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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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