependent motion due to a cylinder moving in an unbounded rotating or stratified fluid J. Fluid Mech. 28, 545570.
Condie, S. A. & Ivey, G. N. 1988 Convectively driven coastal currents in a rotating basin. J. Mar. Res. 46, 473494.
Gill, A. E. 1976 Adjustment under gravity in rotating channel. J. Fluid Mech. 77, 603601.
Herman, A. J., Rhines, P. B. & Johnson, E. R. 1989 Nonlinear Rossby adjustment in a channel: beyond Kelvin waves. J. Fluid Mech. 205, 469502.
Hogg, N. G. 1985 Multilayer hydraulic control with application to the Alboran Sea circulation. J. Phys. Oceanogr. 15, 454466.
Imberger, J. 1972 Two-dimensional sink flow of a stratified fluid contained in duct. J. Fluid Mech. 53, 329349.
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, 303475.
Imberger, J., Thompson, R. & Fandry, C. 1976 Selective withdrawal from a finite rectangular tank. J. Fluid Mech. 78, 489512.
Ivey, G. N. & Blake, S. 1985 Axisymmetric withdrawal and inflow in a density stratified container. J. Fluid Mech. 161, 115137.
Ivey, G. N. & Imberger, J. 1978 Field investigation of selective withdrawal. J. Hydraul. Div. ASCE 9, 12251236.
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. 411423.
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, 17911798.
McDonald, N. R. & Imberger, J. 1991 A line sink in a rotating stratified fluid. J. Fluid Mech. 233, 349368 (referred to herein as MI).
Monismith, S. G., Imberger, J. & Billi, G. 1988 Unsteady selective withdrawal from a line sink. J. Hydraul. Div. ASCE 114, 11341152.
Monismith, S. G. & Maxworthy, T. 1989 Selective withdrawal and spin up of a rotating stratified fluid. J. Fluid Mech. 199, 377401.
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.
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.
Whitehead, J. A. 1985 A laboratory study of gyres and uplift near the Strait of Gibraltar. J. Geophys. Res. C 90, 70457060.
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, 884889.
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