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The propagation of planetary waves over a random topography

Published online by Cambridge University Press:  29 March 2006

Richard E. Thomson
Richard E. Thomson
Affiliation:
Environment Canada, Marine Sciences Directorate, Pacific Region, 1230 Government Street, Victoria, B.C.
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Abstract

The purpose of this paper is to consider the effect of one-dimensional random depth variation on the propagation of planetary waves in a homogeneous layer of fluid having a free upper surface. We begin by determining the dispersion relation for the coherent part of the wave using the vorticity equation for the transport stream function and a previously described perturbation method. Then, from the resulting first-order expressions for the wavenumber, we obtain the phase speeds for the two possible planetary-wave solutions. These are compared with the corresponding phase speeds of planetary waves over a smoothly varying topography; the validity limits of the approximations are discussed. For the most physically realizable situation, of random depth correlation lengths much shorter than a typical wavelength, we find that the phase speed of the shorter (longer) wave component is less (greater) over a randomly varying topography than over a smoothly varying topography. In the case of the shorter waves, greatest relative changes in phase speed occur when the associated fluid motions are at right angles to the ‘strike’ of the roughness elements, while for both long and short waves there is no relative change in phase speed if fluid motions are parallel to the roughness contours. Moreover, both types of waves are shown to lose energy in the direction of energy propagation as a result of scattering. Numerical values are then obtained using hydrographic charts of the western North Pacific, and show that the randomness may significantly decrease the phase speed of the shorter planetary-wave component. Finally, we give a brief descriptive explanation of the results based on the effect of the topography on the wave restoring mechanism.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 70 , Issue 2 , 29 July 1975 , pp. 267 - 285
Copyright
© 1975 Cambridge University Press

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References

Buchwald, V. T. 1972 Energy and energy flux in planetary waves. Proc. Roy. Soc. A 328, 37–48.Google Scholar
Chase, T. E., Menard, H. W. & Mammerickx, J. 1970 Bathymetry of the North Pacific. Scripps Inst. Oceanog. & Inst. Mar. Res., University of California, Rep.Google Scholar
Keller, J. B. 1967 The velocity and attenuation of waves in random medium. In Electromagnetic Scattering (ed. R. L. Rowell & R. S. Stein), pp. 823–834. Gordon & Breach.
Keller, J. B. & Veronis, G. 1969 Rossby waves in the presence of random currents. J. Geophys. Res. 74, 1941–1951.Google Scholar
Longuet-Higgins, M. S. 1964 Planetary waves on a rotating sphere. Proc. Roy. Soc. A 279, 446–473.Google Scholar
Longuet-Higgins, M. S. 1965 Planetary waves on a rotating sphere, II. Proc. Roy. Soc. A 284, 40–54.Google Scholar
Rhines, P. B. 1969a Slow oscillations in an ocean of varying depth. Part 1. Abrupt topography. J. Fluid Mech. 37, 161–189.Google Scholar
Rhines, P. B. 1969b Slow oscillations in an ocean of varying depth. Part 2. Islands and seamounts. J. Fluid Mech. 37, 191–205.Google Scholar
Rhines, P. B. 1970 Wave propagation in a periodic medium with application to the ocean. Rev. Geophys. 8, 303–319.Google Scholar
Rhines, P. B. & Bretherton, F. 1973 Topographic Rossby waves in a rough-bottomed ocean. J. Fluid Mech. 61, 583–607.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, 38–55.Google Scholar
Thomson, R. E. 1973 Energy and energy flux in planetary waves in a variable depth ocean. Geophys. Fluid Dyn. 5, 385–399.Google Scholar
Veronis, G. 1966 Rossby waves with bottom topography. J. Mar. Res. 24, 338–349.Google Scholar