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Slow oscillations in an ocean of varying depth Part 1. Abrupt topography

Published online by Cambridge University Press:  29 March 2006

P. B. Rhines
P. B. Rhines
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Present address: Department of Meteorology, Massachusetts Institute of Technology.
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Abstract

This paper is part of a study of quasigeostrophic waves, which depend on the topography of the ocean floor and the curvature of the earth.

In a homogeneous, β-plane ocean, motion of the fluid across contours of constant f/h releases relative vorticity (f is the Coriolis parameter and h the depth). This well-known effect provides a restoring tendency for either Rossby waves (with h constant) or topographic waves over a slope. The long waves in general obey an elliptic partial differential equation in two space variables. Because the equation has been integrated in the vertical direction, the exact inviscid bottom boundary condition appears in variable coefficients.

When the depth varies in only one direction the equation is separable at the lowest order in ω, the frequency upon f. With a simple slope, |[xdtri ]h/h| = constant, the transition from Rossby to topographic waves occurs at |[xdtri ]h| ∼ h/Re, where Re is the radius of the earth. Isolated topographic features are considered in §2. It is found that a step of fractional height δ on an otherwise flat ocean floor reflects the majority of incident Rossby waves when δ > 2ω. In the ocean ω is usually small, due to continental barriers, so even slight depth variations are important. A narrow ridge does not act as a great obstruction but calculations show, for example, that the Mid-Atlantic Ridge is broad enough to reflect all but the lowest mode Rossby waves in the North Atlantic.

Besides isolating oceanic plains from one another, steps and ridges support trapped topographic waves of greatest frequency ∼ δ/2, analogous to the potential well solutions in quantum mechanics. These waves cannot carry energy along abrupt topography, but they disperse more rapidly over broader slopes; the phase and group speeds may be hundreds of cm/sec. The continentalshelf waves found by Robinson are an example of the latter case. There are many such wave guides, where the f/h contours are crowded, in the deep ocean.

The theory suggests that measurement of Rossby waves will rarely be possible at the coast of a continent.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 37 , Issue 1 , 5 June 1969 , pp. 161 - 189
Copyright
© 1969 Cambridge University Press

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References

Ball, F. K. 1963a An exact theory of simple finite shallow water oscillations on a rotating earth. Proc. 1st Australasian Conf. on Hydraulics and Fluid Mechanics. Oxford: Pergamon Press.
Ball, F. K. 1963b Some general theorems concerning the finite motion of a shallow rotating liquid lying on a paraboloid. J. Fluid Mech. 17, 240.Google Scholar
Ball, F. K. 1965a The effect of rotation on the simpler modes of motion of a liquid in an elliptic paraboloid. J. Fluid Mech. 22, 529.Google Scholar
Ball, F. K. 1965b Second-class motions of a shallow liquid. J. Fluid Mech. 23, 545.Google Scholar
Beardsley, R. C. 1963 Ph.D. Thesis, M.I.T. Department of Geology and Geophysics.
Bretherton, F. P., Carrier, G. F. & Longuet-Higgins, M. S. 1966 Report on the I.U.T.A.M. Symposium on rotating fluid systems. J. Fluid Mech. 26, 393.Google Scholar
Fultz, D. & Kaylor, D. 1959 The propagation of frequency in experimental baroclinic waves in a rotating annular ring. The Atmosphere and the Sea in Motion. New York: Rockefeller Inst. Press.
Greenspan, H. P. 1964 On the transient motion of a contained rotating fluid. J. Fluid Mech. 20, 673.Google Scholar
Greenspan, H. P. 1965 On the general theory of contained rotating fluid motions. J. Fluid Mech. 22, 449.Google Scholar
Greenspan, H. P. & Howard, L. N. 1963 On the time-dependent motion of a rotating fluid. J. Fluid Mech. 17, 385.Google Scholar
Groves, G. W. 1965 On the dissipation of tsunamis and barotropic motions by tidal currents. J. Mar. Res. 23, 251.Google Scholar
Hamon, B. V. 1966 Continental shelf waves and the effects of atmospheric pressure and wind stress on sea level. J. Geophys. Res. 71, 2883.Google Scholar
Holton, J. R. 1965 The influence of viscous boundary layers on transient motions in a stratified rotating fluid. J. Atmos. Sci. 22 (I), 402; 22 (II), 535.Google Scholar
Hough, S. S. 1898 On the application of harmonic analysis to the dynamical theory of the tides II. On the general integration of Laplace's dynamical equations. Phil. Trans. Roy. Soc. A 191, 139.Google Scholar
Ibbetson, A. & Phillips, N. A. 1967 Some laboratory experiments on Rossby waves in a rotating annulus. Tellus, 19, 81.Google Scholar
Kasahara, A. 1966 The dynamical influence of orography on the large-scale motion of the atmosphere. J. Atmos. Sci. 23, 259.Google Scholar
Lamb, H. 1932 Hydrodynamics, sixth ed. New York: Dover Publications.
Lighthill, M. J. 1967 On waves generated in dispersive systems by travelling forcing effects, with applications to the dynamics of rotating fluids. J. Fluid Mech. 27, 725.Google Scholar
Longuet-Higgins, M. S. 1964 Planetary waves on a rotating sphere. Proc. Roy. Soc. A 279, 446.Google Scholar
Longuet-Higgins, M. S. 1965a Planetary waves on a rotating sphere, II. Proc. Roy. Soc. A 284, 40.Google Scholar
Longuet-Higgins, M. S. 1965b The response of a stratified ocean to stationary or moving wind-systems. Deep Sea Res. 12, 923.Google Scholar
Longuet-Higgins, M. S. 1967a On the trapping of wave energy round islands. J. Fluid Mech. 29, 781.Google Scholar
Longuet-Higgins, M. S. 1968a On the trapping of waves along a discontinuity of depth in a rotating ocean. J. Fluid Mech. 31, 417.Google Scholar
Longuet-Higgins, M. S. 1968b Double Kelvin waves with continuous depth profiles J. Fluid Mech. 34, 49.Google Scholar
Morse, P. & Feshbach, H. 1953 Methods of Theoretical Physics, Vol. ii. New York: McGraw-Hill.
Mysak, L. A. 1967 On the theory of continental shelf waves. J. Mar. Res. 25, 205.Google Scholar
Pedlosky, J. 1965 A note on the intensification of the oceanic circulation. J. Mar. Res. 23, 207.Google Scholar
Pedlosky, J. & Greenspan, H. P. 1967 A simple laboratory model for the oceanic circulation. J. Fluid Mech. 27, 291.Google Scholar
Phillips, N. 1963 Geostrophic motion. Rev. Geo. 1, 123.Google Scholar
Phillips, N. 1965 Elementary Rossby waves. Tellus, 17, 295.Google Scholar
Phillips, N. 1966 Large-scale eddy motion in the Western Atlantic. J. Geophys. Res. 71, 3883.Google Scholar
Rattray, M. 1964 Time-dependent motion in the ocean; a unified two-layer beta-plane approximation. Studies in Oceanography (Hidaka Anniversary Volume), University of Tokyo Press.
Rattray, M. & Charnell, R. L. 1966 Quasigeostrophic free oscillations in enclosed basins. J. Mar. Res. 24, 82.Google Scholar
Rhines, P. 1967 Ph.D. Thesis, Cambridge University.
Reid, R. O. 1956 Effect of Coriolis force on edge waves (I) investigation of the normal modes. J. Mar. Res. 16, 109.Google Scholar
Robinson, A. 1964 Continental shelf waves and the response of sea level to weather systems. J. Geophys. Res. 69, 367.Google Scholar
Robinson, A. & Stommel, H. 1959 Amplification of transient response of the ocean to storms by the effect of bottom topography. Deep Sea Res. 5, 312.Google Scholar
Rossby, C. G. 1939 Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centres of action. J. Mar. Res. 2, 38.Google Scholar
Veronis, G. 1963 On the approximation involved in transforming the equation of motion from a spherical surface to a β-plane. J. Mar. Res. 21, 110; 21, 199.Google Scholar
Veronis, G. 1966 Rossby waves with bottom topography. J. Mar. Res. 24, 338.Google Scholar
von Arx, W. 1957 An experimental approach to problems in physical oceanography. Physics and Chemistry of the Earth, volume 2. Ahrens ed. London: Pergamon.
Wunsch, C. 1967 The long-period tides. Rev. Geophys. 5, 447.Google Scholar